Grobner-Shirshov Bases Theory andShirshov Algorithm

Leonid BokutSobolev Institute of Mathematics, Russia and South China Normal University, China

Yuqun ChenSouth China Normal University, China

Ministry of Education and Science of the Russian Federations

National Research University-Novosibirsk State University (NRU-NSU)

Department of Mechanics and Mathematics

Master education program for teaching foreign student in English

“Modern trends in discrete mathematics and combinatorial optimization”

The development is done in the framework of the Program of NSU as a national Research

University

Grobner-Shirshov bases theory and Shirshov algorithm

Authors: Leonid Bokut and Yuqun Chen

Novosibirsk, 2013

c© 2013 Leonid Bokut and Yuqun Chen

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Forewords

This small book presents the principals of the theory of Grobner-Shirshov bases for

associative and Lie algebras. The Buchberger and Shirshov algorithms for commutative,

non-commutative and Lie polynomials are given in the introductory Chapter 1 without

proofs. All the proofs presented in the following Chapters 2-6. Chapters 7 and 8 are de-

voted to Grobner-Shirshov bases for pre-Lie algebras and associative dialgebras. Finally,

the last Chapter 9 contains the history of Grobner-Shirshov bases in the works of A.I. Shir-

shov, MSU and Sobolev Institute of mathematics, NSU, 1953-1962, and the influence by

Alexander Gennadievich Kurosh and Anatolii Ivanovich Malcev to these activities. Also

we include some survey of Grobner-Shirshov bases theory for algebras, groups, semigroups

and categories.

The book is written based on the authors’ many years of teaching this subject at

the Novosibirsk State University and the South China Normal University (Guangzhou).

We are grateful to all participants of the algebra seminars at NSU, Sobolev Institute of

Mathematics and SCNU for many helps and support, especially to G.P. Kukin, Yu.N.

Maltsev, I.V. Lvov, V.K. Kharchenko, E.I. Zelmanov, P.S. Kolesnikov, E.S. Chibrikov,

Yongshan Chen, Chanyan Zhong, Yu Li and Qiuhui Mo.

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Contents

Forewords i

Chapter 1 Introduction 2

§1.1 The Euclidean g.c.d. algorithm . . . . . . . . . . . . . . . . . . . . . . . . 2

§1.2 The Gauss elimination algorithm . . . . . . . . . . . . . . . . . . . . . . . 4

§1.3 Systems of polynomial equations . . . . . . . . . . . . . . . . . . . . . . . 6

§1.4 Examples of systems of noncommutative (NC) polynomial equations . . . 9

§1.5 Systems of Lie polynomials equations . . . . . . . . . . . . . . . . . . . . 13

§1.5.1 Words and Shirshov’s bracketing (arrangement of parentheses) . . 13

§1.5.2 Lie monomials and Lie polynomials . . . . . . . . . . . . . . . . . . 15

§1.5.3 Serre system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

§1.5.4 Lyndon-Shirshov words . . . . . . . . . . . . . . . . . . . . . . . . 17

§1.5.5 Lyndon-Shirshov Lie monomials . . . . . . . . . . . . . . . . . . . 17

§1.5.6 Lie compositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Chapter 2 Free Commutative, Associative and Lie Algebras 21

§2.1 Commutative polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 21

§2.2 Noncommutative polynomials . . . . . . . . . . . . . . . . . . . . . . . . 23

§2.3 Lie polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

§2.3.1 Lazard-Shirshov elimination process . . . . . . . . . . . . . . . . . 26

§2.3.2 Constructing generators for an arbitrary subalgebra in a free Lie

algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

§2.3.3 A proof of Shirshov-Witt Theorem . . . . . . . . . . . . . . . . . . 28

Chapter 3 Grobner Bases for Commutative Algebras 32

§3.1 Buchberger theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

§3.2 Buchberger algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Chapter 4 Grobner-Shirshov Bases for Associative Algebras 37

§4.1 Composition-Diamond lemma for associative algebras . . . . . . . . . . . 37

§4.1.1 PBW-theorem for Lie algebras . . . . . . . . . . . . . . . . . . . . 41

§4.1.2 Normal forms for groups and semigroups . . . . . . . . . . . . . . . 42

§4.2 Composition-Diamond lemma for modules . . . . . . . . . . . . . . . . . . 47

ii

§4.3 Grobner-Shirshov bases for tensor product of free algebras . . . . . . . . . 49

§4.3.1 Composition-Diamond lemma for tensor product . . . . . . . . . . 49

§4.3.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Chapter 5 Grobner-Shirshov Bases for Lie Algebras over a Field 67

§5.1 Lyndon-Shirshov words . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

§5.2 Free Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

§5.3 Composition-Diamond lemma for Lie algebras . . . . . . . . . . . . . . . . 79

§5.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

§5.4.1 Kukin’s construction of a Lie algebra with unsolvable word problem 82

§5.4.2 Grobner-Shirshov basis for the Drinfeld-Kohno Lie algebra Ln . . . 84

Chapter 6 Grobner-Shirshov Bases for Lie Algebras over a Commutative

Algebra 88

§6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

§6.2 Composition-Diamond lemma for Liek[Y ](X) . . . . . . . . . . . . . . . . 90

§6.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Chapter 7 Grobner-Shirshov Bases for Pre-Lie Algebras 109

§7.1 Composition-Diamond lemma for pre-Lie algebras . . . . . . . . . . . . . 109

§7.2 PBW theorem for pre-Lie algebras . . . . . . . . . . . . . . . . . . . . . . 115

Chapter 8 Grobner-Shirshov Bases for Dialgebras 117

§8.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

§8.2 Composition-Diamond lemma for dialgebras . . . . . . . . . . . . . . . . 118

§8.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Chapter 9 A History and a Survey of Grobner-Shirshov Bases Theory 142

Reference 153

Notation 169

Chapter 1 Introduction

The classical Grobner and Grobner–Shirshov bases theory deals with three types of

polynomials: commutative associative, noncommutative associative, and Lie, providing

algorithms for passing from a set of polynomials of a fixed type to another set (of the same

type) with better properties. Buchberger’s algorithm works for commutative polynomials,

while Shirshov’s algorithms work for noncommutative and Lie polynomials.

Two classical cases of Buchberger’s algorithm have been known for ages: the Eu-

clidean algorithm for polynomials in one variable, and the Gauss Elimination algorithm

for linear (degree 1) polynomials in several variables.

A big advantage of Buchberger’s algorithm is that it is finite. It means that for

any given finite set of commutative polynomials in several variables, the computation

terminates after finitely many steps.

Buchberger’s algorithm is a specialization of Shirshov’s algorithms, which, in con-

trast, are infinite in general; nevertheless, they have been used with great success in both

theoretical and applied research.

§1.1 The Euclidean g.c.d. algorithm

Let k be a field, and let f(x) and g(x) be two polynomials in one variable x over

k. Our goal is to present an algorithm that finds gcd(f(x), g(x)), the greatest common

divisor of f(x) and g(x). It is the well-known Euclidean g.c.d. algorithm based on the

Euclidean division algorithm.

Example 1.1 Take

f(x) = x4 + x2 − x− 1, g(x) = x3 − 2x2 + 1.

Do the following reduction of f(x) modulo g(x):

f(x) 7→ f(x)− xg(x) = 2x3 + x2 − 2x− 1.

Here, we use the leading monomial x3 of g(x) to reduce the leading monomial x4 of

f(x), which is the same as computing the remainder of the division of f(x) by g(x) (the

elimination of the leading monomial of g in f). The other elementary step of the algorithm

is the division of a polynomial by using its leading coefficient. In our case,

2x3 + x2 − 2x− 1 7→ f1(x) = x3 +1

2x2 − x− 1

2.

2

The two steps make up a transformation that preserves the g.c.d., which we indicate by

writing f(x), g(x) ' f1(x), g(x).Let us apply the same steps now to f1(x) and g(x):

f1(x) 7→ f1(x)− g(x) =5

2x2 − x− 3

2

7→ f2(x) = x2 − 2

5x− 3

5.

In order to save space below, we write this as

f1(x) 7→ f1(x)|g(x)=0 7→ f2(x).

Thus,

f1(x), g(x) ' f2(x), g(x).

Actually, instead of the two steps

f, g ' f1, g ' f2, g,

we can reduce f(x) in one step by substituting x3 = 2x2 − 1, or in other words, by using

the formal equation g(x) = 0:

f(x) 7→ x4 + x2 − x− 1|x3=2x2−1 = 2x3 + x2 − 2x− 1

7→ 2x3 + x2 − 2x− 1|x3=2x2−1 = 5x2 − 2x− 3

7→ f2(x) = x2 − 2

5x− 3

5.

Using f2(x) = 0, we reduce g(x):

g(x) 7→ x3 − 2x2 + 1|x2= 25x+ 3

5=

2

5x2 − 1

5x− 1

5

7→ x2 − 1

2x− 1

2|x2= 2

5x+ 3

5= − 1

10x+

1

10

7→ g1(x) = x− 1,

and then we obtain that

f2, g ' f2, g1.

Finally, by using g1(x) = 0, we reduce f2(x) to:

f2(x) 7→ x2 − 2

5x− 3

5|x=1 = 0.

The chain of equivalences yields f, g ' x − 1, which shows that gcd(f(x),

g(x)) = x− 1.

3

Later on, we will define Grobner basis GB of a set of polynomials, we will see that

GBx4 + x2 − x− 1, x3 − 2x2 + 1 = x− 1.

In general,

GBf1(x), . . . , fm(x) = gcdf1(x), . . . , fm(x)

for polynomials fi(x) in one indeterminate.

§1.2 The Gauss elimination algorithm

Let k be a field and x1, . . . , xn indeterminates. We now solve a system of linear

equations a11x1 + · · ·+ a1nxn = b1,

. . . . . .

am1x1 + · · ·+ amnxn = bm,

(1.1)

where aij, bi ∈ k.It is well-known that this system can be successfully solved by using the Gauss

Elimination algorithm.

Example 1.2 Solve the following system of equations:2x1 + 3x2 + x3 = 1,

3x1 + 2x2 − x3 = 2.

Instead of these equations, let us deal with the corresponding polynomials

f = 2x1 + 3x2 + x3 − 1, g = 3x1 + 2x2 − x3 − 2.

First of all, we need to order the variables. The method is to set x1 > x2 > x3; then in

our case, x1 is the leading monomial of both f and g. Dividing f and g by their leading

coefficients:

f 7→ f1 = x1 +3

2x2 +

1

2x3 −

1

2,

g 7→ g1 = x1 +2

3x2 −

1

3x3 −

2

3,

we have f, g ' f1, g1, where the equivalence means that the corresponding linear

systems are equivalent (have the same solution sets over k). Using f1, we reduce g1:

g1 7→ g1 − f1 = −5

6x2 −

5

6x3 −

1

6

7→ g2 = x2 + x3 +1

5

4

to obtain f1, g1 ' f1, g2. We may in fact stop here, but in order to have something

really simple, we can further use g2 to eliminate x2 in f1:

f1 7→ f2 = x1 − x3 −4

5.

Thus our original system is equivalent to the systemx1 + x3 =

4

5,

x2 + x3 = −1

5,

which is in row-echelon form. As we shall see later, GBf, g = f2, g2.

In general, the Gauss elimination algorithm produces for any consistent system (one

that has solutions) an equivalent system in row-echelon form:xi1 +c1,i1+1xi1+1+ + · · ·+ c1nxn = d1

xi2 + c2,i2+1xi2+1+ + · · ·+ c2nxn = d2

. . . . . .

xil + cl,il+1xil+1 + · · ·+ clnxn = dl

(1.2)

where 1 ≤ i1 < i2 < · · · < il ≤ n.

The system (1.1) is inconsistent if and only if after finitely many steps we arrive at

0 = dl+1,

where dl+1 6= 0. In this case, the system (1.1) is equivalent to the system

0 = 1.

In terms of the Grobner bases theory, for all tuples of linear polynomials f1, . . . , fmwe have

GBf1, . . . , fm = h1, . . . , hl,

where the fi corresponds to (1.1), and either the hj corresponds to (1.2) or

h1, . . . , hl = 1.

The elimination of xi2 from h1 using h2, of xi3 from h1, h2 using h3, and so on, is

not absolutely necessary, but it gives a unique (reduced) Grobner basis independent of

the choice of the elementary transformation steps, which therefore depends only on the

ordering of the indeterminates. A different ordering leads to a different (reduced) basis.

5

§1.3 Systems of polynomial equations

What will be the common features of the two cases considered above?

(i) We fix a total ordering on monomials, which determines the leading monomial of

each polynomial. For 1-variable polynomials, it is the degree ordering xn > xm ⇔ n > m;

For multivariable linear polynomials, it is enough to order its indeterminates.

(ii) We reduce a given set of polynomials by 1. dividing a polynomial by its lead-

ing coefficient; 2. eliminating the leading monomial of one polynomial from the other

polynomials of the set.

We would like to do something similar to an arbitrary system of polynomial equations

in several indeterminates. However, we will see that reductions in (ii) are insufficient,

and we need a new operation s(f, g), which is called the s-polynomial operation. An

alternative name for this is the composition (f, g)w of f and g with respect to some

uniquely defined monomial w.

Example 1.3 Solve the following system of polynomial equationsf = x21x2 − 1 = 0,

g = x31 + 7x1x2 + 1 = 0.

(1.3)

Let us fix x1 > x2 as before, and use the deg-lex (degree-lexicographic) ordering of

monomials to compare two monomials first by degree and then lexicographically. For

instance,

x31 > x2

1x2 > x1x2 > 1.

Again we will deal with the polynomials f and g rather than the equations f = 0

and g = 0. We aim to reduce the set f, g as much as possible by passing to equivalent

system of equations.

The leading monomials of f and g in (1.3) are f = x21x2 and g = x3

1 respectively. We

cannot use g = 0 to reduce f , neither can we use f = 0 to reduce g, for neither f nor g

is a submonomial of the other.

The great discovery of Buchberger (and Shirshov for noncommutative polynomials)

was that in such cases the system can be simplified further by a new s-polynomial (com-

position) operation. Take the least common multiple w = lcm(f , g), which in our case is

x31x2 = fx1 = gx2, and consider s(f, g) = (f, g)w = fx1 − gx2.

For our system (1.3), we have

s(f, g) = (x21x2 − 1)x1 − (x3

1 + 7x1x2 + 1)x2 = −7x1x22 − x1 − x2

7→ h = x1x22 +

1

7x1 +

1

7x2,

6

thus

f, g ' f, g, h, f = x21x2, g = x3

1, h = x1x22.

Proceeding with s(f, h), we obtain the following:

w1 = lcm(f , h) = x21x

22 = fx2 = hx1,

s(f, h) = (f, h)w1 = (x21x2 − 1)x2 − (x1x

22 +

1

7x1 +

1

7x2)x1

7→ q = x21 + x1x2 + 7x2,

thus

f, g, h ' f, g, h, q, f = x21x2, g = x3

1, h = x1x22, q = x2

1.

The reduction

g 7→ g|q=0 = −x21x2 + 1 7→ x2

1x2 − 1|f=0 = 0,

shows that we may exclude g from the set; we are then left with

f, h, q, f = x21x2, h = x1x

22, q = x2

1.

Next, we reduce f :

f 7→ f |q=0 = −x1x22 − 7x2

2 − 1

7→ x1x22 + 7x2

2 + 1|h=0

7→ f1 = x22 −

1

49x1 −

1

49x2 +

1

7,

thus

f, h, q ' f1, h, q, f1 = x22, h = x1x

22, q = x2

1.

Now we see that h is also redundant:

h 7→ h|f1=0 7→ x21 + x1x2 + 7x2|q=0 7→ 0,

thus

f1, h, q ' f1, q, f1 = x22, q = x2

1,

and we conclude that (1.3) is equivalent tof1 = x2

2 −1

49x1 −

1

49x2 +

1

7= 0,

q = x21 + x1x2 + 7x2 = 0.

(1.4)

We cannot reduce (1.4) any further: since x22 and x2

1 are coprime, the eliminations are

impossible, while s-polynomial operations only introduce redundancies.

7

The theory of Grobner bases guarantees that (1.4) is the unique simplest system

equivalent to (1.3) as long as we fix the ordering of indeterminants x1 > x2 and use the

deg-lex ordering on monomials.

In terms of Grobner bases, we have GBf, g = f1, q.To solve (1.4) let us change the ordering to lex-deg ordering to compare two mono-

mials first lexicographical and then by degree. For example,

x21 > x1x

22 > x1x2 > x1 > x2

2 > x2 > 1.

Then (1.3) is equivalent to

f2 = x1 + x2 − 49x22 − 7 = 0, q1 = (49x2

2 − x2 + 7)2 + (49x22 − x2 + 7)x2 + 7x2 = 0.

Actually GBf, g = f2, q1 in lex-deg ordering of monomials.

This system can be solved in radicals.

Now we are going to put another problem concerning to the system (1.3). Namely

we address to the following equality, or word problem for (1.3):

Problem. Find an algorithm that for a given polynomial F (x1, x2) would answer whether

F (x1, x2) = 0 modulo f = 0 and g = 0.

In other words, we need to determine the validity of the implication for each F , that

is,

f = 0, g = 0⇒ F = 0. (1.5)

To be more precise, it means that F (x1, x2) = h1(x1, x2)f(x1, x2) + h2(x1, x2)g(x1, x2) for

some polynomials h1, h2.

The desired algorithm is: to reduce F (x1, x2) as much as possible by using f1 = 0

and q = 0 (but not f and g!). If zero results, then (1.5) is true; otherwise, (1.5) is false.

Example 1.4 Keeping the same f and g as in (1.3), let

F = x1x32 + α1x1x

22 + α2x1x2 + α3x1 + α4x2 + α5,

where αi ∈ k. For which values of αi is (1.5) valid? The series of reductions

F 7→ F |f1=0 = F1

7→ F1|q=0 = F2

7→ F2|f1=0 = (α2 −1

7)x1x2 + (α3 −

α1

7− 1

73)x1

+(α4 −α1

7− 1

73)x2 + α5

8

quickly yields the answer: (1.5) is valid if and only if

α2 =1

7, α3 = α4 = α1

7+ 1

73 , α5 = 0,

where α1 ∈ k is arbitrary. Note that we cannot reduce F at all just by using the initial

polynomials f and g!

Anyway, we can formulate the Buchberger’s algorithm as follows:

Algorithm. Given any set of polynomials in several indeterminates, add the s-polynomials

to the set, and reduce polynomials by eliminating their leading monomials, also dividing

them by their leading coefficients. Stop when no further reductions are possible (so in

particular, when all s-polynomials reduce to 0).

The main significance is that we can always solve the equality problem for every finite

system of polynomial equations, by using the Buchberger’s algorithm.

§1.4 Examples of systems of noncommutative (NC)

polynomial equations

Let X = x1, x2, . . . be a set (an alphabet) of letters (indeterminates), X∗ the set

of all words (strings) in X including the empty word 1,

1, x1, x2, x21, x1x2, x2x1, x

22, x

21x2, x1x2x1, x

22x1, . . . .

A linear combination of words is called a NC polynomial. Product and sum of NC

polynomials are defined in a natural way. For example,

(x1 + x2)2 − x2

1 − x22 = x1x2 + x2x1,

(x1 + x2 + x3)3 − (x2 + x3)

3 − (x1 + x3)3 − (x1 + x2)

3 + x31 + x3

2 + x33

= x1x2x3 + x1x3x2 + x2x1x3 + x2x3x1 + x3x1x2 + x3x2x1,

(x1 + · · ·+ xn)n −∑

1≤i≤n

(x1 + · · ·+ xi + · · ·+ xn)n

+∑

1≤i1<i2≤n

(x1 + · · ·+ xi1 + · · ·+ xi2 + · · ·+ xn)n − · · ·+ (−1)n−1∑

i

xni

=∑

(i1,i2,...,in)∈Pn

xi1xi2 · · ·xin ,

9

where Pn is the set of all permutations of (1, 2, . . . , n).

In this section we discuss Shirshov algorithm for a reduction of a system of NC poly-

nomial equations in order to try to solve the word (equality) problem (in general, the word

problem for a finite system of NC polynomial equations is algorithmically unsolvable).

Let X is a well-ordered set. Then we define the deg-lex ordering on X∗: first compare

two words by length (degree) and then by comparing them lexicographically.

We will use a Shirshov’s notion of composition (f, g)w of two NC polynomials relative

to a word w, where w = f b = ag, a, b ∈ X∗, f , g are the maximal words of f, g

correspondingly in deg-lex ordering of words, and deg(w) < deg(f) + deg(g). We call w

a least common multiple of f , g, w = lcm(f , g).

Example 1.5 The quaternion system H over real numbers R (due to W.R. Hamilton

1805-1865) is

x21 + 1 = 0, x2

2 + 1 = 0, x23 + 1 = 0, x1x2 − x3 = 0, x2x3 − x1 = 0,

x3x1 − x2 = 0, x2x1 + x3 = 0, x3x2 + x1 = 0, x1x3 + x2 = 0.

Any NC polynomial f(x1, x2, x3) can be uniquely present mod (H) in a form

f = α0 + α1x1 + α2x2 + α3x3 mod(H), αi ∈ R. (1.6)

Let us order x1 < x2 < x3 and the deg-lex ordering of words in X = x1, x2, x3. There

are compositions relative to following words wi = lcm(f , g), 1 ≤ i ≤ 16, where f, g are

from the list.

w1 = x21x2, w2 = x2

1x3, w3 = x22x3, w4 = x2

2x1, w5 = x23x1, w6 = x2

3x2,

w7 = x1x22, w8 = x1x2x1, w9 = x2x

23, w10 = x2x3x2, w11 = x3x

21,

w12 = x2x21, w13 = x3x

22, w14 = x3x2x3, w15 = x1x

23, w16 = x1x3x1.

Let us check first two compositions (f, g are uniquely defined by w).

(f, g)w1 = (x21 + 1)x2 − x1(x1x2 − x3) = x2 + x1x3 ≡ 0 mod(x1x3 = −x2),

(f, g)w2 = (x21 + 1)x3 − x1(x1x3 + x2) = x3 − x1x2 ≡ 0 mod(x1x2 = x3).

So, all compositions are trivial. By Composition-Diamond lemma (Theorem 4.4) any

presentation (1.6) is unique. It is solved the word problem for H: for any two polyno-

mials f(x1, x2, x3), g(x1, x2, x3) one can decide whether f = g mod(H), i.e., f − g =∑αiaihibi, αi ∈ R, hi ∈ H, ai, bi ∈ X∗.

10

Example 1.6 The Grassmann (exterior) system G(X) of NC polynomial equations (due

to H.G. Grassmann 1809-1877) is

x2i = 0, 1 ≤ i ≤ n, xixj + xjxi = 0, 1 ≤ j < i ≤ n.

Let us order xi < xj for i < j, and deg-lex ordering of words in X = x1, . . . , xn. There

are compositions of NC polynomials from G(X) relative to words w = lcm(f , g), where

f, g are from the list.

wiij = xixixj, i > j, wijj = xixjxj, i > j, wijk = xixjxk, i > j > k.

Let us check the last composition

(f, g)wijk= (xixj + xjxi)xk − xi(xjxk + xkxj) = xjxixk − xixkxj

≡ −xjxkxi + xkxixj mod(xixk = −xkxi)

≡ xkxjxi − xkxjxi mod(xjxk = −xkxj, xixj = −xjxi)

≡ 0.

Again by CD-lemma any polynomial has unique presentation as a linear combinations of

words

xi11 · · ·xin

n , 0 ≤ ij ≤ 1.

Example 1.7 For n = 2 and the deg-lex ordering on words in X = x1, x2 with x1 < x2,

the set

f = x1x22 − x2, g = x2

2x1 − x1

admits only three compositions:

w1 =x1x22x1 = fx1 = x1g, (f, g)w1 =x2x1 − x2

1;

w2 =x1x32x1 = fx2x1 = x1x2g, (f, g)w2 =x2

2x1 − x1x2x1;

w3 =x22x1x

22 = gx2

2 = x22f , (g, f)w3 =x3

2 − x1x22.

Consider now the equality problem for the system

f = x1x22 − x2 = 0, g = x2

2x1 − x1 = 0

applying Shirshov algorithm which means adding compositions and reducing the polyno-

mials as before. The first compositions

(f, g)w1 = h = x2x1 − x21,

(f, g)w2 7→ x22x1 − x1x2x1|g=0, h=0 7→ q = x3

1 − x1,

(g, f)w3 7→ x32 − x1x

22|f=0 7→ r = x3

2 − x2

11

lead to the further compositions

(f, h)x1x22x1

= fx1 − x1x2h

7→ x1x2x21 − x2x1|h=0

7→ x41 − x2x1|q=0 7→ x2

1 − x2x1|h=0 7→ 0,

(h, f)x2x1x22

= hx22 − x2f

7→ x21x

22 − x2

2|f=0 7→ t = x22 − x1x2,

and then to the reductions

f 7→ x1x22 − x2|t=0 = f1 = x2

1x2 − x2,

r 7→ x32 − x2|t=0 7→ x1x

22 − x2|t=0 = r1 = x2

1x2 − x2.

Now g is redundant because

g 7→ x22x1 − x1|t=0 7→ x1x2x1 − x1|h=0 7→ x3

1 − x1|q=0 = 0.

The remaining set of four polynomials

f1 = x21x2 − x2, h = x2x1 − x2

1, t = x22 − x1x2, q = x3

1 − x1

admits only the following trivial compositions:

(f1, h)x21x2x1

= f1x1 − x21h 7→ x4

1 − x2x1|q=0, h=0 = 0,

(h, f1)x2x21x2

= hx1x2 − x2f 7→ x31x2 − x2

2|q=0, t=0 = 0,

(q, f1)x31x2

= qx2 − x1f = 0,

(q, f1)x41x2

= qx1x2 − x21f = 0,

(q, f1)x51x2

= qx21x2 − x3

1f = 0,

(f1, r)x21x3

2= f1x

22 − x2

1r 7→ x32 − x2

1x2|r=0, f1=0 = 0,

(f1, t)x21x2

2= fx2 − x2

1t 7→ x31x2 − x2

2|q=0, t=0 = 0,

(h, q)x2x31

= hx21 − x2q 7→ x4

1 − x2x1|q=0, h=0 = 0,

(r, h)x32x1

= rx1 − x22h 7→ x2

2x21 − x2x1|t=0

7→ x1x2x21 − x2x1|h=0 7→ x4

1 − x21|q=0 = 0,

(t, t)x32

= tx2 − x2t 7→ x1x22 − x2x1x2|t=0,h=0 = 0,

as well as some manifestly trivial compositions (q, q)w. Therefore, we have

GSBx1x22 − x2, x

22 − x1 = x2

1x2 − x2, x2x1 − x21, x

22 − x1x2, x

31 − x1.

12

The Shirshov’s Composition-Diamond Lemma (Theorem 4.4) implies that any polynomial

F (x1, x2) can be uniquely presented mod(f = 0, g = 0) as a linear combination of the five

words 1, x1, x21, x2, x1x2.

Let us call a presentation of NC polynomials by these monomials its canonical pre-

sentation. The Composition-Diamond lemma gives the positive solution to the equality

problem: a polynomial is equal to zero mod(f = 0, g = 0) if and only if its canonical

presentation is zero. For instance, let us check whether Fn = xn2x

n1 −xn

1xn2 is equal to zero

mod(f = 0, g = 0):

F0 = 1− 1 = 0;

F1 7→ x2x1 − x1x2|h=0 = x21 − x1x2 6= 0,

which is the canonical presentation of this polynomial,

F2 7→ x22x

21 − x2

1x22|t=0

7→ x1x2x21 − x2

1x1x2|h=0 7→ x41 − x3

1x2|q=0 = x21 − x1x2,

Fn 7→ · · · 7→ x21 − x1x2.

§1.5 Systems of Lie polynomials equations

In this section we will deal with Shirshov algorithm for the word (equality) problem

for systems of Lie polynomial equations. It would take a lot of combinatorial preparations

connected with Lyndon-Shirshov words and Lyndon-Shirshov Lie monomials. All proofs

will be in Chapter 5.

§1.5.1 Words and Shirshov’s bracketing (arrangement of parentheses)

Let X = x1, x2, . . . be an alphabet with x1 < x2 < . . . , X∗ the set of all words

in X. For simplicity we will use also numbers 1, 2, . . . instead of letters x1, x2, . . . . We

denote e the empty word.

Let us order X∗ using lex-antideg ordering to compare two words first lexicographi-

cally and then by “anti-degree”: the shorter word is larger. For example, 21 < 2 < 333 <

33 < 34 < e.

Algorithm of the bracketing (the arrangement of parentheses) of words. Let w be a

word and xi be the minimal letter in w. If w = (xi)ku, k > 0, u 6= xiv, then go to u,

the longest suffix of w that does not begin with the minimal letter of w. If w does not

begin with xi, then join minimal letters of w to the previous ones, order “new letters” in

lex-antideg and continue this process.

13

For example, w = 11212122311312321, w1 = 11(21)(21)22((31)1)(31)23(21), w2 =

11(21)(21)22((31)1)(31)2(3(21)), w3 = 11(21)(21)22((31)1)((31)2)(3(21)). Here our algo-

rithm stops. New “letters with brackets” are 1, (21), ((31)1), ((31)2), (3(21)) and new

“letters without brackets” are 1, 21, 311, 312, 321.

First information in w3. Any new “letter without parentheses” u is so called Lyndon-

Shirshov (LS) word: u = u1u2 > u2u1 lexicographically for any u1, u2 6= 1.

In our case 1, 21, 2, 311, 312, 321 are LS words.

Second information in w3. Word w is a product of LS words in non-decreasing lex-

antideg ordering, w = 1 · 1 · 21 · 21 · 2 · 2 · 311 · 312 · 321, 1 < 21 < 2 < 311 < 312 < 321.

It is an illustration of the

Theorem 1.8 (Shirshov factorization) Any word is a unique product of LS words in non-

decreasing ordering relative to lex-antideg ordering.

Third information in w3. New “letters with brackets” are non-associative LS mono-

mials: (w) = ((u)(v)) is called a LS monomial if w (the associative support of (w)) is a

LS word, (u), (v) are LS monomials, u > v in lex-antideg, and if (u) = ((u1)(u2)), then

u2 ≤ v in lex-antideg.

In our case, monomials 1, (21), 2, ((31)1), ((31)2), (3(21)) are LS monomials. For

example, w1 = ((31)1), (31) > 1, 1 ≤ 1, w2 = ((31)2), (31) > 2, 1 ≤ 2, w3 =

(3(21)), 3 > (21), 2 > 1.

It is an illustration of the

Theorem 1.9 (Lyndon-Shirshov theorem) There is 1-1 correspondence between LS words

and non-associative LS monomials. It means that for any LS word w one may put one

and only one parentheses (w) such that (w) is a LS monomial.

Shirshov’s Algorithm of bracketing a LS word w with a LS subword u, w = aub (the

special bracketing), w = aub → (w)u = (a(u)b), where (u) is a standard bracketing as

before.

– To put the Shirshov’s brackets (the Shirshov’s arrangement parentheses) on u first

as before, w1 = a(u)b.

– To add a minimal (in lex-antideg ordering) “letter” of w1, including (u), to the

previous ones.

– To order new “letters” in lex-antideg and to continue the same process.

14

Example 1.10 w = 2212112111211 = 2u2111211, u = 21211.

w1 = 2((21)((21)1))2111211,

w2 = 2((21)((21)1))(((21)1)1)((21)1),

w3 = 2(((21)((21)1))(((21)1)1))((21)1),

w4 = 2((((21)((21)1))(((21)1)1))((21)1)),

w5 = (2((((21)((21)1))(((21)1)1))((21)1))).

To better understand the special bracketing algorithm, let us start with LS monomial

(w) = (2(21)[((211)(2111))(211)]). Here brackets . . . are the minimal pair of brackets

that cover u. We change /// in the following way . . . → (u)c, c = 2111211. Then

we use Shirshov factorization for c, (u)c → (u)(2111)(211), and now the left normed

bracketing (u)(2111)(211)→ (((u)(2111))(211)). Then (w)u = w5.

§1.5.2 Lie monomials and Lie polynomials

Let X = x1, x2, . . . be an alphabet with x1 < x2 < . . . . Lie monomials in X are

defined as follows:

(i) xi’s are Lie monomials of the degree 1;

(ii) If u, v are Lie monomials, deg(u) = n, deg(v) = m, then [uv] = uv − vu is a Lie

monomial of degree n+m.

Example 1.11

x1, x2, [x1x2] = −[x2x1] (by the anticommutatvity identity [ab] = −[ba]),

[[x2x1]x1] = x2x1x1 − 2x1x2x1 + x1x1x2,

[x2[x2x1]] = x2x2x1 − 2x2x1x2 + x1x2x2,

[x2[x2x1]][[x2x1]x1]] (by the Leibniz identity [[ab]c] = [[ac]b] + [a[bc]])

= [[x2[[x2x1]x1]][x2x1]] + [x2[[x2x1][[x2x1]x1]]]

= x2x2x1x2x1x1 + · · ·+ x1x1x2x1x2x2 + . . . ,

where x2x2x1x2x1x1 >lex · · · >lex x1x1x2x1x2x2 >lex . . . .

A linear combination of Lie monomials over a field k is called a Lie polynomial. Let

Lie(X) be the linear space of Lie polynomials. It is a subspace of NC polynomials k〈X〉.How to check that a NC polynomial is a Lie polynomial? There are several criterions.

One of the most popular is

15

Friedrichs’ criterion (due to K.O. Friedrichs 1901-1982). Let chk = 0. A NC polyno-

mial f(x1, x2, . . . , xn) is a Lie polynomial iff

[x1y1] = 0, . . . , [xnyn] = 0⇒ f(x1 + y1, . . . , xn + yn) = f(x1, . . . , xn) + f(y1, . . . , yn).

Another interesting and important observation about Lie polynomials is as follows.

Zassehnaus-Jacobson Lie polynomial. Let k be a field of characteristic p > 0. Then

(a+ b)p = ap + bp + Λ(a, b),

where Λ(a, b) is a Lie polynomial of Zassenhaus-Jacobson. For example, p = 2, (a+ b)2 =

a2 + b2 + [ab]; p = 3, (a+ b)3 = a3 + b3 + [a[ab]] + [[ab]b].

In general,

(a+ b)p = ap + bp +∑

1≤i≤p−1

1/isi(a, b),

where si is the coefficient on ti−1 of the polynomial

[(ta+ b)p−1a] =: [[

p−1︷ ︸︸ ︷(ta+ b)[. . . [(ta+ b) a] . . . ]]], ta = at, tb = bt.

§1.5.3 Serre system

Serre system L(An) of Lie polynomial equations of type An (due to J.-P. Serre, born

in 1926) are as follows.

Let k be a field with chk 6= 2.

Let H = h1, . . . , hn, X = x1, . . . , xn, Y = y1, . . . , yn. We order alphabet

H ∪X ∪ Y by hi > xj > yk for any i, j, k and hi > hj, xi > xj, yi > yj for any i > j. Let

An is the Cartan matrix (due to Elie Cartan 1869-1951), i.e., aii = 2, ai,i+1 = −1, ai,i−1 =

−1, aij = 0 if |i− j| > 1.

Then L(An) is the Lie system of equations in H ∪X ∪ Y over k:

[hihj] = 0 (i > j), [xiyi] = hi, [xiyj] = 0 (i 6= j),

[hixj]− aijxj = 0, [hiyj] + aijyj = 0,

[xi[xixi−1]] = 0, [[xixi−1]xi−1] = 0, [xixj] = 0 (i > j + 1),

[yi[yiyi−1]] = 0, [[yiyi−1]yi−1] = 0, [yiyj] = 0 (i > j + 1).

In order to use the Shirshov algorithm for Lie polynomials we need

(i) To define Lyndon-Shirshov linear basis of the linear space of Lie polynomials in

H ∪X ∪ Y with an ordering;

16

(ii) To define Lie composition of Lie polynomials from L(An);

(iii) To add all non-trivial compositions to initial set of equations in order to have a

GS basis GSB(L(An));

(iv) To find all LS monomials [w] such that the word w does not contain maximal

words s for all s ∈ GSB(L(An)).

In fact, we have a CD-lemma for Lie algebras in Chapter 5 and solve all the above

problems. Now we outline the solution.

§1.5.4 Lyndon-Shirshov words

Let us remind that a word w is called a LS word if

w = uv >lex vu for any u, v 6= 1.

For example, x2x2x1x2x1x1 is a LS word, and x2x1x1x2x1, x2x2x1x2x2x1 are not.

Proposition 1.12 A deg-lex maximal word of any Lie monomial (and polynomial) is a

LS word.

For example, The deg-lex maximal word of [[[x2x1][x2x1x1x1]][x2x1x1]] is LS word

x2x1x2x1x1x1x2x1x1. For [[[x2x1][x2x1x1]][x2x1x1]], it is x2x1x2x1x1x2x1x1.

§1.5.5 Lyndon-Shirshov Lie monomials

A Lie monomial [w] is called a Lyndon-Shirshov Lie monomial if

(i) the word w is a LS word;

(ii) if [w] = [[u][v]], then [u], [v] are LS Lie monomials and u v in the lex-antideg

ordering;

(iii) if [w] = [[[u1][u2]][v]], then u2 v in the lex-antideg ordering.

For examples, [xi[xixi−1]], [[xixi−1]xi−1] are LS Lie monomials since xi xi−1 and

xi xixi−1. Also the following Lie monomials are Lyndon-Shirshov’s

[[xi+jxi+j−1 . . . xi−1]xi+j−1], [[xi+j . . . xi][xi+j . . . xixi−1]].

Proposition 1.13 The set of LS Lie monomials is a linear basis of Lie(X), the linear

space of Lie polynomials.

Proposition 1.14 There is 1-1 correspondence between LS Lie monomials [w] and LS

words w.

17

We order LS Lie monomials [w] by the deg-lex ordering of LS words w.

Proposition 1.15 For any LS monomial [w], a maximal in deg-lex ordering word of [w],

as NC polynomial, is equal to w.

Proposition 1.16 For any LS word w there are two equivalent ways to put Lie brackets

in order to get a LS monomial [w]:

(i) To join a minimal letter of w to the previous ones, the ordering of new alphabet

by lex-antideg ordering, and continue the process.

(ii) If [w] = [[u][v]], where v is the longest proper LS suffix of w, then u is a LS word.

For examples,

[xnxn−1 . . . x1] = [xn[xn−1 . . . x1]], or by step by step

xnxn−1 . . . x2x1, xnxn−1 . . . x3[x2x1], xnxn−1 . . . [x3[x2x1]], . . . , [xn[xn−1[...[x3[x2x1]]]]].

[xi+k . . . xi+1xixi+k . . . xi+1xixi−1] = [[xi+k . . . xi+1xi][xi+k . . . xi+1xixi−1]], or

xi+k . . . xi+1xixi+k . . . xi+1xixi−1,

xi+k . . . xi+1xixi+k . . . xi+1[xixi−1],

xi+k . . . xi+1xixi+k . . . [xi+1[xixi−1]],

xi+k . . . xi+1xi[xi+k[. . . [xi+1[xixi−1]] . . . ]],

xi+k . . . [xi+1xi][xi+k[. . . [xi+1[xixi−1]] . . . ]],

. . .

xi+k[. . . [xi+1xi] . . . ][xi+k[. . . [xi+1[xixi−1]] . . . ]],

[xi+k[. . . [xi+1xi] . . . ]][xi+k[. . . [xi+1[xixi−1]] . . . ]],

[[xi+k[. . . [xi+1xi] . . . ]][xi+k[. . . [xi+1[xixi−1]] . . . ]]].

There is an algorithm to present any Lie monomial as a linear combination of LS

monomials by induction on degree:

(i) do [[u][v]]→ −[[v][u]], if [u], [v] are LS monomials and u ≺ v,

(ii) do [[[u1][u2]][v]]→ [[[u1][v]][u2]] + [[u1][[u2][v]]], if u2 v.

For example, [[32]1] = [[31]2] + [3[21]], [[12]2] = −[[21]2] = [2[21]], [[[32]1][[12]2]] =

[[[31]2][2[21]]] + [[3[21]][2[21]]] = [[[31][2[21]]]2] + [[31][2[2[21]]]] + [[3[21]][2[21]]].

18

§1.5.6 Lie compositions

To define Lie composition 〈f, g〉w we need to define an associative composition (f, g)w

then to put special Lie brackets on the result.

Let us define and check some compositions for L(An).

1. w = xi+1xixixi−1. Then (f, g)w = fxi−1 − xi+1g, 〈f, g〉w = [fxi−1] − [xi+1g] =

[[[xi+1xi]xi]xi−1]− [xi+1[xi[xixi−1]]] = A−B.

Here B is LS monomial, A is not. So, first of all we must present A as a linear com-

binations of LS monomials using anticommutativity ([ab] = −[ba]) and Leibniz identity

[[ab]c] = [[ac]b] + [a[bc]]). We know in advance that the maximal LS monomial of A is B

with coefficient +1. We have A = [[[xi+1xi−1]xi]xi]+[[xi+1[xixi−1]]xi]+[[xi+1xi][xixi−1]] =

[[[xi+1xi−1]xi]xi] + [[xi+1[xixi−1]]xi] + [[xi+1[xixi−1]]xi] +B.

The first LS monomial is zero mod(L(An), w). So, after reduction,

〈f, g〉w = 2[[xi+1[xixi−1]]xi] mod(S,w)

and we must add [[xi+1[xixi−1]]xi] = 0 to S (since 2 6= 0 in k).

2. w = xi+1xi+1xixixi−1, (f, g)w = fxixi−1 − xi+1xi+1g, 〈f, g〉w = [f [xixi−1]] −[xi+1[xi+1g]].

The rule of special bracketing. In fxixi−1 a minimal letter is xi−1. Join by bracket

this letter to the previous one, [xixi−1], then joint “new letter” [xixi−1] to f .

Then 〈f, g〉w = [[xi+1[xi+1xi]][xixi−1]]− [xi+1[xi+1[xi[xixi−1]]]] = A−B.

Again, B is a LS monomial and A is not. We have A = [[xi+1[xixi−1]][xi+1xi]] +

[xi+1[[xi+1xi][xixi−1]]] = −A1 + A2, A2 = −A1 +B.

So, [f, g]w = −2A1, and we must add to L(An) equation A1 = [[xi+1xi][xi+1[xixi−1]] =

0.

Continuing Shirshov algorithm, we will have the following set of multiple composi-

tions (see §1.5.3):

[[xi+jxi+j−1 . . . xi−1]xi+j−1] = 0, [[xi+j . . . xi][xi+j . . . xixi−1]] = 0,

[[yi+jyi+j−1 . . . yi−1]yi+j−1] = 0, [[yi+j . . . yi][yi+j . . . yiyi−1]] = 0,

j ≥ 1, i ≥ 2, i+ j ≤ n,

where [z1z2 . . . zm] = [z1[z2 . . . zm]]. A GS basis GSB(L(An)) of the system L(An) consists

of initial equations together with the above relations.

19

The set Irr(GSB(L(An)) is h1, . . . , hn ∪ [xi+j . . . xi], n ≥ i + j ≥ i ≥ 1 ∪[yi+j . . . yi], n ≥ i+ j ≥ i ≥ 1.

By CD-lemma for Lie equations,

L(An)→ F = 0

iff F goes to zero by eliminations of maximal monomials of GSB(L(An)). It is an algo-

rithmically solvable problem since GSB(L(An)) is a finite set.

20

Chapter 2 Free Commutative, Associative and Lie

Algebras

We will deal with the following classes (varieties) of linear algebras over a fixed field

k – commutative algebras Com, associative algebras As, Lie algebras Lie:

Com : xy = yx, (xy)z = x(yz),

As : (xy)z = x(yz),

Lie : x2 = 0 (it follows xy = −yx, the anticommutativity identity),

(xy)z + (yz)x+ (zx)y = 0 (the Jacobi identity).

Examples.

1) Com contains polynomial algebras. Any commutative algebra is a quotient algebra

of a polynomial algebra.

2) As contains non-commutative polynomial algebras. Any associative algebra is a

quotient of a non-commutative polynomial algebra.

3) Lie contains Lie polynomial algebras. Any Lie algebra is a quotient of an algebra

of Lie polynomials. For any associative algebra A the algebra A(−) with the Lie product

[xy] = xy−yx is a Lie algebra. Any Lie algebra (over a field!) is a Lie subalgebra of some

associative algebra (Poincare-Birkhoff-Witt theorem, see below).

§2.1 Commutative polynomials

Let X = x1, x2, . . . , xn, . . . be a finite or countable linearly ordered set, x1 > x2 >

· · · > xn > . . . (variables or letters) (actually X may be any well-ordered set).

Remark The above ordering onX (x1 > x2 > . . . ) is common for the case of commutative

variables and follows the ordering of variables in the Gauss elimination algorithm. For

noncommutative variables one uses the opposite ordering x1 < x2 . . . .

By k[X] one denotes the usual polynomial algebra on X,

k[X] = ∑

i=(i1,...,in)

αixi11 . . . x

inn |n ≥ 0, ij ≥ 0, αi ∈ k.

The algebra k[X] satisfies a following universal property: For any commutative alge-

bra C and any map ε : X → C, there exists a unique algebra homomorphism f : k[X]→ C

such that the following triangle is commutative

21

-

?

C

k[X]X i

ε∃! f

where i is the inclusion map.

A monomial u = xi11 . . . x

inn is called a commutative word of the length |u| (degree

deg(u)). Let [X] be the set of all commutative words in X (including 1). Then [X] is a

linear basis of k[X].

From the universal property it follows that any commutative algebra C is a quotient

of some k[X],

C = k[X]/I,

where I is an ideal of k[X]. Let S be a set of generators of I, i.e., I = Id(S) =

∑αiaisi|ai ∈ [X], si ∈ S, αi ∈ k. Then by definition

C = k[X]/Id(S) = k[X|S],

the commutative algebra with generators X and defining relations si = 0, si ∈ S.

Our main problem is, starting with S, to find a special system Sc of defining relations

of C that is called a Grobner basis of the ideal I, or a GB of the C. Sc can be found

using the Buchberger algorithm.

We will use the following theorem.

Theorem 2.1 (Hilbert Basis Theorem) Let X = x1, . . . , xn be a finite set. Then any

ideal I C k[X] is finitely generated,

I = Id(f1, . . . , fm) = m∑

i=1

gifi|gi ∈ k[X], 1 ≤ i ≤ m, m ≥ 1.

This theorem follows by induction on n, using the following lemma.

Lemma 2.2 Let R be a commutative Noetherian ring with 1 (i.e., any ideal of R is finitely

generated). Then the polynomial ring R[x] (on one variable) is a commutative Noetherian

ring as well.

Proof. Let I C R[x] and In (the nth leading ideal) the set of leading coefficients of

elements in I of degree ≤ n. Then clearly In CR, In ⊆ In+1. Moreover, if I ′ CR[x] with

I ⊆ I ′ and with In = I ′n for all n ≥ 0, then I = I ′. Otherwise, take f ∈ I ′ \ I with least

22

possible degree, say, the degree of f to be m. Then f = axm + f1, 0 6= a ∈ I ′m, where

either f1 = 0 or the degree of f1 less than m. Since a ∈ Im, there exists a g = axm+g1 ∈ I,where either g1 = 0 or the degree of g1 less than m. Thus, f − g = f1 − g1 ∈ I ′ which

implies that f1 = g1 since m is the least. This contradicts f 6∈ I.Let L0 ⊆ L1 ⊆ . . . be an ascending chain of ideals of R[x] and Lin the nth leading

ideal of Li. Note that Lij ⊆ Lkm whenever i ≤ k and j ≤ m. Since R is Noetherian, the

ascending chain Lii|i ≥ 0 of ideals of R stabilizes, say at Ljj. For any n with 0 ≤ n ≤j−1 the ascending chain Lin|i ≥ 0 stabilizes, say at Ltnn. Let m = maxj, t1, . . . , tj−1.Then for all i ≥ m and n ≥ 0, Lin = Lmn. Thus Li = Lm. So R[x] is Noetherian.

The algebra k[X] is called a free commutative-associative algebra over X and k.

§2.2 Noncommutative polynomials

Again, let X = x1, . . . , xn, . . . be a finite or countable linearly ordered set, x1 <

x2 < . . . xn < . . . (actually X may be any well-ordered set).

Following P.M. Cohn, by k〈X〉 one denotes the algebra of noncommutative polyno-

mials on X with coefficients in k,

k〈X〉 = ∑

i1,...,im

αi1...imxi1 . . . xim|m ≥ 0, αi1...im ∈ k.

A monomial u = xi1 . . . xim is called a (associative or noncommutative) word in X, the

length |u| (or degree deg(u)) is m.

Let X∗ be the set of all words in X. Then X∗ is a linear basis of k〈X〉.The algebra k〈X〉 satisfies a following universal property: For any associative algebra

A and any map ε : X → A, there exists a unique algebra homomorphism f : k〈X〉 → A

such that the following triangle is commutative

-

?

A

k〈X〉X i

ε∃! f

It follows that any associative algebra A is a quotient of some k〈X〉,

A = k〈X〉/I,

where I is an ideal of k〈X〉. Let S be a set of generators of I, i.e., I = Id(S) =

23

∑αiaisibi|ai, bi ∈ X∗, si ∈ S, αi ∈ k. Then by definition

A = k〈X〉/Id(S) = k〈X|S〉,

the associative algebra with generators X and defining relations si = 0, si ∈ S.

Our main problem is, starting with S, to find a special system Sc of defining relations

of A that is called a Grobner-Shirshov basis of the ideal I, or a GSB of A. Sc can be

found using the Shirshov algorithm for associative algebras.

The algebra k〈X〉 is refereed to as a free ring (free associative algebra on X and k).

The theory of free rings had been developed in the book

P.M. Cohn, Free rings and their relations, Academic Press, 1974 (second edition

1981).

One of the main properties of k〈X〉 is

Theorem 2.3 (Cohn Theorem) Any left (right) ideal I of k〈X〉 is a free left (right) k〈X〉-module, i.e., there exist S ⊂ I such that any polynomial f ∈ I has a unique presentation

f =∑fisi, si ∈ S, fi ∈ k〈X〉.

It may be proved using GS bases theory for modules (actually S is a minimal GS

basis of I as a left k〈X〉-module, see section §4.2).

Another general property of k〈X〉 is

Theorem 2.4 (Kemer Theorem) Any fully characteristic ideal T of k〈X〉 (i.e., an ideal

that is closed under substitutions xi → fi for any fi ∈ k〈X〉) is finitely generated as a

fully characteristic ideal.

The result is proved in the book: A.R. Kemer, Identities of associative algebras,

AMS, 1990.

§2.3 Lie polynomials

Again, let k be a field, X = x1, x2, . . . a well-ordered set, x1 < x2 < . . . . A Lie

polynomial f in X and k is a noncommutative polynomial in X and k (i.e., f ∈ k〈X〉)which is a linear combination of Lie words in X. A Lie word is defined by induction:

1) xi is a Lie word of the length (degree) 1; 2) If u, v are Lie words of length k and l

respectively, then [uv] = uv − vu is a Lie word of length k + l.

Examples of Lie words are xi, [xixj], [[xixj]xk], [xi[xjxk]], [[xixj[[xkxl]], [[. . . [xi1xi2 ] . . . ]xik ]

(a left normed Lie word), [xi1 [. . . [xik−1xik ] . . . ] (a right normed Lie word).

24

The set of all Lie polynomials in X and k will be denoted by Liek(X) or Lie(X) if

k is fixed. It is a Lie algebra under the Lie product [fg] = fg − gf . Actually it is a Lie

subalgebra of k〈X〉(−) generated by X.

Again, the algebra Lie(X) satisfies a following universal property: For any Lie algebra

L and any map ε : X → L, there exists a unique Lie algebra homomorphism f : Lie(X)→L such that the following triangle is commutative

-

?

L

Lie(X)X i

ε∃! f

It follows from PBW Theorem, see Section §4.1.1, also Theorem 5.28.

Again it follows that any Lie algebra L is a quotient of some Lie(X),

L = Lie(X)/I,

where I is an ideal of Lie(X). Let S be a set of generators of I, i.e., I = Id(S) =

∑αi[aisibi]|ai, bi ∈ X∗, si ∈ S, αi ∈ k. Then by definition

C = Lie(X)/Id(S) = Lie(X|S),

the Lie algebra with generators X and defining relations si = 0, si ∈ S.

Our main problem is, starting with S, to find a special system Sc of defining relations

of L that is called a Grobner-Shirshov basis of the ideal I, or a GSB of L. Sc can be

found using the Shirshov algorithm for Lie algebras.

Liek(X) is called a free Lie algebra on X and k.

It is not difficult to prove that any Lie word is a linear combination of left (right)

normed Lie words. So left (right) normed Lie words is a set of linear generators of Lie(X).

But they are linearly dependant.

A natural question arises: What is a linear basis of Lie(X)?

Up to now there are known several such bases:

The Hall basis (M. Hall, 1949), the Hall-Shirshov series of bases (Shirshov, 1953,

1962), the Lyndon-Shirshov basis (Chen, Fox, Lyndon, 1958, Shirshov, 1958).

Let us remark that Lyndon-Shirshov basis is a particular case of Hall-Shirshov series

of bases.

One of the main property of a free Lie algebra is

Theorem 2.5 (Shirshov-Witt Theorem) Any subalgebra of a free Lie algebra is a free Lie

algebra.

25

In a proof of this theorem, Shirshov (1953) used Lazard-Shirshov elimination: In

Lie(X), elements [xixn1 ] = [. . . [xix1] . . . ]x1], i > 1, n ≥ 0 are free generators of the

subalgebra generated by them.

Lazard-Shirshov elimination is used in GS bases theory for Lie algebras together with

Lyndon-Shirshov associative and Lie words.

We prove Theorem 2.5 in the following sections.

§2.3.1 Lazard-Shirshov elimination process

Let A be an index set, N set of natural number, Y = yα,n|α ∈ A, n ∈ N, and

Lie(Y ) the free Lie algebra generated by Y over a filed k. Let δ : Lie(Y ) → Lie(Y )

be a derivation such that yα,n 7→ yα,n+1, i.e., δ is a linear map such that for any f, g ∈Lie(Y ), δ(fg) = δ(f)g+ fδ(g). Let L1 be the direct sum of the linear spaces Lie(Y ) and

Fδ, denoted by Lie(Y ) ⊕ Fδ. In order to obtain a Lie structure on L1, we define a

multiplication [ ] on Lie(Y ) as usual, and for any u ∈ Lie(Y ), [u δ] = δ(u), [δ u] = −δ(u)and [δ δ] = 0. It is easy to prove that (L1, [ ]) is a Lie algebra.

Let X = xα| α ∈ A and I be the Lie ideal, generated by X, of the free Lie

algebra Lie(X ∪ y) over X ∪ y. Obviously I as a vector space is spanned by those

Lie monomials in X ∪y containing at least one xα, α ∈ A. The ideal I has codimension

1 in Lie(X ∪ y), and y is a linear basis of Lie(X ∪ y) modulo I.

Denote xnα (n ∈ N) the Lie product [[· · · [[xα y]y] · · · ]y︸ ︷︷ ︸

n

] in Lie(X ∪ y).

Lemma 2.6 As a Lie subalgebra of Lie(X∪y), the Lie ideal I is generated by xnα| α ∈

A, n ∈ N.

Proof. Let J be the Lie subalgebra, generated by xnα| α ∈ A, n ∈ N, of Lie(X ∪ y).

It is clear that I contains J . Let u be an arbitrary Lie monomial in X ∪ y containing

at least one xα, α ∈ A. It suffices to show that u belongs to J . Let us use induction

on the length of u, denoted by |u|. If |u| = 1, then u = xα and of course it belongs to

J . Suppose that u = [u1u2]. If u1 and u2 contains xα and xβ, respectively, then by the

induction hypothesis both u1 and u2 belong to J , and thus u ∈ J . Otherwise, we may

assume that u2 does not contain any xα, α ∈ A, because of the anti-commutativity law of

Lie algebras. By the induction hypothesis, u1 belongs to J and without loss of generality

let u1 = [xn1α1xn2

α2· · ·xnt

αt] be an arbitrary Lie monomial in xn

α| α ∈ A, n ∈ N. Therefore,

it follows from the Jacobi identity, that u = [u1u2] = [[xn1α1xn2

α2· · ·xnt

αt]y] belong to J . This

completes our proof.

26

Lemma 2.7 The Lie ideal I as a Lie subalgebra of Lie(X ∪ y) is freely generated by

xnα, α ∈ A, n ∈ N.

Proof. Let ϕ : Lie(Y ) → Lie(X ∪ y) be a Lie homomorphism such that yα,n 7→ xnα.

Let ϕ : L1 = Lie(Y ) ⊕ Fδ → Lie(X ∪ y) be a linear map such that δ 7→ y and

u 7→ ϕ(u) if u ∈ Lie(Y ). It is easy to check that ϕ preserves Lie multiplication and thus

it is a Lie homomorphism. Now, let us define a Lie homomorphism ψ : Lie(X∪y)→ L1

by letting xα 7→ yα,0 and y 7→ δ. Then ϕ is a Lie isomorphism since ϕψ = 1 and ψϕ = 1.

It together with lemma 2.6 follows that I ∼= Lie(Y ). This completes our proof.

§2.3.2 Constructing generators for an arbitrary subalgebra in a free Lie al-

gebra

Let H be a linear basis of monomials of a free Lie algebra Lie(X). Define d : X → Nby setting d(xα) = 1 for all α ∈ A and d([xi1 , . . . , xit ]) = t for [xi1 , . . . , xit ] ∈ H. For

any Lie polynomial f in Lie(X), it can be uniquely presented as f =∑αiui, α ∈ k

and ui ∈ H. Then the degree of f is defined by the highest degree of its monomials

ui, denoted by deg(f). Every nonzero Lie polynomial f may also be uniquely written as

f = f (1) + . . . + f (m) where f (i) is a summation of monomials in f with degree i. Then,

f (m) is called the highest part of f , denoted by f ′.

Let L be a Lie subalgebra of Lie(X), and Vi = f ∈ L| deg(f) ≤ i, and Bi be a

linear basis of Vi, satisfying Bi ⊆ Bi+1, for all i. Clearly, ∪∞i=1Vi = L and B := ∪∞

i=1Bi is

a linear basis of L.

Definition 2.8 The set M = bα of elements of a free Lie algebra Lie(X) is called irre-

ducible iff for every b ∈M , its highest part b′ does not belong to the subalgebra generated

by the highest parts b′α’s of the remaining elements bα ∈M , bα 6= b.

Lemma 2.9 There exists an irreducible generating set for the Lie subalgebra L of a free

Lie algebra Lie(X).

Proof. Let < be a well ordering on B such that

B1 < B2 \B1 < . . . < Bi+1 \Bi < . . . .

LetM = b ∈ B|b /∈ the subalgebra generated by all elements in B which are less than b.Claim 1: M is a generating set of L.

In order to prove that M generates L, it suffices to show that B is a subset of 〈M〉,the subalgebra of Lie(X) generated by M . Firstly, the smallest element of B is of course

27

in 〈M〉. Suppose that b ∈ B, and all elements in B less than b belong to 〈M〉. If b is in

M , then b surely belongs to 〈M〉. If b is not in M , then, by the definition of M and the

induction hypothesis, b belongs to 〈M〉.Claim 2: M is an irreducible set.

Assume that there is an element b ∈ M such that b′ belongs to the subalgebra

generated by the highest parts b′α’s of the remaining elements bα ∈ M , bα 6= b. Then b

can be presented by

b′ =∑

αib′i +∑

βj1...jnj[b′j1 . . . b

′jnj

] (2.1)

where bi, bjnk∈ M\b. The summations on the right-hand side of equation (2.1) may

contain factors of degree greater than the degree of b′. Since rewriting the product of two

Hall words, they will either become zero or will keep the the same degree and the same

content. In view of the linear independence of Hall words, all such factors must cancel

each other and hence we may assume the first summation contains only the elements of

the same degree as b and the remaining elements appearing on the right-hand side of

equation (2.1) have degree strictly less than the degree of b. Without loss of generality,

we may also assume that b > bi, for all i. From the above equation, it follows that the

element

b−∑

αibi +∑

βj1...jnj[bj1 . . . bjnj

]

belongs to Vd(b)−1, which contradicts b ∈M . This completes our proof.

§2.3.3 A proof of Shirshov-Witt Theorem

Definition 2.10 A finite subset T = t1, . . . , tn of a free Lie algebra Lie(X) is said to

be algebraically independent if there does not exist a nonzero Lie polynomial F (y1, . . . , yn)

in a free Lie algebra Lie(y1, . . . , yn) such that F (t1, . . . , tn) = 0 in Lie(X). An infinite

subset S of Lie(X) is said to be algebraically independent if its arbitrary finite subset is

algebraically independent.

It’s well known that a set S of Lie(X) is freely generated a Lie subalgebra iff S is alge-

braically independent.

Claim: every irreducible subset of a free Lie algebra Lie(X) is algebraically indepen-

dent.

Let us assume the contrary. There exists a finite irreducible subset S = b1, b2, . . . , bnof Lie(X) which is algebraically dependent. Then we have the following lemma.

Lemma 2.11 The set S ′ = b′1, b′2, . . . , b′n of highest parts of elements in S is an irre-

ducible and algebraically dependent set.

28

Proof. By the assumption that S is algebraically dependent, there exists a nonzero Lie

polynomial F (y1, . . . , yn) in the free Lie algebra Lie(y1, . . . , yn) such that F (b1, . . . , bn) =

0. Setting d : Y → N, yi 7→ deg(bi). The degree of an Hall word u = [yi1yi2 · · · yit ] in

y1, . . . , yn is defined by d(u) =∑

j d(yij). Then F (y1, . . . , yn) can be presented as

F (y1, . . . , yn) = F1(y1, . . . , yn) + F2(y1, . . . , yn)

where F1(y1, . . . , yn) is a summation of all Lie monomials in F (y1, . . . , yn) with highest

degree. Clearly, F1(y1, . . . , yn) 6= 0. Then

F (b1, . . . , bn) = F1(b1, . . . , bn) + F2(b1, . . . , bn)

= F1(b′1 + r1, . . . , b

′n + rn) + F2(b1, . . . , bn)

= F1(b′1, . . . , b

′n) +G(b1, . . . , bn, r1, . . . , rn)

= 0

where G(b1, . . . , bn, r1, . . . , rn) is a Lie polynomial on b1, . . . , bn, r1, . . . , rn. It follows

that F1(b′1, . . . , b

′n) = 0. Therefore S is algebraically dependent. S ′ is irreducible since S

is irreducible. This completes our proof.

Lemma 2.11 allows us to assume that b1, . . . , bn are homogeneous. We may also

assume that deg(b1) ≤ deg(b2) ≤ . . . ≤ deg(bn).

Lemma 2.12 Suppose that c1 = b1, c2 = b2+v2, . . . , cn = bn+vn, where vi ∈ 〈b1, . . . , bi−1〉and each vi is homogeneous with degree deg(bi). Then c1, c2, . . . , cn is an irreducible and

algebraically dependent set of homogeneous elements.

Proof. Clearly, each cj is homogeneous. Let us assume that c1, c2, . . . , cn is reducible.

Like the argument in the proof of Lemma 2.9, ci can be presented as

ci =∑

αjcj +∑

βk1...ktk[ck1 . . . cktk

]

where ci 6= cj, ckl, l = 1, 2 . . . , tk for all j, k. Without loss of generality, we may assume

that i > j for all j. Then, replacing ct by bt + vt(v1 = 0), t = 1, . . . , n, we have

bi + vi =∑

αj(bj + vj) +∑

βk1...ktk[(bk1 + vk1) · · · (bktk

+ vktk)].

Since vj ∈ 〈b1, . . . , bj−1〉 and deg(b1) ≤ deg(b2) ≤ . . . ≤ deg(bn), we have

bi =∑

αsbs +∑

βt1...tlt[bt1 · · · btlt ],

where bi 6= bs, btj , j = 1, 2 . . . , lt for all s, t. This contradicts that b1, b2, . . . , bn is irre-

ducible.

29

We claim that vi ∈ 〈c1, . . . , ci−1〉, i = 2, . . . , n.

Induction on i. If i = 2, then v2 ∈ 〈c1〉 = 〈b1〉. For an arbitrary vi, we have

vi ∈ 〈b1, . . . , bi−1〉 = 〈c1, c2 − v2, . . . , ci−1 − vi−1〉. By the induction hypothesis, vi ∈〈c1, . . . , ci−1〉.

For an arbitrary Hall word u = [yi1yi2 . . . , yit ] in y1, . . . , yn, we denote nyi(u) the

number of the occurrences of yi in u and we let wt(u) = (nyn(u), . . . , ny1(u)). We write

F1(y1, . . . , yn) as

F1(y1, . . . , yn) = F11(y1, . . . , yn) + F12(y1, . . . , yn)

where F11(y1, . . . , yn) is a summation of all Lie monomials in F (y1, . . . , yn) with highest

weight.

Then

F (b1, . . . , bn) = F11(c1, c2 − v2, . . . , cn − vn) + F12(c1, c2 − v2, . . . , cn − vn)

= F11(c1, . . . , cn) +G1(c1, . . . , cn, v1, . . . , vn)

= F11(c1, . . . , cn) +G2(c1, . . . , cn), (by vi ∈ 〈c1, . . . , ci−1〉),

= 0

where G1 and G2 are Lie polynomials in c1, . . . , cn, v1, . . . , vn.Define F2(y1, . . . , yn) by replacing ci with yi in F11(c1, . . . , cn) +G2(c1, . . . , cn). From

the definition of F11(y1, . . . , yn) and the above discussion, it follows that F2(y1, . . . , yn) 6=0 but F2(c1, . . . , cn) = 0. Therefore c1, c2, . . . , cn is algebraically dependent. This

completes our proof.

Lemma 2.13 There exists an irreducible and algebraically dependent subset U = u1, . . . , unof Lie(X) where each Lie polynomial ui is homogeneous in each letter xα, α ∈ A, and

deg(ui) ≤ deg(bi), i = 1, . . . , n.

Proof. Choose an arbitrary generator xα from among the elements b1, . . . , bn. Then

each bi can be written as

bi = bi0 + bi1 + . . .+ biti

where bij is a Lie polynomial, homogeneous in xα and has xα degree j. The assumption

b1, . . . , bn algebraically dependent may lead to b1t1, . . . , bntn

also algebraically dependent

like the proof of Lemma 2.11 (just by setting d(yi) := the xα degree of bi).

If there exists some i such that biti ∈ 〈b1t1, . . . , bi−1ti−1

〉, then there is a Lie polynomial

h(y1, . . . , yi−1) such that biti = h(b1t1, . . . , bi−1ti−1

) and then set bi := bi − h(b1, . . . , bi−1).

30

By Lemma 2.12, b1, b2, . . . , bn is irreducible and algebraically dependent. After re-

peat this kind of process finite times, we may reach the following situation that biti /∈〈b1t1

, . . . , bi−1ti−1〉 for all i even more biti /∈ 〈b1t1

, . . . , bi−1ti−1, bi+1ti+1

, . . . , bntn〉 for all i.

Let v1 = b1t1, . . . , vn = bntn

. Then the set v1, . . . , vn is irreducible and algebraically

dependent and each vi is homogeneous in xα. Enumerating one by one all generators that

occur in the elements b1, . . . , bn, we finally obtain the desired set U = u1, . . . , un.This completes our proof.

Lemma 2.13 allows us to assume that S = b1, b2, . . . , bn is a finite irreducible and

algebraically dependent subset of Lie(X) and each bi is homogeneous in each generator

xα, α ∈ A. We may also assume that deg(b1) ≤ deg(b2) ≤ . . . ≤ deg(bn). In the follow

argument, we show that this assumption will lead to a contradiction. The ordered n-

tuple (λ1; . . . ;λn) where λi is the degree of bi, will be called the height the set S =

b1, b2, . . . , bn. Let M be the set of all possible heights of S = b1, b2, . . . , bn and we

order it lexicographically. If (λ1; . . . ;λn) = (1; . . . ; 1), then b1, b2, . . . , bn are free generators

and thus b1, b2, . . . , bn is algebraically independent which contradicts the assumption in

the very beginning of this section. Suppose that (λ1; . . . ;λn) > (1; . . . ; 1), then there exists

a generator xβ such that bi 6= xβ for all i, and xβ occurs in bj for some j. Then all elements

bi, i = 1, . . . , n belong to the ideal of Lie(X) generated by X\xβ. By Lemma 2.7, bi

can be regarded as Lie monomials in new free generator xnα = [[· · · [[xα xβ]xβ] · · · ]xβ︸ ︷︷ ︸

n

], α ∈

A, n ∈ N. Now in this free subalgebra of Lie(X), we define the degree function by

deg(xnα) = 1. Since each bi is homogeneous in each generator including xβ, Lie polynomials

bi are homogeneous in new generators xnα and hence the set S = b1, b2, . . . , bn is still

irreducible and algebraically dependent. For each i, the degree of bi in new generators

is less or equal to its degree in original generators xα, α ∈ A and there exists at least

one j such that the degree of bj in new generators is strictly smaller than its degree in

xα, α ∈ A. By Lemma 2.13, we may get an irreducible and algebraically dependent subset

U = u1, . . . , un of L(xnα|α ∈ A, n ∈ N) where each Lie polynomial ui is homogeneous in

each letter xnα, α ∈ A, n ∈ N and the degree of ui in xn

α, α ∈ A, n ∈ N is less or equal to

the degree bi in xnα, α ∈ A, n ∈ N, i = 1, . . . , n. By the induction hypothesis on the set S,

we may get a contradiction to the assumption that there exists a finite irreducible subset

of Lie(X) which is algebraically dependent.

The proof of Theorem 2.5. By Lemma 2.9, L always has an irreducible generating set

M . From the above argument, we know that M is algebraically independent. Therefore

L is freely generated by M . This completes our proof of Theorem 2.5.

31

Chapter 3 Grobner Bases for Commutative

Algebras

§3.1 Buchberger theorem

Let k[X] be a polynomial algebra on X = x1, x2, . . . , xn and a field k, x1 > x2 >

· · · > xn.

Remark 2.1 We choose ordering x1 > x2 > . . . following ordering of variables in the

Gauss elimination algorithm and a tradition of Grobner bases theory for commutative

polynomials. For noncommutative and nonassociative variables we will choose x1 < x2 <

. . . .

Let us define the deg-lex ordering on the set of commutative words [X] to compare

two words first by degree and then lexicographically.

For example, x31 > x2

1x2 > x1x2x3 > x21. In general,

xi11 x

i22 . . . x

inn > xj1

1 xj22 . . . x

jnn iff

∑ik >

∑jk, or∑

ik =∑

jk and (i1, i2, . . . , in) > (j1, j2, . . . , jn) lexicographically.

This ordering is a monomial ordering in the sense

(i) If u > v, then wu > wv for any w ∈ [X],

(ii) [X] is a well-ordered set, i.e., a linearly ordered set with minimal (descending

chain) condition – any chain u1 > u2 > · · · > ur > . . . is finite.

Another example of monomial ordering on [X] is lex-deg ordering to compare two

commutative words lexicographically with a condition that a proper beginning of a word

is less than the word.

Let us fix a monomial ordering on [X]. A polynomial f ∈ k[X] has a maximal

(leading) word f with a leading coefficient αf6= 0,

f = αff +

∑ui<f

αiui, ui ∈ [X], αf, αi ∈ k.

If αf

= 1, f is called a monic polynomial.

It is clear that fg = f g.

Let us define a least common multiple of two words as usual:

lcm(xi11 · · ·xin

n , xj11 · · ·xjn

n ) = xmaxi1,j11 · · ·xmaxin,jn

n .

Then

lcm(u, v) = au = bv, a, b ∈ [X].

32

If lcm(u, v) = uv then it is called a trivial lcm, otherwise a non-trivial lcm. If u = bv

then lcm(u, v) = u is called an inclusion lcm.

Definition 3.1 Let f, g be monic polynomials and w = lcm(f , g) = af = bg. Then

polynomial

(f, g)w = af − bg

is called an s-polynomial (a composition) of f and g.

It is clear that (f, g)w ∈ Id(f, g) and (f, g)w < w. If w is a trivial lcm, then

(f, g)w = gf − fg = (g − g)f − (f − f)g, and the composition is trivial mod(f, g;w) in a

sense of the following definition.

Definition 3.2 Let S ⊂ k[X] be a monic set of polynomials. S is called a Grobner

basis if any composition (f, g)w of polynomials from S is trivial mod(S,w), denoted by

(f, g)w ≡ 0 mod(S,w), i.e.,

(f, g)w =∑

αiaisi, si ∈ S, ai ∈ [X], αi ∈ k, aisi < w for all i.

From the above, it is equivalent to that (f, g)w ≡ 0 mod(S,w) for a non-trivial

w = lcm(f , g).

It is clear the case if (f, g)w goes to zero using the Elimination of Leading Words

(ELW’s) of S (in the leading monomials of polynomials), i.e., transformations h→ h−αas,where s ∈ S and αas is the leading term (a monomial with a coefficient) of h in the

standard presentation of h. Then h− αas < h.

Theorem 3.3 (Buchberger Theorem) Let S ⊂ k[X] be a monic set and < a monomial

ordering on [X]. Then the following conditions are equivalent.

(i) S is a Grobner basis.

(ii) If f ∈ Id(S) then f = as, s ∈ S, a ∈ [X].

(iii) Irr(S) = u ∈ [X]|u 6= as, s ∈ S, a ∈ [X] is a k-linear basis of k[X|S], the commu-

tative algebra with generators X and defining relations S.

Proof. (i)⇒(ii). Let f =∑m

i=1 αiaisi, si ∈ S, ai ∈ [X], αi ∈ k. We may assume

that all similar terms (si = sj, ai = aj) are reduced. Let us choose the leading words

wi = aisi. Let

w1 = w2 = · · · = wl > wl+1 ≥ · · · ≥ wm, l ≥ 1.

33

We will use induction on (w1, l), w1 ≥ f , l ≥ 1 with the lex-ordering of the pairs. If w1 = f

or l = 1, the claim is clear. Otherwise l > 1 and w1 = a1s1 = a2s2. Since w1 is a common

multiple of s1, s2, it must contain w = lcm(s1, s2) as a subword, w1 = blcm(s1, s2) such

that a1s1 = bw|s1→s1 , a2s2 = bw|s2→s2 . Then

a1s1 − a2s2 = b(s1, s2)w ≡ 0 mod(S,w1).

It means that we may rewrite the initial equality for f in a way

f = (α1 + α2)a2s2 + · · ·+ αlalsl + . . . .

If l > 2 or α1 + α2 6= 0, we may apply the induction (since we decrease l). Otherwise we

decrease w1.

(ii)⇒(iii). For any S, the ELW’s of S in all monomials of a polynomial gives rise a

following presentation of an arbitrary polynomial h

h =∑

αiui +∑

βjajsj, (3.1)

where each αi, βj ∈ k, ui ∈ Irr(S), sj ∈ S, ajsj ≤ h. It means that Irr(S) is a set of

linear generators of k[X|S].

If a linear combination of S-irreducible words ui’s belongs to Id(S) then by (ii) some

of ui contains s for some s ∈ S, a contradiction.

(iii)⇒(i). If h = (f, g)w as above then all αi = 0 and (f, g)w ≡ 0 mod(S,w).

Corollary 3.4 A monic set S ⊆ k[X] is a Grobner basis if and only if any composition

of polynomial from S goes to zero by ELW’s of S.

Proof. (⇐) It was proved above. (⇒) Let h = (f, g)w and f, g ∈ S. By ELW’s of S

one can present h as before. By Theorem 3.3 all αi’s are zero. It means that h goes to

zero by ELW’s of S.

§3.2 Buchberger algorithm

Let X = x1, . . . , xn, S = s1, . . . , sm ⊂ k[X] and I = Id(S) the ideal of k[X]

generated by S. If each composition of elements from S is trivial mod(S,w), then S is a

Grobner basis. Otherwise let us join to S a nontrivial composition h = (f, g)w, f, g ∈ S.

Actually we present first h in a form (3.1) and to join to S an S-irreducible part of

h, namely h′ =∑αiui, ui ∈ Irr(S). Let S ⊂ S ′ = S ∪ h′. It is easy to see that

34

Id(s1, . . . , sm) is a proper subset of Id(s1, . . . , sm, s′), since h′ 6∈ Id(s1, . . . , sm). Then we

continue the process and would have a serious of generator sets of Id(S),

S ⊂ S ′ ⊂ S ′′ ⊂ . . . (3.2)

such that the ideals generated by the leading monomials of S, S ′, S ′′, . . . constitute an

ascending chain of ideals of k[X]. By Hilbert basis theorem the chain must be finite, so

is (3.2).

Definition 3.5 A Grobner basis S in k[X] is called reduced if for any s ∈ S, supp(s) ⊆Irr(S\s), where supp(s) = u1, u2, . . . , um if s =

∑mi=1 αiui each 0 6= αi ∈ k, ui ∈ [X].

A Grobner basis S in k[X] minimal, if there are no compositions of inclusion of

polynomials from S, i.e., f 6= ag, a ∈ [X], for any different f, g ∈ S.

Clearly, a reduced Grobner basis is minimal.

Corollary 3.6 For any ideal I ⊂ k[X] and any monomial ordering < on [X] there exists

only one reduced Grobner basis of I.

Proof. Clearly, there exists a minimal Grobner-Shirshov basis S ⊂ k[X] for the ideal

I = Id(S).

For any s ∈ S, by using (3.1), we have s = s′+s′′, where supp(s′) ⊆ Irr(S\s), s′′ ∈Id(S \ s). Since S is a minimal Grobner-Shirshov basis, we have s = s′ for any s ∈ S.

Let S ′ = s′|s ∈ S. It is clear that S ′ ⊆ Id(S) = I. For any f ∈ Id(S), by Theorem

3.3, f = as = as′ for some a ∈ [X], s ∈ S. From this and Theorem 3.3 again, it follows

that S ′ = s′|s ∈ S is a reduced Grobner-Shirshov basis for the ideal I.

Suppose that S, R are two reduced Grobner-Shirshov bases for the ideal I. For any

s ∈ S, by Theorem 3.3, s = ar, r = bs1 for some a, b ∈ [X], r ∈ R, s1 ∈ S and hence

s = abs1. Since s ∈ supp(s) ⊆ Irr(S \ s), we have s = s1. It follows that a = b = 1

and so s = r.

If s 6= r then 0 6= s− r ∈ I = Id(S) = Id(R). By Theorem 3.3, s− r = a1r1 = b1s2

for some a1, b1 ∈ [X] with r1, s2 < s = r. This means that s2 ∈ S \ s and r1 ∈ R \ r.Noting that s− r ∈ supp(s)∪ supp(r), we have either s− r ∈ supp(s) or s− r ∈ supp(r).If s− r ∈ supp(s) then s− r ∈ Irr(S \ s) which contradicts s− r = b1s2; if s− r ∈supp(r) then s− r ∈ Irr(R \ r) which contradicts s− r = a1r1. This shows that s = r

and then S ⊆ R. Similarly, R ⊆ S.

35

Corollary 3.7 For any ideal I = Id(S) ⊂ k[X] the membership problem

for any f(X) ∈ k[X] to answer whether f(X) ∈ Id(S) or not

is algorithmically solvable.

Proof. We may assume that S is a finite Grobner basis of I. Induction on u = f ≥ 1.

If u = 1, it is clear. Otherwise if u does not contain s, s ∈ S, the answer is negative. If

u = as, s ∈ S, a ∈ [X] then f1 = f − αas has less leading monomial and is belonged to

I. By induction we are done.

Corollary 3.8 Gauss elimination algorithm for solving systems of linear equations is a

particular case of Buchbereger algorithm.

Proof. Let

Ax = b, A ∈Matm,n(k), x = (x1, . . . , xn)Tr, b = (b1, . . . , bm)Tr

be a system of linear equations over k. Let fi = aix − bi, 1 ≤ i ≤ m be the correspond-

ing system of linear polynomials. Then Buchberger algorithm for fi gives the Gauss

elimination algorithm for the system, and visa versa.

Theorem 3.9 By Grobner bases of ideals I, J in k[X] one may algorithmically find

Grobner bases of I + J, I ∩ J , and rad(I) := f ∈ k[X]|fm ∈ I for some m ≥ 1.

36

Chapter 4 Grobner-Shirshov Bases for Associative

Algebras

In this chapter, we give a proof of Shirshov CD-lemma for associative algebras.

§4.1 Composition-Diamond lemma for associative algebras

Let k be a field, k〈X〉 the free associative algebra over k generated by X and X∗

the free monoid generated by X, where the empty word is the identity which is denoted

by 1. For a word w ∈ X∗, we denote the length (degree) of w by |w| (deg(w)). Let X∗

be a well-ordered set. Let f ∈ k〈X〉 with the leading word f and f = αf − rf where

0 6= α ∈ k. We call f monic if α = 1.

A well ordering > on X∗ is called monomial if it is compatible with the multiplication

of words, that is, for u, v ∈ X∗, we have

u > v ⇒ w1uw2 > w1vw2, for all w1, w2 ∈ X∗.

A standard example of monomial ordering on X∗ is the deg-lex ordering to compare two

words first by degree and then lexicographically, where X is a well-ordered set.

Let f and g be two monic polynomials in k〈X〉 and < a monomial ordering on X∗.

Then, there are two kinds of compositions:

(i) If w is a word such that w = f b = ag for some a, b ∈ X∗ with |f | + |g| > |w|,then the polynomial (f, g)w = fb − ag is called the intersection composition of f and g

with respect to w.

(ii) If w = f = agb for some a, b ∈ X∗, then the polynomial (f, g)w = f − agb is

called the inclusion composition of f and g with respect to w.

In the composition (f, g)w, w is called an ambiguity, or a lcm(f , g).

Let S ⊂ k〈X〉 such that every s ∈ S is monic. Let h ∈ k〈X〉 and w ∈ X∗. Then h is

called trivial modulo (S,w), denoted by h ≡ 0 mod(S,w), if h =∑αiaisibi, where each

αi ∈ k, ai, bi ∈ X∗, si ∈ S and aisibi < w.

Elements asb, a, b ∈ X∗, s ∈ S are called S-words.

Definition 4.1 A monic set S ⊂ k〈X〉 is called a GS basis in k〈X〉 with respect to the

monomial ordering < if any composition of polynomials in S is trivial modulo S and

corresponding w.

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A set S is called a minimal GS basis in k〈X〉 if S is a GS basis in k〈X〉 such that

there is no inclusion compositions in S, i.e., for any f, g ∈ S with f 6= g, one has f 6= agb

for any a, b ∈ X∗.

Denote

Irr(S) = u ∈ X∗|u 6= asb, s ∈ S, a, b ∈ X∗.

A GS basis S in k〈X〉 is called reduced if for any s ∈ S, supp(s) ⊆ Irr(S \ s),where supp(s) = u1, u2, . . . , un if s =

∑ni=1 αiui each 0 6= αi ∈ k, ui ∈ X∗.

The following lemma plays a key role for proving CD-lemma for associative algebras.

Lemma 4.2 If S is a GS basis in k〈X〉 and w = a1s1b1 = a2s2b2, where a1, b1, a2, b2 ∈X∗, s1, s2 ∈ S, then a1s1b1 ≡ a2s2b2 mod(S,w).

Proof. There are three cases to consider.

Case 1. Suppose that subwords s1 and s2 of w are disjoint, say, |a2| ≥ |a1| + |s1|.Then, a2 = a1s1c and b1 = cs2b2 for some c ∈ X∗, and so, w1 = a1s1cs2b2. Now,

a1s1b1 − a2s2b2 = a1s1cs2b2 − a1s1cs2b2

= a1s1c(s2 − s2)b2 + a1(s1 − s1)cs2b2.

Since s2 − s2 < s2 and s1 − s1 < s1, we conclude that

a1s1b1 − a2s2b2 =∑

i

αiuis1vi +∑

j

βjujs2vj

for some αi, βj ∈ k, S-words uis1vi and ujs2vj such that uis1vi, uj s2vj < w.

Case 2. Suppose that the subword s1 of w contains s2 as a subword. Then s1 =

as2b, a2 = a1a and b2 = bb1, that is, w = a1as2bb1 for some S-word as2b. We have

a1s1b1 − a2s2b2 = a1s1b1 − a1as2bb1 = a1s1 − as2bb1 = a1(s1, s2)s1b1.

By the triviality of compositions it follows that a1s1b1 ≡ a2s2b2 mod(S,w).

Case 3. s1 and s2 have a nonempty intersection as a subword of w. We may assume

that a2 = a1a, b1 = bb2, w = s1b = as2, |w| < |s1|+ |s2|. Then, similar to the Case 2, we

have a1s1b1 ≡ a2s2b2 mod(S,w).

Remark: A short proof of Lemma 4.2 uses a notation of lcm(u, v) of two words:

lcm(u, v) =

ub = av, a, b ∈ X∗, deg(ub) < deg(u) + deg(v) (an intersection lcm)

avb, a, b ∈ X∗ (an inclusion lcm)

ucv, c ∈ X∗ (a trivial lcm)

38

Note that lcm(u, v) 6= lcm(v, u) in general. Then a general composition (f, g)w, w =

lcm(f , g) of two monic polynomials f, g is defined as follows

(f, g)w = lcm(f , g)|f 7→f − lcm(f , g)|g 7→g.

If w is a trivial lcm, then (f, g)w ≡ 0 mod(f, g;w).

Now, in (i) ⇒ (ii), if w = a1s1b1 = a2s2b2, then up to order of s1, s2, we have

w1 = cwb, c, b ∈ X∗, w = lcm(s1, s2) and a1s1b1 = ws1 7→s1 , a2s2b2 = ws2 7→s2 .

As the result,

a1s1b1 − a2s2b2 = c(s1, s2)wb ≡ 0 mod(S, cwb = w1).

The result (i)⇒ (ii) follows by induction on (w1, l).

Lemma 4.3 Let S ⊂ k〈X〉 be a subset of monic polynomials. Then for any f ∈ k〈X〉,

f =∑ui≤f

αiui +∑

ajsjbj≤f

βjajsjbj

where each αi, βj ∈ k, ui ∈ Irr(S) and ajsjbj an S-word. So, Irr(S) is a set of linear

generators of the algebra f ∈ k〈X|S〉.

Proof. The result follows by induction on f .

Theorem 4.4 (CD-lemma for associative algebras) Let < be a monomial ordering on X∗,

S ⊂ k〈X〉 a monic set and Id(S) the ideal of k〈X〉 generated by S. Then the following

statements are equivalent.

(i) S is a Grobner-Shirshov basis in k〈X〉.

(ii) f ∈ Id(S)⇒ f = asb for some s ∈ S and a, b ∈ X∗.

(iii) Irr(S) = u ∈ X∗|u 6= asb, s ∈ S, a, b ∈ X∗ is a linear basis of the algebra k〈X|S〉.

Proof. (i) ⇒ (ii). Let S be a GS basis and 0 6= f ∈ Id(S). Then, we have

f =∑n

i=1 αiaisibi where each αi ∈ k, ai, bi ∈ X∗, si ∈ S. Let

wi = aisibi, w1 = w2 = · · · = wl > wl+1 ≥ · · · .

We will use the induction on (w1, l) to prove that f = asb for some s ∈ S and a, b ∈ X∗.

If l = 1, then f = a1s1b1 = a1s1b1 and hence the result holds. Assume that l ≥2. Then, we have w1 = a1s1b1 = a2s2b2. It follows from Lemma 4.2 that a1s1b1 ≡

39

a2s2b2 mod(S,w1). Now the result follows from the induction whenever α1 + α2 6= 0, or

l > 2, or both α1 + α2 = 0 and l = 2. This shows (ii).

(ii) ⇒ (iii). By Lemma 4.3, Irr(S) generates k〈X|S〉 as a linear space. Suppose

that∑i

αiui = 0 in k〈X|S〉, where 0 6= αi ∈ k, ui ∈ Irr(S). It means that∑i

αiui ∈ Id(S)

in k〈X〉. Then∑i

αiui = uj ∈ Irr(S) for some j which contradicts (ii).

(iii) ⇒ (i). For any f, g ∈ S , by using Lemma 4.3 and (iii), we have (f, g)w ≡0 mod(S,w). Therefore, S is a GS basis.

Shirshov algorithm. If a monic subset S ⊂ k〈X〉 is not a GS basis then one can add to

S all nontrivial compositions making them monic. Continuing this process repeatedly, we

finally obtain a GS basis Sc that contains S and generates the same ideal, Id(Sc) = Id(S).

Sc is called GS completion of S. Using the reduction algorithm (elimination of leading

words of polynomials), we may get a minimal GS basis Sc.

Let K be a commutative ring with identity 1.

Remark: Both associative and Lie (Theorem 5.39) CD-lemmas are valid if we replace

the base field k by an arbitrary commutative ring K with identity since we assume that

all GS bases consist of monic polynomials. For example, let (L, [ ]) be a Lie algebra over

K and L a free K-module with a well-ordered K-basis ai|i ∈ I. Then with the deg-lex

ordering on ai|i ∈ I∗, its universal enveloping associative algebra UK(L) has usual GS

basis aiaj − ajai =∑αt

ijat|i > j, i, j ∈ I, where αtij ∈ K and [ai, aj] =

∑αt

ijat in L,

and by CD-lemma for associative algebras over K, L ⊂ UK(L) and ai1 · · · ain|i1 ≤ · · · ≤in, n ≥ 0, i1, . . . , in ∈ I is a K-basis of UK(L), see Theorem 4.7.

In fact, by the same reason, all CD-lemmas in this book are valid if we replace the

base field k by an arbitrary commutative ring K with identity. If this is the case, then the

(iii) in a CD-lemma should be: K(X|S) is a free K-module with a K-basis Irr(S). But

in general case, Shirshov algorithm does not work: if S is a monic set, then S ′, a set of

adding all non-trivial compositions to S, is not a monic set in general and the algorithm

may stop without any result.

The following theorem gives a linear basis of the ideal Id(S) if S ⊂ k〈X〉 is a GS

basis.

Theorem 4.5 Suppose that S ⊂ k〈X〉 is a Grobner-Shirshov basis. Then for any u ∈X∗ \ Irr(S), by Lemma 4.3, there exists u ∈ kIrr(S) with u < u (if u 6= 0) such that

40

u− u ∈ Id(S) and the set u− u|u ∈ X∗ \ Irr(S) is a linear basis of the ideal Id(S) of

k〈X〉.

Proof. Suppose that 0 6= f ∈ Id(S). Then by CD-lemma for associative algebras,

f = a1s1b1 = u1 for some s1 ∈ S and a1, b1 ∈ X∗ which implies that f = u1 ∈ X∗\Irr(S).

Let f1 = f−α1(u1−u1), where α1 is the coefficient of the leading term of f and u1 < u1 or

u1 = 0. Then f1 ∈ Id(S) and f1 < f . By induction on f , the set u− u|u ∈ X∗ \ Irr(S)generates Id(S) as a linear space. It is clear that the set u− u|u ∈ X∗ \ Irr(S) is linear

independent.

By a similar proof of Corollary 3.6, we have the following theorem.

Theorem 4.6 Let I be an ideal of k〈X〉 and > a monomial ordering on X∗. Then there

exists uniquely the reduced Grobner-Shirshov basis S for I.

§4.1.1 PBW-theorem for Lie algebras

Let (L, [ ]) be a Lie algebra over a field k with a well-ordered linear basis X = xi|i ∈I. Let S = [xixj] = xi, xj|i > j, i, j ∈ I be the multiplication table of L, where for

any i, j ∈ I, xi, xj = Σtαtijxt, α

tij ∈ k. Then U(L) = k〈X|S(−)〉 is called the universal

enveloping associative algebra of L, where S(−) = xixj − xjxi = xi, xj|i > j, i, j ∈ I.

Theorem 4.7 (PBW Theorem) Let the notation be as above. Then with the deg-lex order-

ing on X∗, S(−) is a Grobner-Shirshov basis in k〈X〉. Then by CD-lemma for associative

algebras, the set Irr(S(−)) consists of the elements

xi1 . . . xin , i1 ≤ · · · ≤ in, i1, . . . , in ∈ I, n ≥ 0

which forms a linear basis of U(L).

The result is valid for a Lie algebra that is a free K-module over a commutative ring

K (see Remark after Theorem 4.4).

Theorem 4.8 (PBW Theorem in a Shirshov form) Let L = Lie(X|S), S ⊂ Lie(X) ⊂k〈X〉, S a Grobner-Shirshov basis in Lie(X) and U(L) = k〈X|S〉. Then S is a GS basis

in k〈X〉 and a linear basis of U(L) consists of words u = u1 · · ·un, where u1 · · · un

in lex-ordering, n ≥ 0, each ui is S-irreducible associative Lyndon-Shirshov word.

41

§4.1.2 Normal forms for groups and semigroups

Let X be a set, S ⊆ X∗ ×X∗, ρ(S) the congruence on X∗ generated by S,

A = sgp〈X|S〉 = X∗/ρ(S)

the quotient semigroup and k(X∗/ρ(S)) the semigroup algebra. We identify the set u =

v|(u, v) ∈ S with S. Then it is easy to see that

σ : k〈X|S〉 → k(X∗/ρ(S)),∑

αi(ui + Id(S)) 7→∑

αi(uiρ(S))

is an algebra isomorphism.

Shirshov completion Sc of S consists of “semigroup relations”, Sc = ui− vi, i ∈ I.Then Irr(Sc) is a linear basis of k〈X|S〉 and so σ(Irr(Sc)) is a linear basis of k(X∗/ρ(S)).

This shows that Irr(Sc) is exactly normal forms of elements of the semigroup sgp〈X|S〉.Thus, in order to find normal forms of the semigroup sgp〈X|S〉, it suffices to find a

GS basis Sc in k〈X|S〉. In particular, let G = gp〈X|S〉 be a group, where S = (ui, vi) ∈F (X)× F (X)|i ∈ I and F (X) is the free group on a set X. Then G has a presentation

as a semigroup

G = sgp〈X ∪X−1|S, xεx−ε = 1, ε = ±1, x ∈ X〉, X ∩X−1 = ∅.

It is clear that in a group G = gp〈X|S〉, if for any x ∈ X there exists an n(x) ∈ Nsuch that the relation xn(x) = 1 is in S, then

G = gp〈X|S〉 = sgp〈X|S〉.

We give some examples.

1. Symmetric group Sn and alternating group An

Let Sn be the symmetric group of the permutations of 1, 2, · · · , n. Then the subset

An of all even permutations in Sn is a normal subgroup of Sn. One calls An the alternating

group of degree n. The following presentations of Sn and An are given in the monograph

of N. Jacobson (see [115], p71):

Sn = gp〈x1, . . . , xn−1 | x2i = 1, xjxi = xixj (j − 1 > i, 1 ≤ i, j ≤ n),

xi+1xixi+1 = xixi+1xi (1 ≤ i ≤ n− 2)〉.

under the isomorphism Sn → Sn, xi 7→ (i, i + 1), i = 1, 2, · · · , n − 1, where (ij) is the

transposition.

42

An = gp〈xi (1 ≤ i ≤ n− 2) ; x31 = 1, (xi−1xi)

3 = x2i = 1 (2 ≤ i ≤ n− 2),

(xixj)2 = 1 (1 ≤ i < j − 1, j ≤ n− 2)〉

under the isomorphism An → An, xi 7→ (12)((i+ 1)(i+ 2)), i = 1, 2, · · · , n− 2.

We will give proofs of these statements.

In Sn, let us define words xij = xixi−1 · · ·xj, where i > j; and xii = 1, xi,i+1 = 1.

Lemma 4.9 Let S be the set of defining relations in Sn together with xi+1jxi+1 =

xixi+1j|1 ≤ j < i, 1 ≤ i, j ≤ n. Then S is a Grobner-Shirshov basis of Sn in deg-

lex ordering.

Theorem 4.10 Let the notations be as above. Then

(i) the set N = x1i1x2i2 · · ·x(n−1)in−1|1 ≤ ij ≤ j + 1, j = 1, 2, · · · , n − 1 is a normal

form for Sn in generators xi = (i, i + 1), i = 1, 2, · · · , n − 1 relative to the deg-lex

ordering;

(ii) the map φ : xi 7→ (i, i+1), i = 1, 2, · · · , n− 1 induces an isomorphism of Sn and Sn.

Proof. By Lemma 4.9 and Theorem 4.4,

Irr(S) = x1i1x2i2 · · ·x(n−1)in−1| 1 ≤ ij ≤ j + 1, j = 1, 2, · · · , n− 1

which gives |Sn| = n!. On the other hand, φ induces a homomorphism φ : Sn → Sn and

|Sn| = n!. So, φ is an isomorphism.

We now give a presentation of the group An as a semigroup:

An = sgp〈x−11 , xi (1 ≤ i ≤ n− 2); x1x

−11 = x−1

1 x1 = x31 = 1, (xi−1xi)

3 = x2i = 1

(2 ≤ i ≤ n− 2), (xixj)2 = 1 (1 ≤ i < j − 1, j ≤ n− 2)〉

Let us order the generators in the following way:

x−11 < x1 < x2 < · · · < xn−2

Let X = x−11 , x1, x2, · · · , xn−2. Now we define the words

xji = xjxj−1...xi,

where j > i > 1 and xj1ε = xjxj−1...x1ε, ε = ±1.

43

Lemma 4.11 For ε = ±1, the set S of the following relations is a Grobner-Shirshov

basis of An in deg-lex ordering.

1) x2ε1 = x−ε

1

2) x2i = 1, (i > 1)

3) xjxi = xixj, (j − 1 > i ≥ 2)

4) xjxε1 = x−ε

1 xj, (j > 2)

5) xjixj = xj−1xji, (j > i ≥ 2)

6) xj1εxj = xj−1xj1−ε, (j > 2)

7) x2x1εx2 = x−ε

1 x2x−ε1

8) xε1x

−ε1 = 1

Now, we can state a presentation of An by generators and defining relations.

Theorem 4.12 Let

An = gp〈xi (1 ≤ i ≤ n− 2) ; x31 = 1, (xi−1xi)

3 = xi2 = 1 (2 ≤ i ≤ n− 2),

(xixj)2 = 1 (1 ≤ i < j − 1, j ≤ n− 2)〉.

Then

(i) every element w of An has a unique representation w = x1j1x2j2 ...xn−2jn−2, where

xtt = xt (t > 1), xii+1 = 1, xi1 = xixi−1 · · ·xε1, 1 ≤ ji ≤ i+ 1, 1 ≤ i ≤ n− 2, ε = ±1

(here we use xi1 instead of xi1ε).

(ii) the map φ : xi 7→ (12)((i+ 1)(i+ 2)), i = 1, 2, · · · , n− 2 induces an isomorphism of

An and An.

Proof. By Lemma 4.11, it is easy to see that every element w of An has a normal

form w = x1j1x2j2 ...xn−2jn−2 ∈ Irr(S), ji ≤ i + 1. Here x1j1 may have 3 possibilities

1, x1, x−11 ; x2j2 4 possibilities, and generally, xiji

i + 2 possibilities, 1 ≤ i ≤ n − 2. So

|An| = n!/2. On the other hand, φ is a homomorphism and |An| = n!/2. Hence φ is an

isomorphism.

2. Chinese monoid

44

The Chinese monoid CH(X,<) over a well-ordered set (X,<) has the following

presentation CH(X) = sgp〈X|S〉, where X = xi|i ∈ I and S consists of the following

relations:

xixjxk = xixkxj = xjxixk for every i > j > k,

xixjxj = xjxixj, xixixj = xixjxi for every i > j.

Theorem 4.13 With the deg-lex ordering on X∗ the following relations (1)-(5) forms a

Grobner-Shirshov basis of the Chinese monoid CH(X).

(1) xixjxk − xjxixk,

(2) xixkxj − xjxixk,

(3) xixjxj − xjxixj,

(4) xixixj − xixjxi,

(5) xixjxixk − xixkxixj,

where xi, xj, xk ∈ X, i > j > k.

Let Λ be the set which consists of words on X of the form un = w1w2 · · ·wn, n ≥ 0,

where

w1 = xt111

w2 = (x2x1)t21xt22

2

w3 = (x3x1)t31(x3x2)

t32xt333

· · ·

wn = (xnx1)tn1(xnx2)

tn2 · · · (xnxn−1)tn(n−1)xtnn

n

with xi ∈ X, x1 < x2 < · · · < xn and all exponents non-negative.

Corollary 4.14 The set Λ is normal forms of elements of the Chinese monoid CH(X).

3. Free inverse semigroup

45

Let S be a semigroup. s ∈ S is called an inverse of t ∈ S if sts = s, tst = t. An

inverse semigroup is a semigroup in which every element t has a unique inverse, denoted

by t−1.

Let X−1 = x−1|x ∈ X with X ∩X−1 = ∅. Denote X ∪X−1 by Y . We define the

formal inverses for elements of Y ∗ by the rules

1−1 = 1, (x−1)−1 = x (x ∈ X),

(y1y2 · · · yn)−1 = y−1n · · · y−1

2 y−11 (y1, y2, · · · , yn ∈ Y ).

It is well known that

FI(X) = sgp〈Y | aa−1a = a, aa−1bb−1 = bb−1aa−1, a, b ∈ Y ∗〉

is the free inverse semigroup (with identity) generated by X.

We give definitions in Y ∗ of formal idempotent, (prime) canonical idempotent and

ordered (prime) canonical idempotent. Let < be a well ordering on Y and then deg-lex

ordering on Y ∗.

(i) The empty word 1 is an idempotent.

(ii) If h is an idempotent and x ∈ Y , then x−1hx is both an idempotent and a prime

idempotent.

(iii) If e1, e2, · · · , em (m > 1) are prime idempotents, then e = e1e2 · · · em is an idempotent.

(iv) An idempotent w ∈ Y ∗ is called canonical if w has no subword of the form x−1exfx−1,

where x ∈ Y , both e and f are idempotents.

(v) A canonical idempotent w ∈ Y ∗ is called ordered if for any subword e = e1e2 · · · em

of w, fir(e1) < fir(e2) < · · · < fir(em), where each ei is an idempotent, m > 2 and

fir(u) is the first letter of u ∈ Y ∗.

Theorem 4.15 Let S be the set of the following two kinds of polynomials in k〈Y 〉:

• ef − fe, where e, f are ordered prime canonical idempotents such that ef > fe;

• x−1e′xf ′x−1−f ′x−1e′, where x ∈ Y, x−1e′x and xf ′x−1 are ordered prime canonical

idempotents.

Then, with deg-lex ordering on Y ∗, S is a Grober-Shirshov basis for free inverse

semigroup sgp〈Y |S〉.

46

Theorem 4.16 Normal forms of elememts of the free inverse semigroup sgp〈Y |S〉 are

u0e1u1 · · · emum ∈ Y ∗,

where m ≥ 0, u1, · · · , um−1 6= 1, u0u1 · · ·um has no subword of form yy−1 for y ∈ Y ,

e1, · · · , em are ordered canonical idempotents, and the first (last, respectively) letters of

ei (1 ≤ i ≤ m) are not equal to the first (last, respectively) letter of ui (ui−1, respectively).

The above normal form (Bokut, Chen, Zhao [41]) is an analogous to “semi-normal”

forms that were given in Poliakova-Schein [172], 2005.

§4.2 Composition-Diamond lemma for modules

Let X, Y be sets and modk〈X〉〈Y 〉 a free left k〈X〉-module with the basis Y . Then

modk〈X〉〈Y 〉 = ⊕y∈Y k〈X〉y is called a “double-free” module. We now define the GS basis

in modk〈X〉〈Y 〉. Suppose that < is a monomial ordering on X∗, < a well ordering on Y

and X∗Y = uy|u ∈ X∗, y ∈ Y . We define an ordering < on X∗Y as follows: for any

w1 = u1y1, w2 = u2y2 ∈ X∗Y ,

w1 < w2 ⇔ u1 < u2 or u1 = u2, y1 < y2

Let S ⊂ modk〈X〉〈Y 〉 with each s ∈ S monic. Then we define the composition in S

only the inclusion composition which means f = ag for some a ∈ X∗, where f, g ∈ S and

(f, g)f = f − ag. If (f, g)f =∑αiaisi, where αi ∈ k, ai ∈ X∗, si ∈ S and aisi < f , then

this composition is called trivial modulo (S, f).

Theorem 4.17 (CD-lemma for modules) Let S ⊂ modk〈X〉〈Y 〉 be a non-empty set with

each s ∈ S monic and < the ordering on X∗Y as before. Then the following statements

are equivalent:

(i) S is a Grobner-Shirshov basis in modk〈X〉〈Y 〉.

(ii) If 0 6= f ∈ k〈X〉S, then f = as for some a ∈ X∗, s ∈ S.

(iii) Irr(S) = w ∈ X∗Y |w 6= as, a ∈ X∗, s ∈ S is a linear basis for the factor

modk〈X〉〈Y |S〉 = modk〈X〉〈Y 〉/k〈X〉S.

Outline of the proof. Let u ∈ X∗Y, u = uXyu, uX ∈ X∗, yu ∈ Y. Define

47

cm(u, v) = aXu = bXv, lcm(u, v) = u = dXv,

where yu = yv.

Up to the ordering of u, v, one has cm(u, v) = c · lcm(u, v).

The composition of two monic f, g ∈ modk〈X〉(Y ) is

(f, g)|lcm(f ,g) = lcm(f , g)|f 7→f − lcm(f , g)|g 7→g.

If a1s1 = a2s2 for monic s1, s2, then a1s1 − a2s2 = c · (s1, s2)lcm(s1,s2).

It gives an analogy of Lemma 4.2 for modules and (i)⇒ (ii) of Theorem 4.17.

Let S ⊂ k〈X〉 and A = k〈X|S〉. Then, for any left A-module AM , we can regard

AM as a k〈X〉-module in a natural way: for any f ∈ k〈X〉, m ∈M , fm := (f+Id(S))m.

We note that AM is an epimorphic image of some free A-module. Now, we assume

that AM = modA〈Y |T 〉 = modA〈Y 〉/AT , where T ⊂ modA〈Y 〉. Let T1 = ∑fiyi ∈

modk〈X〉〈Y 〉|∑

(fi + Id(S))yi ∈ T and R = SX∗Y ∪ T1. Then, as k〈X〉-modules, AM =

modk〈X〉〈Y |R〉.

Assume that S is a GS basis for the left ideal k〈X〉S with no compositions at all

between different elements of S. It means that S is a minimal GS basis of the left ideal

k〈X〉S. Then, k〈X〉S is a free k〈X〉-module with the basis S. This shows the Cohn

Theorem 2.3.

As an application of CD-lemma for modules, we give GS bases for the Verma modules

over a Lie algebras of coefficients of a free Lie conformal algebras. We find a linear basis

for such a module.

Let B be a set of symbols. Let the locality function N : B × B → Z+ be a constant,

i.e. N(a, b) ≡ N for any a, b ∈ B. Let X = b(n)| b ∈ B, n ∈ Z and L = Lie(X|S) be

a Lie algebra over a field k of characteristic 0 generated by X with the relation S, where

S = ∑

s

(−1)s

(N

s

)[b(n− s)a(m+ s)] = 0| a, b ∈ B, m, n ∈ Z.

For any b ∈ B, let b =∑n

b(n)z−n−1 ∈ L[[z, z−1]]. Then, it is well-known that they

generate a free Lie conformal algebra C with data (B, N) (see [189]). Moreover, the

coefficient algebra of C is just L. Let B be a linearly ordered set. Define an ordering on

X in the following way:

a(m) < b(n)⇔ m < n or (m = n and a < b).

48

We use the deg-lex ordering on X∗. Then, it is clear that without loss of generality one

may assume the leading term of each polynomial in S is b(n)a(m) such that

n−m > N or (n−m = N and (b > a or (b = a and N is odd))).

Lemma 4.18 With the deg-lex ordering on X∗, S is a GS basis in Lie(X).

Corollary 4.19 Let U = U(L) be an universal enveloping algebra of L. Then a linear

basis of U consists of monomials

a1(n1)a2(n2) · · · ak(nk), ai ∈ B, ni ∈ Z

such that for any 1 ≤ i < k,

ni − ni+1 ≤

N − 1 if ai > ai+1 or (ai = ai+1 and N is odd)

N otherwise.

Proof. Viewing U as a k〈X〉-module, we have

UU = modU〈X| S(−)〉 = modk〈X〉〈I| S(−)X∗I〉.

Since S is a GS basis in Lie(X), S(−) is a GS basis in k〈X〉 by Theorem 5.37. Therefore,

S(−)X∗I is a GS basis in the free module modk〈X〉〈I〉. Now, the result follows from

Theorem 4.17.

§4.3 Grobner-Shirshov bases for tensor product of free

algebras

Let X and Y be sets and k〈X〉⊗k〈Y 〉 be the tensor product algebra. In this section,

we will give a Composition-Diamond lemma for the algebra k〈X〉 ⊗ k〈Y 〉. We will also

prove a theorem on the pair of algebras (k[X] ⊗ k〈Y 〉, k〈X〉 ⊗ k〈Y 〉) following the spirit

of Eisenbud, Peeva and Sturmfels’ theorem [97] on (k[X], k〈X〉).

§4.3.1 Composition-Diamond lemma for tensor product

Let X and Y be two well-ordered sets, T = yx = xy|x ∈ X, y ∈ Y . With the deg-

lex ordering (y > x for any x ∈ X, y ∈ Y ) on (X ∪ Y )∗, T is clearly a Grobner-Shirshov

basis in k〈X ∪ Y 〉. Then, by Theorem 4.4,

N = X∗Y ∗ = Irr(T ) = u = uXuY |uX ∈ X∗ and uY ∈ Y ∗

49

is a set of normal words of the tensor product

k〈X〉 ⊗ k〈Y 〉 = k〈X ∪ Y | T 〉.

Let kN be a k-linear space spanned by N . For any u = uXuY , v = vXvY ∈ N , we

define the multiplication of the normal words as follows

uv = uXvXuY vY ∈ N.

It is clear that kN is exactly the tensor product algebra k〈X〉 ⊗ k〈Y 〉, that is, kN =

k〈X ∪ Y |T 〉 = k〈X〉 ⊗ k〈Y 〉.Now, we order the set N . For any u = uXuY , v = vXvY ∈ N ,

u > v ⇔ |u| > |v| or (|u| = |v| and (uX > vX or (uX = vX and uY > vY ))),

where |u| = |uX | + |uY | is the length of u. Obviously, > is a monomial ordering on N .

Such an ordering is also called the deg-lex ordering on N = X∗Y ∗. Throughout this

section, we will adopt the above ordering unless it is otherwise stated.

For any polynomial f ∈ k〈X〉 ⊗ k〈Y 〉, f has a unique presentation of the form

f = αf f +∑

αiui,

where f , ui ∈ N, f > ui, αf , αi ∈ k.The proof of the following lemma are straightforward and we hence omit the details.

Lemma 4.20 Let f ∈ k〈X〉 ⊗ k〈Y 〉 be a monic polynomial. Then ufv = ufv for any

u, v ∈ N .

We give here the definition of compositions. Let f and g be two monic polynomials

of k〈X〉 ⊗ k〈Y 〉 and w = wXwY ∈ N . Then we have the following compositions.

1. Inclusion

1.1 X-inclusion only

Suppose that wX = fX = agXb for some a, b ∈ X∗, and fY , gY are disjoint. Then

there are two compositions according to wY = fY cgY and wY = gY cfY for c ∈ Y ∗,

respectively:

(f, g)w1 = fcgY − fY cagb, w1 = fX fY cgY

and

(f, g)w2 = gY cf − agbcfY , w2 = fX gY cfY .

1.2 Y -inclusion only

50

Suppose that wY = fY = cgY d for c, d ∈ Y ∗ and fX , gX are disjoint. Then there are

two compositions according to wX = fXagX and wX = gXafX for a ∈ X∗, respectively:

(f, g)w1 = fagX − fXacgd, w1 = fXagXfY

and

(f, g)w2 = gXaf − cgdafX w2 = gXafXfY .

1.3 X, Y -inclusion

Suppose that wX = fX = agXb for some a, b ∈ X∗ and wY = fY = cgY d for some

c, d ∈ Y ∗. Then

(f, g)w = f − acgbd.

The transformation f 7→ (f, g)w = f − acgbd is said to be the elimination of leading

word (ELW) of g in f .

1.4 X, Y -skew-inclusion

Suppose that wX = fX = agXb for some a, b ∈ X∗ and wY = gY = cfY d for some

c, d ∈ Y ∗. Then

(f, g)w = cfd− agb.

2. Intersection

2.1 X-intersection only

Suppose that wX = fXa = bgX for some a, b ∈ X∗ with |fX | + |gX | > |wX |, and

fY , gY are disjoint. Then there are two compositions according to wY = fY cgY and

wY = gY cfY for c ∈ Y ∗, respectively:

(f, g)w1 = facgY − fY cbg, w1 = wX fY cgY

and

(f, g)w2 = gY cfa− bgcfY , w2 = wX gY cfY .

2.2 Y -intersection only

Suppose that wY = fY c = dgX for some c, d ∈ Y ∗ with |fY | + |gY | > |wY |, and

fX , gX are disjoint. Then there are two compositions according to wX = fXagX and

wX = gXafX for a ∈ X∗, respectively:

(f, g)w1 = fcagX − fXadg, w1 = fXagXwY

and

(f, g)w2 = gXafc− dgafX , w2 = gXafXwY .

2.3 X, Y -intersection

51

If wX = fXa = bgX for some a, b ∈ X∗ and wY = fY c = dgY for some c, d ∈ Y ∗

together with |fX |+ |gX | > |wX | and |fY |+ |gY | > |wY |, then

(f, g)w = fac− bdg.

2.4 X, Y -skew-intersection

If wX = fXa = bgX for some a, b ∈ X∗ and wY = cfY = gY d for some c, d ∈ Y ∗

together with |fX |+ |gX | > |wX | and |fY |+ |gY | > |wY |, then

(f, g)w = cfa− bgd.

3. Both inclusion and intersection

3.1 X-inclusion and Y -intersection

There are two subcases to consider.

If wX = fX = agXb for some a, b ∈ X∗ and wY = fY c = dgY for some c, d ∈ Y ∗ with

|fY |+ |gY | > |wY |, then

(f, g)w = fc− adgb.

If wX = fX = agXb for some a, b ∈ X∗ and wY = cfY = gY d for some c, d ∈ Y ∗ with

|fY |+ |gY | > |wY |, then

(f, g)w = cf − agbd.

3.2 X-intersection and Y -inclusion

There are two subcases to consider.

If wX = fXa = bgX for some a, b ∈ X∗ with |fX |+ |gX | > |wX | and wY = fY = cgY d

for some c, d ∈ Y ∗, then

(f, g)w = fa− bcgd.

If wX = fXa = bgX for some a, b ∈ X∗ with |fX |+ |gX | > |wX | and wY = cfY d = gY

for some c, d ∈ Y ∗, then

(f, g)w = cfad− bg.

From Lemma 4.20, it follows that for any case of compositions

(f, g)w < w.

If Y = ∅, then the compositions of f, g are the same as in k〈X〉.Let S be a monic subset of k〈X〉 ⊗ k〈Y 〉 and f, g ∈ S. A composition (f, g)w is said

to be trivial modulo (S,w), denoted by

(f, g)w ≡ 0 mod(S,w),

52

if (f, g)w =∑

i αiaisibi, where ai, bi ∈ N, si ∈ S, αi ∈ k and aisibi < w for any i.

We call S a Grobner-Shirshov basis in k〈X〉 ⊗ k〈Y 〉 if all compositions of elements

in S are trivial modulo S and corresponding w.

Lemma 4.21 Let S be a Grobner-Shirshov basis in k〈X〉 ⊗ k〈Y 〉 and s1, s2 ∈ S. If

w = a1s1b1 = a2s2b2 for some ai, bi ∈ N, i = 1, 2, then a1s1b1 ≡ a2s2b2 mod(S,w).

Proof. There are four cases to consider.

Case 1 Inclusion

1.1 X-inclusion only

Suppose that wX1 = s1

X = as2Xb, a, b ∈ X∗ and s1

Y , s2Y are disjoint. Then

aX2 = aX

1 a and bX2 = bbX1 . There are two subcases to be considered: wY1 = s1

Y cs2Y and

wY1 = s2

Y cs1Y , where c ∈ Y ∗.

For wY1 = s1

Y cs2Y , we have w1 = sX

1 s1Y cs2

Y , aY2 = aY

1 s1Y c, bY1 = cs2

Y bY2 , w =

a1w1bX1 b

Y2 = a1s1acs2b2 and hence

a1s1b1 − a2s2b2 = a1s1bX1 cs2

Y bY2 − aX1 aa

Y1 s1

Y cs2bbX1 b

Y2

= a1(s1cs2Y − s1

Y cas2b)bX1 b

Y2

= a1(s1, s2)w1bX1 b

Y2

≡ 0 mod(S,w).

For wY1 = s2

Y cs1Y , we have w1 = sX

1 s2Y cs1

Y , aY1 = aY

2 s2Y c, bY2 = cs1

Y bY1 , w =

aX1 a

Y2 w1b1 and hence

a1s1b1 − a2s2b2 = aX1 a

Y2 s2

Y cs1b1 − aX1 aa

Y2 s2bb

X1 cs1

Y bY1

= aX1 a

Y2 (s2

Y cs1 − as2bcs1Y )b1

= aX1 a

Y2 (s1, s2)w1b1

≡ 0 mod(S,w).

1.2 Y -inclusion only

This case is similar to 1.1.

1.3 X, Y -inclusion

We may assume that s2 is a subword of s1, i.e., w1 = s1 = acs2bd, a, b ∈ X∗, c, d ∈ Y ∗,

aX2 = aX

1 a, bX2 = bbX1 , aY

2 = aY1 c and bY2 = dbY1 . Thus, a2 = a1ac, b2 = bdb1, w = a1w1b1

53

and hence

a1s1b1 − a2s2b2 = a1s1b1 − a1acs2bdb1

= a1(s1 − acs2bd)b1

= a1(s1, s2)w1b1

≡ 0 mod(S,w).

1.4 X,Y -skew-inclusion

Assume that wX1 = s1

X = as2Xb, a, b ∈ X∗ and wY

1 = s2Y = cs1

Y d, c, d ∈ Y ∗. Then

aX2 = aX

1 a, bX2 = bbX1 , aY

1 = aY2 c and bY1 = dbY2 . Thus, w = aX

1 aY2 w1b

X1 b

Y2 and hence

a1s1b1 − a2s2b2 = aX1 a

Y2 cs1b

X1 db

Y2 − aX

1 aaY2 s2bb

X1 b

Y2

= aX1 a

Y2 (cs1d− as2b)b

X1 b

Y2

= aX1 a

Y2 (s1, s2)w1b

X1 b

Y2

≡ 0 mod(S,w).

Case 2 Intersection

2.1 X-intersection only

We may assume that s1X is at the left of s2

X , i.e., wX1 = s1

Xb = as2X , a, b ∈ X∗

and |s1X | + |s2

X | > |wX1 |. Then aX

2 = aX1 a and bX1 = bbX2 . There are two subcases to be

considered: wY1 = s1

Y cs2Y and wY

1 = s2Y cs1

Y , c ∈ Y ∗.

For wY1 = s1

Y cs2Y , i.e., w1 = s1bcs2

Y , we have aY2 = aY

1 s1Y c, bY1 = cs2

Y bY2 , w =

a1s1acs2b2 = a1w1b2 and

a1s1b1 − a2s2b2 = a1s1bbX2 cs2

Y bY2 − aX1 aa

Y1 s2

Y cs2b2

= a1(s1bcs2Y − as2

Y cs2)b2

= a1(s1, s2)w1b2

≡ 0 mod(S,w).

For wY1 = s2

Y cs1Y , i.e., w1 = s2

Y cs1b, we have aY1 = aY

2 s2Y c, bY2 = cs1

Y bY1 , w =

aX1 a

Y2 w1b

X2 b

Y1 and hence

a1s1b1 − a2s2b2 = aX1 a

Y2 s2

Y cs1bbX2 b

Y1 − aX

1 aaY2 s2b

X2 cs1

Y bY1

= aX1 a

Y2 (s2

Y cs1b− as2cs1Y )bX2 b

Y1

= aX1 a

Y2 (s1, s2)w1b

X2 b

Y1

≡ 0 mod(S,w).

2.2 Y -intersection only

54

This case is similar to 2.1.

2.3 X,Y -intersection

Assume that wX1 = s1

Xb = as2X , wY

1 = s1Y d = cs2

Y , a, b ∈ X∗, c, d ∈ Y ∗, |s1X | +

|s2X | > |wX

1 | and |s1Y | + |s2

Y | > |wY1 |. Then aX

2 = aX1 a, b

X1 = bbX2 , aY

2 = aY1 c, b

Y1 = dbY2 ,

w = a1w1b2 and hence

a1s1b1 − a2s2b2 = a1s1bbX2 db

Y2 − aX

1 aaY1 cs2b2

= a1(s1bd− acs2)b2

= a1(s1, s2)w1b2

≡ 0 mod(S,w).

2.4 X,Y -skew-intersection

Assume that wX1 = s1

Xb = as2X , wY

1 = cs1Y = s2

Y d, |s1X | + |s2

X | > |wX1 |, |s1

Y | +|s2

Y | > |wY1 |, a, b ∈ X∗, c, d ∈ Y ∗. Then aX

2 = aX1 a, b

X1 = bbX2 , aY

1 = aY2 c, b

Y2 = dbY1 ,

w = aX1 a

Y2 w1b

X2 b

Y1 and hence

a1s1b1 − a2s2b2 = aX1 a

Y2 cs1bb

X2 b

Y1 − aX

1 aaY2 s2b

X2 db

Y1

= aX1 a

Y2 (cs1b− as2d)b

X2 b

Y1

= aX1 a

Y2 (s1, s2)w1b

X2 b

Y1

≡ 0 mod(S,w).

Case 3 Both inclusion and intersection

3.1 X-inclusion and Y -intersection

We may assume that wX1 = s1

X = as2Xb, a, b ∈ X∗. Then aX

2 = aX1 a and bX2 = bbX1 .

There two cases to consider: wY1 = s1

Y d = cs2Y and wY

1 = cs1Y = s2

Y d, where c, d ∈ Y ∗,

|s1Y |+ |s2

Y | > |wY1 |.

For wY1 = s1

Y d = cs2Y , we have aY

2 = aY1 c, b

Y1 = dbY2 , w = a1w1b

X1 b

Y2 and hence

a1s1b1 − a2s2b2 = a1s1bX1 db

Y2 − aX

1 aaY1 cs2bb

X1 b

Y2

= a1(s1d− acs2b)bX1 b

Y2

= a1(s1, s2)w1b2

≡ 0 mod(S,w).

For wY1 = cs1

Y = s2Y d, we have aY

1 = aY2 c, b

Y2 = dbY1 , w = aX

1 aY2 w1b

X2 db

Y1 and hence

a1s1b1 − a2s2b2 = aX1 a

Y2 cs1b1 − aX

1 aaY2 s2bb

X1 db

Y1

= aX1 a

Y2 (cs1 − as2bd)b1

= aX1 a

Y2 (s1, s2)w1b1

≡ 0 mod(S,w).

55

3.2 X-intersection and Y -inclusion

Assume that wX1 = s1

Xb = as2X , a, b ∈ Y ∗ with |s1

X | + |s2X | > |wX

1 |. Then

aX2 = aX

1 a, bX1 = bbX2 . There are two subcases to be considered: wY

1 = s1Y = cs2

Y d and

s2Y = cs1

Y d, where c, d ∈ Y ∗.

For wY1 = s1

Y = cs2Y d, we have aY

2 = aY1 c, b

Y2 = dbY1 , w = a1w1b

X2 b

Y1 and hence

a1s1b1 − a2s2b2 = a1s1bbX2 b

Y1 − a1acs2b

X2 db

Y1

= a1(s1b− acs2d)bX2 b

Y1

= a1(s1, s2)w1bX2 b

Y1

≡ 0 mod(S,w).

For wY1 = s2

Y = cs1Y d, we have aY

1 = aY2 c, b

Y1 = dbY2 , w = aX

1 aY2 w1b2 and hence

a1s1b1 − a2s2b2 = aX1 a

Y2 cs1bb

X2 db

Y2 − aX

1 aaY2 s2b2

= aX1 a

Y2 (cs1bd− as2)b2

= aX1 a

Y2 (s1, s2)w1b2

≡ 0 mod(S,w).

Case 4. s1 and s2 are disjoint

For w = wXwY , by symmetry, there are two subcases to be considered: wY =

aY1 s1

Y cs2Y bY2 and wY = aY

2 s2Y cs1

Y bY1 , where wX = aX1 s1

Xas2XbX2 , a ∈ X∗, aX

2 =

aX1 s1

Xa, bX1 = as2XbX2 and c ∈ Y ∗.

For w = aX1 s1

Xas2XbX2 a

Y1 s1

Y cs2Y bY2 = a1s1acs2b2, we have a2 = a1s1ac, b1 = acs2b2

and hence

a1s1b1 − a2s2b2 = a1s1acs2b2 − a1s1acs2b2

= a1(s1 − s1)acs2b2 − a1s1ac(s2 − s2)b2

≡ 0 mod(S,w).

For w = aX1 s1

Xas2XbX2 a

Y2 s2

Y cs1Y bY1 , we have aY

1 = aY2 s2

Y c, bY2 = cs1Y bY1 and hence

a1s1b1 − a2s2b2 = aX1 a

Y2 s2

Y cs1as2XbX2 b

Y1 − aX

1 s1XaaY

2 s2bX2 cs1

Y bY1

= aX1 a

Y2 (s2

Y cs1as2X − s1

Xas2cs1Y )bX2 b

Y1 .

56

Let s1 =∑n

i=1 αiuX1iu

Y1i and s2 =

∑mj=1 βju

X2ju

Y2j, where α1 = β1 = 1. Then

s2Y cs1as2

X − s1Xas2cs1

Y =n∑

i=2

αiuX1ias2cu

Y1i −

m∑j=2

βiuY2jcs1au

X2j

=n∑

i=2

αiuX1ia(s2 − s2)cu

Y1i +

m∑j=2

βjuY2jc(s1 − s1)au

X2j

+n∑

i=2

αiuX1ias2cu

Y1i −

m∑j=2

βjuY2jcs1au

X2j

≡n∑

i=2

m∑j=2

αiβjuX1iau

X2ju

Y2jcu

Y1i −

m∑j=2

n∑i=2

αiβjuY2jcu

Y1iu

X1iau

X2j

≡ 0 mod(S,w1)

where w1 = s2Y cs1as2

X = s1Xas2cs1

Y . Since w = aX1 a

Y2 w1b

X2 b

Y1 , we have

a1s1b1 − a2s2b2 = aX1 a

Y2 (s2

Y cs1as2X − s1

Xas2cs1Y )bX2 b

Y1

≡ 0 mod(S,w).

This completes the proof.

Lemma 4.22 Let S ⊂ k〈X〉 ⊗ k〈Y 〉 with each s ∈ S monic and Irr(S) = w ∈ N |w 6=asb, a, b ∈ N, s ∈ S. Then for any f ∈ k〈X〉 ⊗ k〈Y 〉,

f =∑

aisibi≤f

αiaisibi +∑uj≤f

βjuj,

where each αi, βj ∈ k, ai, bi ∈ N, si ∈ S and uj ∈ Irr(S).

Proof. Let f =∑i

αiui ∈ k〈X〉 ⊗ k〈Y 〉, where 0 6= αi ∈ k and u1 > u2 > · · · . If

u1 ∈ Irr(S), then let f1 = f − α1u1. If u1 6∈ Irr(S), then there exist some s ∈ S and

a1, b1 ∈ N such that f = a1s1b1. Let f1 = f − α1a1s1b1. In both cases, we have f1 < f .

Then the result follows from induction on f .

By summing up the above lemmas, we arrive at the following theorem:

Theorem 4.23 (CD-lemma for tensor product k〈X〉 ⊗ k〈Y 〉) Let S ⊂ k〈X〉 ⊗ k〈Y 〉 with

each s ∈ S monic and < the ordering on N = X∗Y ∗ as before. Then the following

statements are equivalent.

(i) S is a Grobner-Shirshov basis in k〈X〉 ⊗ k〈Y 〉.

(ii) f ∈ Id(S)⇒ f = asb for some a, b ∈ N, s ∈ S.

57

(iii) Irr(S) = w ∈ N |w 6= asb, a, b ∈ N, s ∈ S is a k-linear basis for the factor algebra

k〈X〉 ⊗ k〈Y 〉/Id(S).

Proof. (i) ⇒ (ii). Suppose that 0 6= f ∈ Id(S). Then f =∑αiaisibi for some

αi ∈ k, ai, bi ∈ N, si ∈ S. Let wi = aisibi and w1 = w2 = · · · = wl > wl+1 ≥ · · · . We will

prove that f = asb for some a, b ∈ N, s ∈ S, by using induction on (w1, l). If l = 1, then

the result is clear. If l > 1, then w1 = a1s1b1 = a2s2b2. Now, by (i) and Lemma 4.21, we

see that a1s1b1 ≡ a2s2b2 mod(S,w1). Thus,

α1a1s1b1 + α2a2s2b2 = (α1 + α2)a1s1b1 + α2(a2s2b2 − a1s1b1)

≡ (α1 + α2)a1s1b1 mod(S,w1).

Now (ii) follows.

(ii)⇒ (iii). For any 0 6= f ∈ k〈X〉 ⊗ k〈Y 〉, by Lemma 4.22, we can express f as

f =∑

αiaisibi +∑

βjuj,

where αi, βj ∈ k, ai, bi ∈ N, si ∈ S and uj ∈ Irr(S). Then Irr(S) generates the factor

algebra. Moreover, if 0 6= h =∑βjuj ∈ Id(S), uj ∈ Irr(S), u1 > u2 > · · · and β1 6= 0,

then u1 = h = asb for some a, b ∈ N, s ∈ S by (ii), a contradiction. This shows that

Irr(S) is a linear basis of the factor algebra.

(iii)⇒ (i). For any f, g ∈ S, we have h = (f, g)w ∈ Id(S). The result is now trivial

if (f, g)w = 0. Assume that (f, g)w 6= 0. Then, by Lemma 4.22 and (iii), we have

h =∑

aisibi≤h

αiaisibi.

Now, by noting that h = (f, g)w < w, we see immediately that (i) holds.

Remark: Theorem 4.23 is valid for any monomial ordering on X∗Y ∗.

Remark: Theorem 4.23 is precisely the Composition-Diamond lemma for associative

algebras (Theorem 4.4) when Y = ∅.

§4.3.2 Applications

Now, we give some applications of Theorem 4.23.

Example 4.24 Suppose that for the deg-lex ordering, S1 and S2 are GS bases in k〈X〉 and

k〈Y 〉 respectively. Then for the deg-lex ordering on X∗Y ∗ as before, S1 ∪S2 is a GS basis

in k〈X ∪Y |T 〉 = k〈X〉⊗k〈Y 〉. It follows that k〈X|S1〉⊗k〈Y |S2〉 = k〈X ∪Y |T ∪S1∪S2〉.

58

Proof. The possible compositions in S1 ∪ S2 are X-including only, X-intersection

only, Y -including only and Y -intersection only. Suppose that f, g ∈ S1 and (f, g)w1 ≡0 mod(S1, w1) in k〈X〉. Then in k〈X〉⊗k〈Y 〉, (f, g)w = (f, g)w1c, where w = w1c for any

c ∈ Y ∗. From this, it follows that each composition in S1 ∪ S2 is trivial modulo S1 ∪ S2.

A special case of Example 4.24 is the following.

Example 4.25 Let X,Y be well-ordered sets, k[X] the free commutative associative al-

gebra generated by X. Then S = xixj = xjxi|xi > xj, xi, xj ∈ X is a GS basis in

k〈X〉⊗k〈Y 〉 with respect to the deg-lex ordering. Therefore, k[X]⊗k〈Y 〉 = k〈X∪Y |T∪S〉.

In [97], a GS basis in k〈X〉 is constructed by lifting a commutative Grobner basis and

adding some commutators. Let X = x1, x2, . . . , xn, [X] the free commutative monoid

generated by X and k[X] the polynomial ring. Let

S1 = hij = xixj − xjxi| i > j ⊂ k〈X〉.

Then, consider the natural map γ : k〈X〉 → k[X] which maps xi to xi and the lexicographic

splitting of γ, which is defined as the k-linear map

δ : k[X]→ k〈X〉, xi1xi2 · · ·xir 7→ xi1xi2 · · ·xir if i1 ≤ i2 · · · ≤ ir.

For any u ∈ [X], we present u = xl11 x

l22 · · ·xln

n , where li ≥ 0.

We use any monomial ordering on [X].

Following [97], we define an ordering on X∗ using the ordering x1 < x2 < · · · < xn

as follows: for any u, v ∈ X∗,

u > v ⇔ γ(u) > γ(v) in [X] or (γ(u) = γ(v) and u >lex v).

It is easy to check that this ordering is monomial on X∗ and δ(s) = δ(s) for any s ∈ k[X].

Moreover, for any v ∈ γ−1(u), v ≥ δ(u).

For any m = xi1xi2 · · ·xir ∈ [X], i1 ≤ i2 · · · ≤ ir, denote the set of all the monomials

u ∈ [xi1+1, · · · , xir−1] by U(m).

The proofs of the following lemmas are straightforward.

Lemma 4.26 Let a, b ∈ X∗, a = δ(γ(a)), b = δ(γ(b)) and s ∈ k[X]. If w = aδ(s)b =

δ(γ(ab)s), then, in k〈X〉,

aδ(s)b ≡ δ(γ(ab)s) mod(S1, w).

59

Proof. Suppose that s = s + s′ and h = aδ(s)b − δ(γ(ab)s). Since aδ(s)b = δ(γ(ab)s),

we have h = aδ(s′)b− δ(γ(ab)s′), and h < w. By noting that γ(aδ(s′)b) = γ(δ(γ(ab)s′)),

h ≡ 0 mod(S1, w).

Lemma 4.27 Let f, g ∈ k[X], g = xi1xi2 · · ·xir (i1 ≤ i2 ≤ · · · ≤ ir) and w = δ(f g).

Then, in k〈X〉,

δ((f − f)g) ≡∑

αiaiδ(uig)bi mod(S1, w)

where αi ∈ k, ai ∈ [x ∈ X|x ≤ xi1 ], bi ∈ [x ∈ X|x ≥ xir ], ui ∈ U(g) and γ(∑αiaiuibi) =

f − f .

Theorem 4.28 Let the orderings on [X] and X∗ be defined as above. If S is a minimal

Grobner basis in k[X], then S ′ = δ(us)|s ∈ S, u ∈ U(s) ∪ S1 is a Grobner-Shirshov

basis in k〈X〉.

Proof. We will show that all the possible compositions of elements in S ′ are trivial. Let

f = δ(us1), g = δ(vs2) and hij = xixj − xjxi ∈ S ′.

(i) f ∧ g

Case 1. f and g have a composition of including, i.e., w = δ(us1) = aδ(vs2)b for

some a, b ∈ X∗ and a = δ(γ(a)), b = δ(γ(b)).

If s1 and s2 have no composition in k[X], i.e., lcm(s1s2) = s1s2, then u = u′s2, γ(ab)v =

u′s1 for some u′ ∈ [X]. By Lemma 4.26 and Lemma 4.27, we have

(f, g)w = δ(us1)− aδ(vs2)b

≡ δ(us1)− δ(γ(ab)vs2)

≡ δ(u′s2s1)− δ(u′s1s2)

≡ δ(u′(s1 − s1)s2)− δ(u′(s2 − s2)s1)

≡ 0 mod(S ′, w).

Since, in k[X], S is a minimal Grobner basis, the possible compositions are only

intersection. If s1 and s2 have composition of intersection in k[X], i.e., (s1, s2)w′ =

a′s1− b′s2, where a′, b′ ∈ [X], w′ = a′s1 = b′s2 and |w′| < |s1|+ |s2|, then w′ is a subword

of γ(w). Hence, we deduce that w = δ(tw′) = δ(ta′s1) = δ(tb′s2) and u = ta′, γ(ab)v = tb′

60

for some t ∈ [X]. Then

(f, g)w = δ(us1)− aδ(vs2)b

≡ δ(us1)− δ(γ(ab)vs2)

≡ δ(ta′s1)− δ(tb′s2)

≡ δ(t(a′s1 − b′s2))

≡ δ(t(s1, s2)w′)

≡ 0 mod(S ′, w)

since t(s1, s2)w′ < tw′ = γ(w).

Case 2. If f and g have a composition of intersection, we may assume that f is on

the left of g, i.e., w = δ(us1)a = bδ(vs2) for some a, b ∈ X∗ and a = δγ(a), b = δγ(b).

Similarly to Case 1, we have to consider whether s1 and s2 have compositions in k[X] or

not? One can check that both cases are trivial mod(S ′, w) by Lemma 4.26 and Lemma

4.27.

(ii) f ∧ hij

By noting that hij = xixj can not be a subword of f = δ(us1) since i > j, only

possible compositions are intersection. Suppose that s1 = xi1 · · ·xirxi, (i1 ≤ i2 ≤ · · · ≤ir ≤ i). Then f = δ(us1) = xi1vxi for some v ∈ k〈X〉, v = δγ(v) and w = δ(us1)xj.

If j ≤ i1, then

(f, hij)w = δ(us1)xj − xi1v(xixj − xjxi)

= δ(u(s1 − s1))xj + xi1vxjxi

≡ xjδ(u(s1 − s1)) + xjxi1vxi

≡ xj(δ(u(s1 − s1)) + δ(us1))

≡ xjδ(us1)

≡ 0 mod(S ′, w).

If j > i1, then uxj ∈ U(s1) and

(f, hij)w = δ(us1)xj − xi1v(xixj − xjxi)

= δ(u(s1 − s1))xj + xi1vxjxi

≡ δ(uxj(s1 − s1)) + δ(xi1vxixj)

≡ δ(uxj(s1 − s1)) + δ(uxj s1)

≡ δ(uxjs1)

≡ 0 mod(S ′, w).

61

Thus, the proof is completed.

Now we extend γ and δ as follows.

γ ⊗ 1 : k〈X〉 ⊗ k〈Y 〉 → k[X]⊗ k〈Y 〉, uXuY 7→ γ(uX)uY ,

δ ⊗ 1 : k[X]⊗ k〈Y 〉 → k〈X〉 ⊗ k〈Y 〉, uXuY 7→ δ(uX)uY .

Any polynomial f ∈ k[X] ⊗ k〈Y 〉 has a presentation f =∑αiu

Xi u

Yi , where αi ∈

k, uXi ∈ [X] and uY

i ∈ Y ∗.

Let the orderings on [X] and Y ∗ be any monomial orderings respectively. We order

the set [X]Y ∗ = u = uXuY |uX ∈ [X], uY ∈ Y ∗ as follows. For any u, v ∈ [X]Y ∗,

u > v ⇔ uY > vY or (uY = vY and uX > vX).

Now, we order X∗Y ∗: for any u, v ∈ X∗Y ∗,

u > v ⇔ γ(uX)uY > γ(vX)vY or (γ(uX)uY = γ(vX)vY and uX >lex vX).

This ordering is clearly a monomial ordering on X∗Y ∗.

The following definitions of compositions and GS bases are taken from [165].

Let f, g be monic polynomials of k[X] ⊗ k〈Y 〉, L the least common multiple of fX

and gX .

1. Inclusion

Let gY be a subword of fY , say, fY = cgY d for some c, d ∈ Y ∗. If fY = gY then

fX ≥ gX and if gY = 1 then we set c = 1. Let w = LfY = LcgY d. We define the

composition

C1(f, g, c)w =L

fXf − L

gXcgd.

2. Overlap

Let a non-empty beginning of gY be a non-empty ending of fY , say, fY = cc0, gY =

c0d, fY d = cgY for some c, d, c0 ∈ Y ∗ and c0 6= 1. Let w = LfY d = LcgY . We define the

composition

C2(f, g, c0)w =L

fXfd− L

gXcg.

3. External

Let c0 ∈ Y ∗ be any associative word (possibly empty). In the case that the greatest

common divisor of fX and gX is non-empty and fY , gY are non-empty, we define the

composition

C3(f, g, c0)w =L

fXfc0g

Y − L

gXfY c0g,

62

where w = LfY c0gY .

Let S be a monic subset of k[X]⊗k〈Y 〉. Then S is called a GS basis (standard basis)

if for any element f ∈ Id(S), f contains s as its subword for some s ∈ S.

It is defined as usual that a composition is trivial modulo S and corresponding w. We

also have that S is a GS basis in k[X]⊗ k〈Y 〉 if and only if all the possible compositions

of its elements are trivial. A GS basis in k[X] ⊗ k〈Y 〉 is called minimal if for any s ∈ Sand all si ∈ S \ s, si is not a subword of s.

Similar to the proof of Theorem 4.28, we have the following theorem.

Theorem 4.29 Let the orderings on [X]Y ∗ and X∗Y ∗ be defined as before. If S is a

minimal GS basis in k[X] ⊗ k〈Y 〉, then S ′ = δ ⊗ 1(us)|s ∈ S, u ∈ U(sX) ∪ S1 is a GS

basis in k〈X〉 ⊗ k〈Y 〉, where S1 = hij = xixj − xjxi| i > j.

Proof. We will show that all the possible compositions of elements in S ′ are trivial.

For s1, s2 ∈ S, let f = δ ⊗ 1(us1), g = δ ⊗ 1(vs2), hij = xixj − xjxi ∈ S ′ and

L = lcm(s1X , s2

X).

1. f ∧ gIn this case, all the possible compositions are related to the ambiguities w’s (in the

following, a, b ∈ X∗, c, d ∈ Y ∗).

1.1 X-inclusion only

wX = δ(us1X) = aδ(vs2

X)b, wY = s1Y cs2

Y or wY = s2Y cs1

Y .

1.2 Y -inclusion only

wX = δ(us1X)aδ(vs2

X) or wX = δ(vs2X)aδ(us1

X), wY = s1Y = cs2

Y d.

1.3 X, Y -inclusion

w = δ ⊗ 1(us1) = acδ ⊗ 1(vs2)bd.

1.4 X, Y -skew-inclusion

wX = δ(us1X) = aδ(vs2

X)b, wY = s2Y = cs1

Y d.

2.1 X-intersection only

wX = δ(us1X)a = bδ(vs2

X), wY = s1Y cs2

Y or wY = s2Y cs1

Y .

2.2 Y -intersection only

wX = δ(us1X)aδ(vs2

X) or wX = δ(us2X)aδ(vs1

X), wY = s1Y c = ds2

Y .

2.3 X, Y -intersection

wX = δ(us1X)a = bδ(vs2

X), wY = s1Y c = ds2

Y .

2.4 X, Y -skew-intersection

wX = δ(us1X)a = bδ(vs2

X), wY = cs1Y = s2

Y d.

3.1 X-inclusion and Y -intersection

63

wX = δ(us1X) = aδ(vs2

X)b, wY = s1Y c = ds2

Y or wY = cs1Y = s2

Y d.

3.2 X-intersection and Y -inclusion

wX = δ(us1X)a = bδ(vs2

X), wY = s1Y = cs2

Y d or wY = s2Y = cs1

Y d.

We only check the cases of 1.1, 1.2 and 1.3. Other cases are similarly checked.

1.1 X-inclusion only

Suppose that wX = δ(us1X) = aδ(vs2

X)b, a, b ∈ X∗ and s1Y , s2

Y are disjoint. There

are two cases to consider: wY = s1Y cs2

Y and wY = s2Y cs1

Y , where c ∈ Y ∗. We will only

prove the first case and the second is similar.

If s1 and s2 have no composition in k[X] ⊗ k〈Y 〉, i.e., lcm(s1, s2) = s1s2, then

u = u′sX2 , γ(ab)v = u′sX

1 for some u′ ∈ [X]. By the proof of Theorem 4.28, we have

(f, g)w = δ ⊗ 1(us1)cs2Y − s1

Y caδ ⊗ 1(vs2)b

≡ δ ⊗ 1(us1γ(cs2Y ))− δ ⊗ 1(γ(s1

Y c)γ(ab)vs2)

≡ δ ⊗ 1(u′s2Xs1cs2

Y )− δ ⊗ 1(s1Y cu′s1

Xs2)

≡ δ ⊗ 1(u′s1cs2)− δ ⊗ 1(u′s1cs2)

≡ δ ⊗ 1(u′(s1 − s1)cs2)− δ ⊗ 1(u′s1c(s2 − s2))

≡ 0 mod(S ′, w).

If s1 and s2 have composition of external (the elements of S have no composition of

inclusion because S is minimal and s1 and s2 have no composition of overlap because sY1

and sY2 are disjoint ) in k[X]⊗ k〈Y 〉, i.e., C3(s1, s2, c)w′ = L

s1X s1γ(cs2

Y )− Ls2

X γ(s1Y c)s2 =

t2s1γ(cs2Y ) − t1γ(s1

Y c)s2 where gcd(s1X , s2

X) = t 6= 1, s1X = tt1, s2

X = tt2 and L =

tt1t2, w′ = Lγ(s1

Y cs2Y ), then w′ is a subword of γ(w). Therefore, we have w = δ⊗1(mw′)

and u = mt2, γ(ab)v = mt1 since ut1 = γ(ab)vt2 and gcd(t1, t2) = 1. Then

(f, g)w = δ ⊗ 1(us1)cs2Y − s1

Y caδ ⊗ 1(vs2)b

≡ δ ⊗ 1(us1γ(cs2Y ))− δ ⊗ 1(γ(s1

Y c)γ(ab)vs2)

≡ δ ⊗ 1(mt2s1γ(cs2Y ))− δ ⊗ 1(mt1γ(s1

Y c)s2)

≡ δ ⊗ 1(mC3(s1, s2, c)w′)

≡ 0 mod(S ′, w)

since mC3(s1, s2, c)w′ < mw′ = γ(w).

1.2 Y -inclusion only

Suppose that wY = s1Y = cs2

Y d, c, d ∈ Y ∗ and δ(us1X), δ(vs2

X) are disjoint. Then

there are two compositions according to wX = δ(us1X)aδ(vs2

X) and wX = δ(vs2X)aδ(us1

X)

64

for a ∈ X∗. We only prove the first.

(f, g)w = δ ⊗ 1(us1)aδ(vs2X)− δ(us1

X)acδ ⊗ 1(vs2)d

≡ δ ⊗ 1(us1γ(a)vs2X − us1

Xγ(a)vγ(c)s2γ(d))

≡ δ ⊗ 1(uγ(a)v(s1s2X − s1

Xγ(c)s2γ(d)))

≡ δ ⊗ 1(uγ(a)vC1(s1, s2, γ(c))w′)

≡ 0 mod(S ′, w),

where w′ = s1X s2

X s1Y = s1

X s2Xγ(c)s2

Y γ(d) and uγ(a)vC1(s1, s2, γ(c))w′ < uγ(a)vw′ =

γ(w).

1.3 X,Y -inclusion

We may assume that g is a subword of f , i.e., w = δ⊗1(us1) = acδ⊗1(vs2)bd, a, b ∈X∗, c, d ∈ Y ∗. Then us1

X = γ(ab)vs2X = mL for some m ∈ [X], us1

Y = γ(c)s2Y γ(d).

(f, g)w = δ ⊗ 1(us1)− acδ ⊗ 1(vs2)bd

≡ δ ⊗ 1(us1 − γ(ac)vs2γ(bd))

≡ δ ⊗ 1(mL

s1Xs1 −m

L

s2Xγ(c)s2γ(d))

≡ δ ⊗ 1(mC1(s1, s2, γ(c))w′)

≡ 0 mod(S ′, w),

where w′ = Lγ(c)s2Y γ(d) and mC1(s1, s2, c)w′ < mw′ = γ(w).

2. f ∧ hij

Similar to the proof of Theorem 4.28, they only have compositions of X-intersection.

Suppose that s1X = xi1 · · ·xirxi, (i1 ≤ i2 ≤ · · · ≤ ir ≤ i). Then f = δ ⊗ 1(us1) =

xi1vxis1Y for some v ∈ k〈X〉, and v = δγ(v) and w = δ ⊗ 1(us1)xj s1

Y .

If j ≤ i1, then

(f, hij)w = δ ⊗ 1(us1)xj − xi1vs1Y (xixj − xjxi)

= δ ⊗ 1(u(s1 − s1))xj + xi1vxjxis1Y

≡ xjδ ⊗ 1(u(s1 − s1)) + xjxi1vxis1Y

≡ xj(δ ⊗ 1(u(s1 − s1)) + δ ⊗ 1(us1))

≡ xjδ ⊗ 1(us1)

≡ 0 mod(S ′, w).

65

If j > i1, then uxj ∈ U(s1) and

(f, hij)w = δ ⊗ 1(us1)xj − xi1vs1Y (xixj − xjxi)

= δ ⊗ 1(u(s1 − s1))xj + xi1vxjxis1Y

≡ δ ⊗ 1(uxj(s1 − s1)) + δ ⊗ 1(xi1vxixj s1Y )

≡ δ ⊗ 1(uxj(s1 − s1)) + δ ⊗ 1(uxj s1)

≡ δ ⊗ 1(uxjs1)

≡ 0 mod(S ′, w).

This completes the proof.

As an application of the above result, we have now constructed a GS basis for the

tensor product k〈X〉⊗k〈Y 〉 by lifting a given GS basis in the tensor product k[X]⊗k〈Y 〉in which k[X] is a commutative algebra.

66

Chapter 5 Grobner-Shirshov Bases for Lie Algebras

over a Field

§5.1 Lyndon-Shirshov words

We start with the Lyndon-Shirshov associative words.

Let X = xi|i ∈ I be a well-ordered set with xi > xp if i > p for any i, p ∈ I. Let

X∗ be the free monoid generated by X. For u = xi1xi2 · · ·xik ∈ X∗, let

xβ = min(u) = minxi1 , xi2 , · · · , xik,

fir(u) = xi1 ,

length of u : |u| = k.

Definition 5.1 Let u = xi1xi2 · · ·xik ∈ X∗. Then u is called Weak-ALSW if fir(u) >

min(u) or |u| = 1, where ALSW means an “associative Lyndon-Shirshov word”.

Let u be a Weak-ALSW, min(u) = xβ and |u| ≥ 2. We define

X ′(u) = xji = xi xβ · · ·xβ︸ ︷︷ ︸

j

|i > β, j ≥ 0.

Note that xji = xi xβ · · ·xβ︸ ︷︷ ︸

j

is just a symbol.

Now, we order X ′(u) by the following way:

xj1i1> xj2

i2⇔ i1 > i2 or (i1 = i2, j2 > j1).

Suppose that u, v are Weak-ALSW’s and min(v) ≥ min(u) = xβ. Then we define

v′u = xm1i1· · ·xmt

itin (X ′(u))∗ ⇔ v = xi1 xβ · · ·xβ︸ ︷︷ ︸

m1

· · ·xit xβ · · ·xβ︸ ︷︷ ︸mt

in X∗,

where xij > xβ, mj ∈ N , 1 ≤ j ≤ t. For the sake of simpler notation, we use u′ instead

of u′u.

Throughout this section and next section, we assume that x1 < x2 < x3 < · · · .

Example 5.2 Let u = x2x1, v = x3x2. Then v′u = x03x

02, v′ = x1

3.

The following lemma is obvious.

67

Lemma 5.3 Let u be a Weak-ALSW, xβ = min(u), u = vw, v, w 6= 1 and w 6= xβw1.

Then u′ = v′uw′u.

Example 5.4 Let v = x3x2x1, w = x2x2 and u = vw = x3x2x1x2x2. Then u′ = x03x

12x

02x

02,

v′u = x03x

12, w

′u = x0

2x02 and u′ = v′uw

′u.

Recall that without specific explanation, we always use the lexicographic ordering

both on (X ′(u))∗ and X∗ (i.e., w > wt if t 6= 1 and zxit1 > zxjt2 if xi > xj).

Lemma 5.5 Let xβ = min(uv), fir(u) 6= xβ and fir(v) 6= xβ. Then u > v ⇔ u′uv > v′uv.

Proof. Assume that u > v. Then there are two cases to consider.

Case 1:

u = xi1 xβ · · ·xβ︸ ︷︷ ︸l1

· · ·xis−1 xβ · · ·xβ︸ ︷︷ ︸ls−1

xis xβ · · ·xβ︸ ︷︷ ︸ls

· · ·

v = xi1 xβ · · ·xβ︸ ︷︷ ︸l1

· · ·xis−1 xβ · · ·xβ︸ ︷︷ ︸ls−1

yz · · · , where xis > y.

(a) If y = xβ, then

u′uv = xl1i1· · ·xls−2

is−2x

ls−1

is−1xls

is· · ·

v′uv = xl1i1· · ·xls−2

is−2x

l′s−1

is−1· · · , where l′s−1 > ls1 .

So, u′uv > v′uv.

(b) If y > xβ, then

u′uv = xl1i1· · ·xls−1

is−1xls

is· · ·

v′uv = xl1i1· · ·xls−1

is−1yn · · · .

So, u′uv > v′uv.

Case 2:

u = xi1 xβ · · ·xβ︸ ︷︷ ︸l1

· · ·xis xβ · · ·xβ︸ ︷︷ ︸ls

v = xi1 xβ · · ·xβ︸ ︷︷ ︸l1

· · ·xis xβ · · ·xβ︸ ︷︷ ︸ls

yz · · · .

(a) If y = xβ, then

u′uv = xl1i1· · ·xls

is

v′uv = xl1i1· · ·xl′s

is· · · , where l′s > ls.

So, u′uv > v′uv.

68

(b) If y > xβ, then v′uv = u′uvyn · · · and so, u′uv > v′uv.

Conversely, assume that u′uv > v′uv. We will prove that u > v. There are also two

cases to consider.

Case 1: u′uv = xl1i1· · ·xls

is, v′uv = xl1

i1· · ·xls

isyn · · · .

Case 2: u′uv = xl1i1· · ·xls−1

is−1xls

is· · · , v′uv = xl1

i1· · ·xls−1

is−1xi′s

ls′ · · · , where xis > xi′s or

(xis = xi′s and l′s > ls).

In both cases, it is clear that u > v.

Definition 5.6 Let u ∈ X∗. Then u is called an ALSW if

(∀v, w ∈ X∗, v, w 6= 1) u = vw ⇒ vw > wv.

Remark: Let u, v ∈ X∗ and the v′u ∈ (X ′(u))∗ be as before. We denote by |v| the length

of v in X∗ and |v′u|X′ the length of v′u in (X ′(u))∗.

Lemma 5.7 Let u be a Weak-ALSW with |u| ≥ 2. Then u is an ALSW in X∗ if and

only if u′ is an ALSW in (X ′(u))∗.

Proof. “ =⇒ ” If |u′|X′ = 1, then u′ is an ALSW. Suppose that |u′|X′ > 1 and

u′ = v′uw′u. Then u = vw. Since u is an ALSW, vw > wv which implies (vw)′u > (wv)′u

by Lemma 5.5. Therefore, by Lemma 5.3, v′uw′u > w′

uv′u and so, u′ is an ALSW.

“ ⇐= ” Let u = vw and xβ = min(u). If fir(w) = xβ, then vw > wv. If

fir(w) 6= xβ, then

u′ = v′uw′u ⇒ v′uw

′u > w′

uv′u ⇒ (vw)′u > (wv)′u ⇒ vw > wv.

Hence, u is an ALSW.

Remark For a Weak-ALSW u, it is clear that |u′|X′ < |u| if |u| > 1. For an ALSW u,

we denote by u′′ = (u′)′ and u(k) = (u′)(k−1) for k > 0 generally. From this, it follows that

Xk(u) = Xk−1(u′).

Lemma 5.8 For u ∈ X∗, u is an ALSW if and only if (∃k ≥ 0), s.t., |u(k)|Xk(u) = 1.

Proof. We apply induction on |u|. If |u| = 1, then there is nothing to do. Assume that

|u| > 1. Since |u′|X′ < |u| and

|u(k)|Xk(u) = |(u′)(k−1)|Xk−1(u′),

by induction and by Lemma 5.7, the result follows.

69

Example 5.9 Let u = x5x4x5x3. Then

u′ = x05x

04x

15, u

′′ = (x05)

1(x15)

0 and u′′′ = ((x05)

1)1.

Therefore, by Lemma 5.8, u is an ALSW.

Lemma 5.10 Let u ∈ X∗. Then u is an ALSW if and only if

(∀v, w ∈ X∗, v, w 6= 1) u = vw ⇒ u > w.

Proof. “ =⇒ ” Induction on |u|. If |u| = 1, then the result clearly holds. Suppose that

|u| ≥ 2, xβ = min(u) and u = vw, v, w 6= 1. If w = xβw1, then u > w. If w 6= xβw1, then

u′ = v′uw′u. Since u′ is an ALSW, by induction, u′ > w′

u. Hence, by Lemma 5.5, u > w.

“⇐= ” Since u = vw > w > wv, we have u is an ALSW.

Lemma 5.11 Suppose that u is an ALSW, xβ = min(u) and |u| > 1. Then uxβ is an

ALSW.

Proof. Follows from Lemma 5.10.

Lemma 5.12 Let u and v be ALSW’s. Then uv is an ALSW if and only if u > v.

Proof. “ =⇒ ” Suppose that uv is an ALSW. Then, by Lemma 5.10, u > uv > v.

“ ⇐= ” We use induction on |uv|. Suppose that u > v. If |uv| = 2 or v = xβ =

min(uv), then the result is obvious. Otherwise, we can get that u′uv > v′uv, where u′uv, v′uv

are ALSW’s. By induction, u′uvv′uv = (uv)′ is an ALSW and so is uv.

Lemma 5.13 For any u ∈ X∗, there exists a unique decomposition u = u1u2 · · ·uk, where

ui is an ALSW, 1 ≤ i ≤ k, and u1 ≤ u2 ≤ · · · ≤ uk.

Proof. To prove the existence, we use induction on |u|. If |u| = 1 then it is trivial.

Let |u| > 1 and xβ = min(u). If u = xβv, then v has the required decomposition and so

does u. Otherwise, u is a Weak-ALSW. Thus, u′ has the decomposition and so does u,

by Lemmas 5.5 and 5.7.

To prove the uniqueness, we let u = u1 · · ·uk = w1 · · ·ws be the decompositions such

that ui, wj are ALSW’s for any i, j; u1 ≤ · · · ≤ uk and w1 ≤ · · · ≤ ws. If u = xβv,

then u1 = w1 = xβ and the result follows from the induction on |u|. Otherwise, u is a

Weak-ALSW and u′ = u′1u · · ·u′ku = w′1u · · ·w′

su are the decompositions of u′. Now, by

induction again, the result follows.

Remark: In Lemma 5.13, the word uk is the longest ALSW end of u.

70

Example 5.14 Let u = x1x1x2x1x2x1x1. Then

u = x1︸︷︷︸u1

x1︸︷︷︸u2

x2x1x2x1x1︸ ︷︷ ︸u3

= u1u2u3

is the decomposition of u.

Lemma 5.15 Let u be an ALSW and |u| ≥ 2. If u = vw, where w is the longest ALSW

proper end of u, then v is an ALSW.

Proof. Suppose that v is not an ALSW. Then, by Lemma 5.13, we can assume that

v = v1v2 · · · vm (m > 1),

where each vi is an ALSW and v1 ≤ v2 ≤ · · · ≤ vm. If vm > w, then vmw is an ALSW

and |vmw| > |w|, a contradiction. If vm ≤ w, then we get another decomposition of u

which contradicts the uniqueness in Lemma 5.13. Thus, v must be an ALSW.

Example 5.16 Let u = x5x4x5x4x3x5x3. Then

u = x5x4︸︷︷︸v

x5x4x5x3︸ ︷︷ ︸w

= vw

and u, v, w are all ALSW’s.

Now, for an ALSW u, we introduce two bracketing ways.

One is up-to-down bracketing which is defined inductively by

[xi] = xi, [u] = [[v][w]],

where u = vw and w is the longest ALSW proper end of u.

Example 5.17 Let u = x2x2x1x1x2x1. Then

u → [[x2x2x1x1][x2x1]]→ [[x2[x2x1x1]][x2x1]]→ [[x2[[x2x1]x1]][x2x1]].

The other is down-to-up bracketing. Let us explain it on a sample word

u = x2x2x1x1x2x1.

Join the minimal letter x1 to the previous letters:

u 7→ x2[x2x1]x1[x2x1].

71

Form a new alphabet of the nonassociative words x2, [x2x1] and x1 ordered lexicograph-

ically, i.e.,

x2 > [x2x1] > x1.

Join the minimal letter x1 to the previous letters:

x2[x2x1]x1[x2x1] 7→ x2[[x2x1]x1][x2x1].

Form a new alphabet

x2 > [x2x1] > [[x2x1]x1].

Join the minimal letter [[x2x1]x1] to the previous letter:

x2[[x2x1]x1][x2x1] 7→ [x2[[x2x1]x1]][x2x1].

Form a new alphabet

[x2[[x2x1]x1]] > [x2x1].

Finally, join the minimal letter [x2x1] to the previous letter:

[x2[[x2x1]x1]][x2x1] 7→ [[x2[[x2x1]x1]][x2x1]] = [u].

Remark: We denote by [ ] the down-to-up bracketing and by [[ ]] the up-to-down brack-

eting. Clearly, if u = xixβ · · ·xβ, then [[u]] = [u].

Lemma 5.18 Let u be an ALSW with xβ = min(u). Then

(i) [[u′]] |xji 7→[[xixβ ]···xβ ]= [[u]].

(ii) [u′] |xji 7→[[xixβ ]···xβ ]= [u].

Proof. Induction on |u′|. If |u′| = 1, then u = xixβ · · ·xβ and the results hold. Assume

that |u′| > 1.

Let u′ = v′uw′u, where w′

u is the longest ALSW proper end of u′. Then u = vw and w

is the longest ALSW proper end of u. By induction, we can get

[[u′]] |xji 7→[[xixβ ]···xβ ]= [[[v′u]][[w

′u]]] |xj

i 7→[[xixβ ]···xβ ]= [[[v]][[w]]] = [[u]].

This shows (i).

Now we prove (ii). Since |u′| > 1, there exists ALSW’s v′u, w′u such that v′u > w′

u and

[u′] = [[v′u][w′u]]. By Lemma 5.5, we have v > w and [u] = [[v][w]]. So

[u′] |xji 7→[[xixβ ]···xβ ]= [[v′u][w

′u]] |xj

i 7→[[xixβ ]···xβ ]= [[v][w]] = [u].

72

Lemma 5.19 [u] = [[u]] for any ALSW u.

Proof. Induction on |u|. If |u| = 1, then u = xi and [xi] = [[xi]] = xi. Assume that

|u| > 1. Since |u′|X′ < |u|, by induction, we can get [u′] = [[u′]]. By Lemma 5.18,

[u] = [u′] |xji 7→[[xixβ ]···xβ ]= [[u′]] |xj

i 7→[[xixβ ]···xβ ]= [[u]].

§5.2 Free Lie algebras

Now we give the definition of a non-associative Lyndon-Shirshov word.

Definition 5.20 Let < be the ordering on X∗ as before and (u) a non-associative word.

Then (u) is called a non-associative Lyndon-Shirshov word, denoted by NLSW, if

(i) u is an ALSW,

(ii) if (u) = ((v)(w)), then both (v) and (w) are NLSW’s,

(iii) in (ii) if (v) = ((v1)(v2)), then v2 ≤ w in X∗.

Remark In Definition 5.20 (ii), v > w by Lemma 5.12.

Theorem 5.21 Let u be an ALSW. Then there exists a unique bracketing way such that

(u) is a NLSW.

Proof. (Existence). Let u be an ALSW. We will prove that up-to-down bracketing is

one of bracketing way such that [[u]] is a NLSW. Induction on |u|. If |u| = 1, then nothing

to do. Suppose that |u| > 1 and u = vw where w is the longest ALSW proper end of u.

Then, [[u]] = [[[v]][[w]]]. By induction, both [[v]] and [[w]] are NLSW’s. Now, we assume

that [[v]] = [[v1]][[v2]] and v2 > w. Then, v2w is an ALSW, a contradiction. So, v2 ≤ w

and hence, [[u]] is a NLSW.

(Uniqueness). Assume that ( ) is a bracketing way such that (u) is a NLSW. We use

induction on |u| to show (u) = [[u]]. If |u| = 1, then (u) = [[u]] clearly. Suppose that

|u| > 1. Since |u′|X′ < |u|, (u′) = [[u′]] by induction. By substitution and Lemma 5.18,

(u′) |xji 7→[[xixβ ]···xβ ]= [[u′]] |xj

i 7→[[xixβ ]···xβ ], i.e., (u) = [[u]].

73

Let X∗∗ be the set of all non-associative words (u) in X. If (u) is a NLSW, then we

denote it by [u].

From now on, let k〈X〉 be the free associative algebra generated by X. We consider

( ) as Lie bracket in k〈X〉, i.e., for any a, b ∈ k〈X〉, (ab) = ab − ba. Denote by Lie(X)

the subLie-algebra of k〈X〉 generated by X.

Given a polynomial f ∈ k〈X〉, it has the leading word f ∈ X∗ according to the above

ordering on X∗ such that

f =∑u∈X∗

f(u)u = αf +∑

αiui,

where f, ui ∈ X∗, f > ui, α, αi, f(u) ∈ k. We call f the leading term of f . Denote the

set u|f(u) 6= 0 by suppf and deg(f) by |f |. f is called monic if α = 1.

Note that if |u| = |v| and u < v, then the lexicographic ordering which we use before

is the same as the degree-lexicographic ordering on X∗.

Theorem 5.22 Let the ordering < be as before. Then, for any (u) ∈ X∗∗, (u) has a

representation:

(u) =∑

αi[ui],

where each αi ∈ k, [ui] is a NLSW and |ui| = |u|. Even more, if (u) = ([v][w]), then

ui > minv, w.

Proof. Induction on |u|. If |u| = 1, then (u) = [u] and the result holds. Suppose that

|u| > 1 and (u) = ((v)(w)). Then, by induction,

(v) =∑

αi[vi] and (w) =∑

βj[wj],

where αi, βj ∈ k, [vi], [wj] are NLSW’s, |vi| = |v| and |wj| = |w|. Without loss of

generality, we may assume that (u) = ([v][w]) with v > w because of ([v][w]) = −([w][v]).

If |v| = 1, then

(u) = ([v][w])

is a NLSW. Suppose that |v| > 1 and [v] = [[v1][v2]].

There are two subcases

(a) If v2 ≤ w, then (u) = (([v1][v2])[w]) is a NLSW.

(b) If v2 > w, then

(u) = (([v1][v2])[w]) = (([v1][w])[v2]) + ([v1]([v2][w])).

74

By induction,

([v1][w]) =∑

γi[ti], ti > minv1, w = w

([v2][w]) =∑

γj′[t′j], t′j > minv2, w = w.

Then,

(u) =∑

γi([ti][v2]) +∑

γj′([v1][t

′j]).

By noting that

minti, v2 and mint′j, v1 > minv, w = w,

the result follows from the inverse induction on minv, w.

Example 5.23 Let (u) = (((x3x2)(x2x1))(x2x1x1)). Then

(u) = (((x3(x2x1))x2)(x2x1x1)) + ((x3(x2(x2x1)))(x2x1x1)),

(((x3(x2x1))x2)(x2x1x1)) = (((x3(x2x1))(x2x1x1))x2) + ((x3(x2x1))(x2(x2x1x1)))

= (((x3(x2x1x1))(x2x1))x2) + ((x3((x2x1)(x2x1x1)))x2)

+((x3(x2x1))(x2(x2x1x1))),

((x3(x2(x2x1)))(x2x1x1)) = ((x3(x2x1x1))(x2(x2x1))) + (x3((x2(x2x1))(x2x1x1)))

= ((x3(x2x1x1))(x2(x2x1))) + (x3((x2(x2x1x1))(x2x1))

+(x3(x2((x2x1)(x2x1x1)))),

and hence,

(u) = (((x3(x2x1x1))(x2x1))x2) + ((x3((x2x1)(x2x1x1)))x2)

+((x3(x2x1))(x2(x2x1x1))) + ((x3(x2x1x1))(x2(x2x1)))

+(x3((x2(x2x1x1))(x2x1)) + (x3(x2((x2x1)(x2x1x1))))

is a linear combination of NLSW’s.

Lemma 5.24 Let [u] be a NLSW. Then [u] = u.

Proof. We use induction on |u|. If |u| = 1, then the result holds immediately. Let

|u| > 1 and [u] = [[v][w]]. Then, by induction, [v] = v and [w] = w. Suppose that

[v] = v +∑vi<v

αivi, [w] = w +∑

wj<w

βjwj,

75

where αi, βj ∈ k, v, vi, w, wj ∈ X∗. It is easy to see that |vi| = |v| and |wj| = |w| for any

i, j. Then,

[u] = [(v +∑vi<v

αivi)(w +∑

wj<w

βjwj)]

= (v +∑vi<v

αivi)(w +∑

wj<w

βjwj)− (w +∑

wj<w

βjwj)(v +∑vi<v

αivi)

= vw +∑

wj<w

βjvwj +∑vi<v

αiviw +∑

αiβjviwj

− wv −∑

wj<w

βjwjv −∑vi<v

αiwvi −∑

βjαiwjvi.

Since

vw > vwj, viw, viwj, wv and wv > wvi, wjv, wjvi,

we have [u] = u.

Remark. By the proof of Lemma 5.24, if we consider [u] as a polynomial in k〈X〉, then

each r ∈ supp([u]) has the same length as u, moreover, cont(r) = cont(u), where, for

example, cont(u) = xi1 , · · · , xit if u = xi1 · · ·xit ∈ X∗.

Lemma 5.25 NLSW’s are k-linear independent.

Proof. Supposek∑

i=1

αi[ui] = 0,

where each αi ∈ k, [ui] is a NLSW and u1 > u2 > · · · > uk. If α1 6= 0, then∑i

αi[ui] =

u1 6= 0, a contradiction. Then, all αi must be 0.

By Theorem 5.22 and Lemma 5.25, we have the following corollary.

Corollary 5.26 NLSW’s is a linear basis of Lie(X).

From Corollary 5.26 and Lemma 5.24, we have

Corollary 5.27 For any f ∈ Lie(X), f is an ALSW.

Theorem 5.28 Lie(X) is the free Lie algebra generated by X.

Proof. Let L be a Lie algebra and g : X −→ L a mapping. Then, we define a mapping

g1 : Lie(X) −→ L; [xi1 · · ·xin ] 7−→ [g(xi1) · · · g(xin)],

76

where [xi1 · · ·xin ] is NLSW. It is easy to check g1 is a unique Lie homomorphism such

that g1i = g.

-

?

L

Lie(X)X i

g∃! g1

The following theorem plays a key role in proving the Composition-Diamond lemma

for Lie algebras (see Theorem 5.39).

Theorem 5.29 (A.I. Shirshov) Let u, v be ALSW’s, u = avb, a, b ∈ X∗. Then

(i) [u] = [a[vc]d], where b = cd, c, d ∈ X∗.

(ii) Let

[u]v = [u]|[vc] 7→[[[v][c1]]···[ck]], (5.1)

where c = c1 · · · ck, cj is an ALSW and c1 ≤ c2 ≤ · · · ≤ ck. Then, in k〈X〉,

[u]v = u.

Moreover,

[u]v = a[v]b+∑

i

αiai[v]bi,

where each αi ∈ k and aivbi < avb.

Proof. (i) Induction on |u|. If |u| = 1, then u = v = xi and the result holds. Assume

that |u| > 1. If v = xi, then [u] = [a[xi]d] and the result holds. Now, we consider the case

of |v| > 1. Let xβ = min(u) and b = xeβ b, where e ≥ 0 and fir(b) 6= xβ. Then

u = avb = avxeβ b = avb,

where v = vxeβ is also an ALSW, by Lemma 5.11. Then, by induction, for u′ = a′uv

′ub

′u,

we have [u′] = [a′u[v′uc

′u]d

′u], b

′u = c′ud

′u. By substitution

xji 7→ [[xixβ] · · ·xβ], we obtain

[u] = [a[vc]d] = [a[vxeβ c]d] = [a[vc]d], where c = xe

β c.

(ii) If c = 1, then [u]v = [u] and the results hold clearly. Otherwise, by Lemma 5.13,

we may assume that

c = xβ · · ·xβcl+1 · · · ck,

77

where each ci is an ALSW and xβ < cl+1 ≤ · · · ≤ ck.

Then

[u]v = [u]|[vxeβ c] 7→[[[v]xβ ]···xβ [cl+1]···[ck]] and [u]v = [u]|[vc] 7→[[[v][cl+1]]···[ck]].

Now, we use induction on |u|. If |u| = 1, then this is a trivial case. Suppose that |u| > 1

and |v| > 1. Then, by (i),

u = avcd, u′ = a′uv′uc

′ud

′u

and by induction,

[u′]v′u = a′u[v′u]c

′ud

′u +

∑i∈I1

αiai′u[v

′u]bi

′u,

where each ai′uv

′ubi

′u < u′. Now, it is easy to check that

[[xixβ] · · ·xβ] =∑m≥0

(−1)m

(j

m

)xm

β xixj−mβ and xm

β xixj−mβ < xix

jβ (m > 0).

Now, by substitution xji 7→ [[xixβ] · · ·xβ], we obtain

[u]v = a[v]cd+∑i∈I2

αiai[v]bi,

where each aivbi < avcd. Also, by substitution [v] 7→ [[[v]xβ] · · ·xβ], we have

[u]v = a[v]xeβ cd+

∑j∈I

βjaj[v]bj = a[v]b+∑j∈I

βjaj[v]bj,

where each ajvbj < avb.

Remark. By the proof of Theorem 5.29, if we consider [u]v as a polynomial in k〈X〉,then for any w ∈ supp([u]v), cont(w) = cont(u).

Definition 5.30 Let S ⊂ Lie(X) with each s ∈ S monic, a, b ∈ X∗ and s ∈ S. If asb

is an ALSW, then we call [asb]s = [asb]s|[s] 7→s a special normal S-word (or special normal

s-word) while [asb]s is called a relative nonassociative Lyndon-Shirshov word, denoted by

RNLSW, where [asb]s is defined by (5.1) (see Theorem 5.29).

A Lie S-word (asb) is called a normal S-word if (asb) = asb.

Clearly, any special normal s-word is a normal s-word.

Corollary 5.31 Let u, v be ALSW’s, f ∈ Lie(X), f = v and u = avb, a, b ∈ X∗. Then,

for the normal f -word [afb]v = [avb]v|[v] 7→f , we have

[afb]v = afb+∑

i

αiaifbi,

where each αi ∈ k, ai, bi ∈ X∗, aif bi < u.

78

§5.3 Composition-Diamond lemma for Lie algebras

In this section, we give the Composition-Diamond lemma for Lie algebras.

Throughout this section, we extend the lexicographic ordering on X∗ mentioned in

section §5.2 to the deg-lex ordering < on X∗.

Lemma 5.32 Let ac, cb be ALSW’s, where a, b, c ∈ X∗ and c 6= 1. Then w = acb is also

an ALSW.

Proof. We use induction on |w| = n. If n = 3, then w = xixjxk is an ALSW, because

ac and cb are ALSW’s implies that xi > xj > xk. In the inductive case n > 3, suppose

min(w) = xβ, b = xeβ b, e ≥ 0, fir(b) 6= xβ and c = cxe

β. Then

w = acb and w′ = a′wc′wb

′w.

It is clear that a′wc′w, c

′wb

′w are ALSW’s. By induction, w′ is an ALSW and so is w.

Definition 5.33 Let f and g be two monic Lie polynomials in Lie(X) ⊂ k〈X〉. Then,

there are two kinds of Lie compositions:

(i) If w = f = agb for some a, b ∈ X∗, then the polynomial 〈f, g〉w = f − [agb]g is called

the composition of inclusion of f and g with respect to w.

(ii) If w is a word such that w = f b = ag for some a, b ∈ X∗ with deg(f)+deg(g) >deg(w),

then the polynomial 〈f, g〉w = [fb]f − [ag]g is called the composition of intersection of

f and g with respect to w.

By Lemma 5.32, in the Definition 5.33 (i) and (ii), w is an ALSW.

Definition 5.34 Let S ⊂ Lie(X) be a nonempty subset, h a Lie polynomial and w ∈X∗. We shall say that h is trivial modulo (S,w), denoted by h ≡Lie 0 mod(S,w), if

h =∑i

αi[aisibi]si, where each αi ∈ k, ai, bi ∈ X∗, si ∈ S, [aisibi]si

is a RNLSW and

aisibi < w.

Definition 5.35 Let S ⊂ Lie(X) be a nonempty set of monic Lie polynomials. Then S

is called a Grobner-Shirshov set (basis) in Lie(X) if any composition 〈f, g〉w with f, g ∈ Sis trivial modulo (S,w), i.e., 〈f, g〉w ≡Lie 0 mod(S,w).

Lemma 5.36 Let f, g be monic Lie polynomials. Then

〈f, g〉w − (f, g)w ≡ass 0 mod(f, g, w).

79

Proof. If 〈f, g〉w and (f, g)w are compositions of intersection, where w = f b = ag, then,

by Corollary 5.31, we may assume that

〈f, g〉w = [fb]f − [ag]g = fb+∑I1

αiaifbi − ag −∑I2

βjajgbj,

where aif bi, aj gbj < fb = ag = w. It follows that

〈f, g〉w − (f, g)w ≡ass 0 mod(f, g, w).

Similarly, for the case of the compositions of inclusion, we have the same conclusion.

Theorem 5.37 Let S ⊂ Lie(X) ⊂ k〈X〉 be a nonempty set of monic Lie polynomials.

Then S is a Grobner-Shirshov basis in Lie(X) if and only if S is a Grobner-Shirshov

basis in k〈X〉.

Proof. Note that, by the definitions, for any f, g ∈ S, they have composition in Lie(X)

if and only if so do in k〈X〉.Suppose that S is a Grobner-Shirshov basis in Lie(X). Then, for any composition

〈f, g〉w, we have

〈f, g〉w =∑I1

αi[aisibi]si,

where [aisibi]siare RNLSW’s and aisibi < w. By Corollary 5.31,

〈f, g〉w =∑I2

βjcjsjdj,

where each cj sjdj < w. Thus, by Lemma 5.36, we get

(f, g)w ≡ass 0 mod(S,w).

Hence, S is a Grobner-Shirshov basis in k〈X〉.Conversely, assume that S is a Grobner-Shirshov basis in k〈X〉. Then, for any

composition 〈f, g〉w in S, by Lemma 5.36, we obtain

〈f, g〉w ≡ass (f, g)w ≡ass 0 mod(S,w).

Therefore, we can assume by Theorem 4.4 that

〈f, g〉w =∑I1

αiaisibi,

80

where aisibi < w and w > a1s1b1 > a2s2b2 > . . .. By noting that 〈f, g〉w ∈ Lie(X), 〈f, g〉w =

a1s1b1 is an ALSW which shows that [a1s1b1]s1 is a RNLSW. Let h1 = 〈f, g〉w−α1[a1s1b1]s1 .

Clearly, h1 < 〈f, g〉w. Then, by Corollary 5.31, we have

h1 ≡ass 0 mod(S,w).

Now, by induction on 〈f, g〉w, we have

〈f, g〉w =∑I2

αi[cisidi]si,

where each [cisidi]siis a RNLSW and cisidi < w. This proves that S is a Grobner-Shirshov

basis in Lie(X).

Lemma 5.38 Let S ⊂ Lie(X) with each s ∈ S monic. Let

Irr(S) = [u] | [u] is a NLSW, u 6= asb, s ∈ S, a, b ∈ X∗.

Then, for any h ∈ Lie(X), h has a representation:

h =∑

[ui]∈Irr(S), ui≤h

αi[ui] +∑

sj∈S, aj sjbj≤h

βj[ajsjbj]sj.

Proof. We can assume that h =∑i

αi[ui], where each [ui] is a NLSW, 0 6= αi ∈ k and

u1 > u2 > · · · . If [u1] ∈ Irr(S), then let h1 = h − α1[u1]. If [u1] 6∈ Irr(S), then there

exists s ∈ S and a1, b1 ∈ X∗ such that u1 = a1s1b1. Now, let

h1 = h− α1[a1s1b1]s1 ∈ Lie(X).

Hence, in both cases, we have h1 < h. Now, the result follows from induction on h.

Theorem 5.39 (CD-lemma for Lie algebras over a field) Let S ⊂ Lie(X) ⊂ k〈X〉 be

nonempty set of monic Lie polynomials. Let IdLie(S) be the Lie-ideal of Lie(X) generated

by S. Then the following statements are equivalent.

(i) S is a Grobner-Shirshov basis in Lie(X).

(ii) f ∈ IdLie(S) =⇒ f = asb, for some s ∈ S and a, b ∈ X∗.

(ii’) f ∈ IdLie(S) =⇒ f = α1[a1s1b1]s1 +α2[a2s2b2]s2 + · · · , where αi ∈ k and f = a1s1b1 >

a2s2b2 > · · · .

(iii) Irr(S) = [u] | [u] is a NLSW, u 6= asb, s ∈ S, a, b ∈ X∗ is a k-basis for Lie(X|S).

81

Proof. (i) =⇒ (ii). By noting that IdLie(S) ⊆ Idass(S), where Idass(S) is the ideal of

k〈X〉 generated by S, and by using Theorems 5.37 and 4.4, the result follows.

(ii) =⇒ (iii). Suppose that∑

[ui]∈Irr(S)

αi[ui] = 0 in Lie(X|S) with u1 > u2 > · · · ,

that is,∑

[ui]∈Irr(S)

αi[ui] ∈ IdLie(S). Then each αi must be 0. Otherwise, say α1 6= 0. Then,

by (ii), we know that∑i

αi[ui] = u1 which implies that [u1] 6∈ Irr(S), a contradiction.

On the other hand, for any f ∈ Lie(X), by Lemma 5.38, we have

f + IdLie(S) =∑

i

αi([ui] + IdLie(S)).

(iii) =⇒ (i). For any composition 〈f, g〉w with f, g ∈ S, we have 〈f, g〉w ∈ IdLie(S).

Then, by (iii) and by Lemma 5.38,

〈f, g〉w =∑

βj[ajsjbj]sj,

where each βj ∈ k, [ajsjbj]sjis normal S-word and ajsjbj < w. This proves that S is a

Grobner-Shirshov basis in Lie(X).

(ii)⇐⇒ (ii′). This part is clear.

Example 5.40 (Shirshov) For an arbitrary polynomial f ∈ Lie(X), it is clear that fis a Grobner-Shirshov basis in Lie(X). From this it follows that the word problem is

solvable for each one-relator Lie algebra Lie(X|f).

§5.4 Applications

§5.4.1 Kukin’s construction of a Lie algebra with unsolvable word problem

Let P = sgp〈x, y|ui = vi, i ∈ I〉 be a semigroup. Consider the Lie algebra

AP = Lie(x, x, y, y, z|S),

where S consists of the following relations:

1) [xx] = 0, [xy] = 0, [yx] = 0, [yy] = 0,

2) [xz] = −[zx], [yz] = −[zy],

3) bzuic = bzvic, i ∈ I.

Here, bzuc means the left normed bracketing.

In this section, we give a Grobner-Shirshov basis for Lie algebra AP and by using

this result we give another proof for Kukin’s theorem, see Corollary 5.42.

82

Let the ordering x > y > z > x > y and > the deg-lex ordering on x, y, x, y, z∗.Let ρ be the congruence on x, y∗ generated by (ui, vi), i ∈ I. Let

3)′ bzuc = bzvc, (u, v) ∈ ρ with u > v.

Theorem 5.41 With the above notation, the set S1 = 1), 2), 3)′ is a Grobner-Shirshov

basis in Lie(x, y, x, y, z).

Proof. For any u ∈ x, y∗, by induction on |u|, bzuc = zu. All possible compositions

in S1 are intersection of 2) and 3)′, and inclusion of 3)′ and 3)′.

For 2)∧ 3)′, w = xzu, (u, v) ∈ ρ, u > v, f = [xz] + [zx], g = bzuc − bzvc. We have

([xz] + [zx], bzuc − bzvc)w = [fu]f − [xg]g

≡ b([xz] + [zx])uc − [x(bzuc − bzvc)]

≡ bzxuc+ bxzvc ≡ bzxuc − bzxvc ≡ 0 mod(S1, w).

For 3)′ ∧ 3)′, w = zu1 = zu2e, e ∈ x, y∗, (ui, vi) ∈ ρ, ui > vi, i = 1, 2. We have

(bzu1c − bzv1c, bzu2c − bzv2c)w ≡ (bzu1c − bzv1c)− b(bzu2c − bzv2c)ec

≡ bbzv2cec − bzv1c ≡ bzv2ec − bzv1c ≡ 0 mod(S1, w).

Thus, the set S1 = 1), 2), 3)′ is a Grobner-Shirshov basis in Lie(x, y, x, y, z).

Corollary 5.42 (Kukin [138]) Let u, v ∈ x, y∗. Then

u = v in the semigroup P ⇔ bzuc = bzvc in the Lie algebra AP .

Proof. Suppose that u = v in the semigroup P . Without loss of generality, we

may assume that u = au1b, v = av1b for some a, b ∈ x, y∗ and (u1, v1) ∈ ρ. For

any r ∈ x, y, by the relations 1), we have [xr] = 0 and so bzxcc = b[zx]cc =

[bzccx], bzycc = [bzccy] for any c ∈ x, y∗. From this it follows that in AP , bzuc =

bzau1bc = bbzau1cbc = bbzu1←−a cbc = bzu1

←−a bc = bzv1←−a bc = bzav1bc = bzvc, where

←−−−−−−−−xi1xi2 · · ·xin = xinxin−1 · · ·xi1 and xi1xi2 · · ·xin = xi1 xi2 · · · xin , xij ∈ x, y. Moreover,

3)′ holds in AP .

Suppose that bzuc = bzvc in the Lie algebra AP . Then both bzuc and bzvc have

the same normal form in AP . Since S1 is a Grobner-Shirshov basis in AP by Theorem

5.41, both bzuc and bzvc can be reduced to the same normal form of the form bzcc for

some c ∈ x, y∗ only by the relations 3)′. This implies that in P , u = c = v.

By the above corollary, if the semigroup P has the undecidable word problem then

so does the Lie algebra AP .

83

§5.4.2 Grobner-Shirshov basis for the Drinfeld-Kohno Lie algebra Ln

In this section we give a Grobner-Shirshov basis for the Drinfeld-Kohno Lie algebra

Ln.

Definition 5.43 Let n > 2 be an integer. The Drinfeld-Kohno Lie algebra Ln over Z is

defined by generators tij = tji for distinct indices 1 ≤ i, j ≤ n− 1, and relations

tijtkl = 0,

tij(tik + tjk) = 0,

where i, j, k, l are distinct.

Clearly, Ln has a presentation LieZ(T |S), where T = tij| 1 ≤ i < j ≤ n− 1 and S

consists of the following relations

tijtkl = 0, k < i < j, k < l, l 6= i, j (5.2)

tjktij + tiktij = 0, i < j < k (5.3)

tjktik − tiktij = 0, i < j < k (5.4)

Now we order T : tij < tkl if either i < k or i = k and j < l. Let < be the deg-lex

ordering on T ∗.

Theorem 5.44 Let S = (5.2), (5.3), (5.4) be as before, < the deg-lex ordering on T ∗.

Then S is a Grobner-Shirshov basis for Ln.

Proof. We list all the possible ambiguities. Denote (i)∧ (j) the composition of the type

(i) and type (j). For convenience, we denote (5.2), (5.3), (5.4) by (1), (2), (3) respectively.

For (1) ∧ (n), 1 ≤ n ≤ 3, the possible ambiguities w’s are:

(1) ∧ (1) tijtkltmr, (k < i < j, k < l, l 6= i, j, m < k < l, m < r, r 6= k, l),

(1) ∧ (2) tijtkltmk, ( k < i < j, k < l, l 6= i, j, m < k < l),

(1) ∧ (3) tijtkltml, (k < i < j, k < l, l 6= i, j, m < k < l).

For (2) ∧ (n), 1 ≤ n ≤ 3, the possible ambiguities w’s are:

(2) ∧ (1) tjktijtmr, ( m < i < j < k, m < r, r 6= i, j),

(2) ∧ (2) tjktijtmi, ( m < i < j < k),

(2) ∧ (3) tjktijtmj, ( m < i < j < k).

84

For (3) ∧ (n), 1 ≤ n ≤ 3, the possible ambiguities w’s are:

(3) ∧ (1) tjktiktmr, ( m < i < j < k, m < r, r 6= i, k),

(3) ∧ (2) tjktiktmi, ( m < i < j < k),

(3) ∧ (3) tjktiktmk, ( m < i < j < k).

We claim that all compositions are trivial relative to S.

Here, we only prove cases (1)∧ (1), (1)∧ (2), (2)∧ (1), (2)∧ (2), and the other cases

can be proved similarly.

For (1) ∧ (1), let f = tijtkl, g = tkltmr, k < i < j, k < l, l 6= i, j, m < k < l, m <

r, r 6= k, l. Then w = tijtkltmr and

(f, g)w = (tijtkl)tmr − tij(tkltmr)

= (tijtmr)tkl mod(S,w).

There are three subcases to consider: r 6= i, j, r = i, r = j.

Subcase 1. If r 6= i, j, then

(tijtmr)tkl ≡ 0 mod(S,w).

Subcase 2. If r = i, then

(tijtmr)tkl = (tijtmi)tkl

≡ tkl(tmjtmi)

≡ (tkltmj)tmi + tmj(tkltmi)

≡ 0 mod(S,w).

Subcase 3. If r = j, then

(tijtmr)tkl = (tijtmj)tkl

≡ −tkl(tmjtmi)

≡ −(tkltmj)tmi − tmj(tkltmi)

≡ 0 mod(S,w).

For (1)∧ (2), let f = tijtkl, g = tkltmk + tmltmk, k < i < j, k < l, l 6= i, j, m < k <

l. Then w = tijtkltmk and

(f, g)w = (tijtkl)tmk − tij(tkltmk + tmltmk)

= (tijtmk)tkl − tij(tmltmk)

≡ −tij(tmltmk)

≡ −(tijtml)tmk − tml(tijtmk)

≡ 0 mod(S,w).

85

For (2)∧(1), let f = tjktij + tiktij, g = tijtmr, m < i < j < k, m < r, r 6= i, j. Then

w = tjktijtmr and

(f, g)w = (tjktij + tiktij)tmr − tjk(tijtmr)

≡ (tjktmr)tij + (tiktmr)tij mod(S,w).

There are two subcases to consider: r 6= k, r = k.

Subcase 1. If r 6= k, then

(tjktmr)tij + (tiktmr)tij ≡ 0 mod(S,w).

Subcase 2. If r = k, then

(tjktmr)tij + (tiktmr)tij = (tjktmk)tij + (tiktmk)tij

≡ −tij(tmktmj)− tij(tmktmi)

≡ (tijtmj)tmk + (tijtmi)tmk

≡ −tmk(tmjtmi) + tmk(tmjtmi)

≡ 0 mod(S,w).

For (2) ∧ (2), let f = tjktij + tiktij, g = tijtmi + tmjtmi, m < i < j < k. Then

w = tjktijtmi and

(f, g)w = (tjktij + tiktij)tmi − tjk(tijtmi + tmjtmi)

= (tjktmi)tij + (tiktmi)tij + tik(tijtmi)− tjk(tmjtmi)

≡ tij(tmktmi)− tik(tmjtmi)− tjk(tmjtmi)

≡ −(tijtmi)tmk + (tiktmi)tmj − (tjktmj)tmi

≡ −tmk(tmjtmi)− (tmktmi)tmj + (tmktmj)tmi

≡ 0 mod(S,w).

So S is a Grobner-Shirshov basis for Ln.

Let L be a Lie algebra over a commutative ring K, L1 an ideal of L and L2 a

subalgebra of L. Then we call L a semidirect product of L1 and L2 if L = L1 ⊕ L2 as

K-modules.

By Theorems 5.39 and 5.44, we have immediately the following corollaries.

Corollary 5.45 The Drinfeld-Kohno Lie algebra Ln is a free Z-module with a Z-basis

Irr(S) = [tik1tik2 · · · tikm ] | tik1tik2 · · · tikm is an ALSW in T ∗, m ∈ N.

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Corollary 5.46 Ln is an iterated semidirect product of free Lie algebras.

Proof. Let Ai be the free Lie algebra generated by tij | i < j ≤ n− 1. Clearly,

Ln = A1 ⊕ A2 ⊕ · · · ⊕ An−2

as Z-modules, and from the relations (1), (2), (3), we have

Ai / Ai + Ai+1 + · · · + An−2.

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Chapter 6 Grobner-Shirshov Bases for Lie Algebras

over a Commutative Algebra

In this chapter, we establish the Composition-Diamond lemma for Lie algebras over

a polynomial algebra, i.e., for “double free” Lie algebras. It provides a Grobner-Shirshov

bases theory for Lie algebras over a commutative algebra.

Let k be a field, K a commutative associative k-algebra with identity, and L a Lie

K-algebra. Let LieK(X) be the free Lie K-algebra generated by a set X. Then, of course,

L can be presented as K-algebra by generators X and some defining relations S,

L = LieK(X|S) = LieK(X)/Id(S).

In order to define a Grobner-Shirshov basis for L, we first present K in a form

K = k[Y |R] = k[Y ]/Id(R),

where k[Y ] is a polynomial algebra over the field k, R ⊂ k[Y ]. Then the Lie K-algebra Lhas the following presentation as a k[Y ]-algebra

L = Liek[Y ](X|S,Rx, x ∈ X)

Now by definition, a Grobner-Shirshov basis for L = LieK(X|S) is Grobner-Shirshov

basis of the ideal Id(S,Rx, x ∈ X) in the “double free” Lie algebra Liek[Y ](X).

As an application of Composition-Diamond lemma (Theorem 6.14), a Grobner-Shirshov

basis of L gives rise to a linear basis of L as a k-algebra.

We give applications of Grobner-Shirshov bases theory for Lie algebras over a com-

mutative algebra K (over a field k) to the Poincare-Birkhoff-Witt theorem. A Lie algebra

over a commutative ring is called special if it is embeddable into an (universal enveloping)

associative algebra. Otherwise it is called non-special. There are known classical exam-

ples by A.I. Shirshov [196] and P. Cartier [67] of Lie algebras over commutative algebras

over GF (2) that are not embeddable into associative algebras. Shirshov and Cartier used

ad hoc methods to prove that some elements of corresponding Lie algebras are not zero

though they are zero in the universal enveloping algebras, i.e., they proved non-speciality

of the examples. Here we find Grobner-Shirshov bases of these Lie algebras and then

use Composition-Diamond lemma (Theorem 6.14) to get the result, i.e., we give a new

conceptual proof.

P.M. Cohn [93] gave the following examples of Lie algebras

Lp = LieK(x1, x2, x3|y3x3 = y2x2 + y1x1)

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over truncated polynomial algebras

K = k[y1, y2, y3|ypi = 0, 1 ≤ i ≤ 3],

where k is a filed of characteristic p > 0. He conjectured that Lp is non-special Lie algebra

for any p. Lp is called the Cohn’s Lie algebra. Using Theorem 6.14 we will prove that L2,

L3 and L5 are non-special Lie algebras. We present an algorithm that one can check for

any p, whether Cohn’s Lie algebras is non-special.

We give new class of special Lie algebras in terms of defining relations (Theorem ??).

For example, any one relator Lie algebra LieK(X|f) with a k[Y ]-monic relation f over

a commutative algebra K is special (Corollary ??). It gives an extension of the list of

known special Lie algebras (ones with valid PBW Theorems) (see P.-P. Grivel [107]). Let

us give this list:

1. L is a free K-module (G. Birkhoff [11], E. Witt [213]),

2. K is a principal ideal domain (M. Lazard [143, 144]),

3. K is a Dedekind domain (P. Cartier [67]),

4. K is over a field k of characteristic 0 (P.M. Cohn [93]),

5. L is K-module without torsion (P.M. Cohn [93]),

6. 2 is invertible in K and for any x, y, z ∈ L, [x[yz]] = 0 (Y. Nouaze and P. Revoy

[168]),

7. No nonzero element of K annihilates an element of the center of LK (A.I. Shirshov

[196]).

As a last application we prove that every finitely or countably generated Lie algebra

over an arbitrary commutative algebra K can be embedded into a two-generated Lie

algebra over K.

§6.1 Preliminaries

We start with some concepts and results from the literature concerning with the

Grobner-Shirshov bases theory of a free Lie algebra Liek(X) generated by X over a field

k.

Let X = xi|i ∈ I be a well-ordered set with xi > xj if i > j for any i, j ∈ I. Let

X∗ be the free monoid generated by X. For u = xi1xi2 · · ·xim ∈ X∗, let the length of u

be m, denoted by |u| = m.

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We use two linear orderings on X∗:

(i) (lex-ordering) 1 > t if t 6= 1 and, by induction, if u = xiu′ and v = xjv

′ then

u > v if and only if xi > xj or xi = xj and u′ > v′;

(ii) (deg-lex ordering) u v if |u| > |v|, or |u| = |v| and u > v.

We regard Liek(X) as the Lie subalgebra of the free associative algebra k〈X〉, which

is generated by X under the Lie bracket [u, v] = uv − vu. Given f ∈ k〈X〉, denote by f

the leading word of f with respect to the deg-lex ordering; f is monic if the coefficient of

f is 1.

We denote the set of all NLSW’s on X by NLSW (X).

In fact, NLSW’s may be defined as Hall–Shirshov words relative to lex-deg ordering.

Considering any NLSW [w] as a polynomial in k〈X〉, we have [w] = w (see Lemma

5.24). This implies that if f ∈ Liek(X) ⊂ k〈X〉 then f is an ALSW.

Lemma 6.1 Let w = aub be as in Theorem 5.29. Then [uc] = [u[c1][c2] . . . [cn]], that is

[w] = [a[. . . [u[c1]] . . . [cn]]d].

Lemma 6.2 Suppose that w = aubvc, where w, u, v ∈ ALSW (X). Then there is some

bracketing

[w]u,v = [a[u]b[v]d]

in the word w such that

[w]u,v = w.

More precisely,

[w]u,v =

[a[up]uq[vs]vl] if [w] = [a[up]q[vs]l],

[a[u[c1] · · · [ct]v · · · [cn]]up] if [w] = [a[u[c1] · · · [ct] · · · [cn]]p] with v a subword of ct.

§6.2 Composition-Diamond lemma for Liek[Y ](X)

Let Y = yj|j ∈ J be a well-ordered set and [Y ] = yj1yj2 · · · yjl|yj1 ≤ yj2 ≤ · · · ≤

yjl, l ≥ 0 the free commutative monoid generated by Y . Then [Y ] is a k-linear basis of

the polynomial algebra k[Y ].

Let the set X be a well-ordered set, and let the lex-ordering < and the deg-lex

ordering ≺X on X∗ be defined as before.

Let Liek[Y ](X) be the “double” free Lie algebra, i.e., the free Lie algebra over the

polynomial algebra k[Y ] with generating set X.

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From now on we regard Liek[Y ](X) ∼= k[Y ] ⊗ Liek(X) as the Lie subalgebra of

k[Y ]〈X〉 ∼= k[Y ]⊗ k〈X〉 the free associative algebra over polynomial algebra k[Y ], which

is generated by X under the Lie bracket [u, v] = uv − vu.Let

TA = u = uY uX |uY ∈ [Y ], uX ∈ ALSW (X)

and

TN = [u] = uY [uX ]|uY ∈ [Y ], [uX ] ∈ NLSW (X).

By Chapter 5, we know that the elements of TA and TN are one-to-one corresponding

to each other.

Remark: For u = uY uX ∈ TA, we still use the notation [u] = uY [uX ] where [uX ] is a

NLSW on X.

Let kTN be the linear space spanned by TN over k. For any [u], [v] ∈ TN , define

[u][v] =∑

αiuY vY [wX

i ]

where αi ∈ k, [wXi ]’s are NLSW’s and [uX ][vX ] =

∑αi[w

Xi ] in Liek(X).

Then k[Y ]⊗ Liek(X) ∼= kTN as k-algebra and TN is a k-basis of k[Y ]⊗ Liek(X).

We define the deg-lex ordering on

[Y ]X∗ = uY uX |uY ∈ [Y ], uX ∈ X∗

as follows: for any u, v ∈ [Y ]X∗,

u v if (uX X vX) or (uX = vX and uY Y vY ),

where Y and X are the deg-lex ordering on [Y ] and X∗ respectively.

Remark: By abuse of the notation, from now on, in a Lie expression like [[u][v]] we will

omit the external brackets, [[u][v]] = [u][v].

Clearly, the ordering is “monomial” in a sense of [u][w] [v][w] whenever wX 6= uX

for any u, v, w ∈ TA.

Considering any [u] ∈ TN as a polynomial in k-algebra k[Y ]〈X〉, we have [u] = u ∈ TA.

For any f ∈ Liek[Y ](X) ⊂ k[Y ]⊗ k〈X〉, one can present f as a k-linear combination

of TN -words, i.e., f =∑αi[ui], where [ui] ∈ TN . With respect to the ordering on

[Y ]X∗, the leading word f of f in k[Y ]〈X〉 is an element of TA. We call f k-monic if the

coefficient of f is 1. On the other hand, f can be presented as k[Y ]-linear combinations

of NLSW (X), i.e., f =∑fi(Y )[uX

i ], where fi(Y ) ∈ k[Y ], [uXi ] ∈ NLSW (X) and

uX1 X uX

2 X . . .. Clearly fX = uX1 and fY = f1(Y ). We call f k[Y ]-monic if the

f1(Y ) = 1. It is easy to see that k[Y ]-monic implies k-monic.

Equipping with the above concepts, we rewrite Lemma 5.24 as follows.

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Lemma 6.3 (Shirshov) Suppose that w = aub where w, u ∈ TA and a, b ∈ X∗. Then

[w] = [a[uc]d],

where [uc] ∈ TN and b = cd.

Represent c in a form c = c1c2 . . . cn, where c1, . . . , cn ∈ ALSW (X) and c1 ≤ c2 ≤. . . ≤ cn. Then

[w] = [a[u[c1][c2] . . . [cn]]d].

Moreover, the leading word of [w]u = [a[· · · [[[u][c1]][c2]] . . . [cn]]d] is exactly w, i.e.,

[w]u = w.

We still use the notion [w]u as the special bracketing of w relative to u.

Let S ⊂ Liek[Y ](X) and Id(S) be the k[Y ]-ideal of Liek[Y ](X) generated by S. Then

any element of Id(S) is a k[Y ]-linear combination of polynomials of the following form:

(u)s = [c1][c2] · · · [cn]s[d1][d2] · · · [dm], m, n ≥ 0

with some placement of parentheses, where s ∈ S and ci, dj ∈ ALSW (X). We call such

(u)s an s-word (or S-word).

Now, we define two special kinds of S-words.

Definition 6.4 Let S ⊂ Liek[Y ](X) be a k-monic subset, a, b ∈ X∗ and s ∈ S. If

asb ∈ TA, then by Lemma 6.3 we have the special bracketing [asb]s of asb relative to s.

We define [asb]s = [asb]s|[s] 7→s to be a special normal s-word (or special normal S-word).

Definition 6.5 Let S ⊂ Liek[Y ](X) be a k-monic subset and s ∈ S. We define the

normal s-word, denoted by bucs, where u = asb, a, b ∈ X∗ (u is an associative S-word),

inductively.

(i) s is normal of s-length 1;

(ii) If bucs is normal with s-length k and [v] ∈ NLSW (X) such that |v| = l, then [v]bucswhen v > buc

X

s and bucs[v] when v < bucX

s are normal of s-length k + l.

From the definition of the normal s-word, we have the following lemma.

Lemma 6.6 For any normal s-word bucs = (asb), a, b ∈ X∗, we have bucs = asb ∈ TA.

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Remark: It is clear that for an s-word (u)s = [c1][c2] · · · [cn]s[d1][d2] · · · [dm], (u)s is

normal if and only if (u)s = c1c2 · · · cnsd1d2 · · · dm.

Now we give the definition of compositions.

Definition 6.7 Let f, g be two k-monic polynomials of Liek[Y ](X). Denote the least com-

mon multiple of fY and gY in [Y ] by L = lcm(fY , gY ).

If gX is a subword of fX , i.e., fX = agXb for some a, b ∈ X∗, then the polynomial

C1〈f, g〉w =L

fYf − L

gY[agb]g

is called the inclusion composition of f and g with respect to w, where w = LfX = LagXb.

If a proper prefix of gX is a proper suffix of fX , i.e., fX = aa0, gX = a0b, a, b, a0 6= 1,

then the polynomial

C2〈f, g〉w =L

fY[fb]f −

L

gY[ag]g

is called the intersection composition of f and g with respect to w, where w = LfXb =

LagX .

If the greatest common divisor of fY and gY in [Y ] is not 1, then for any a, b, c ∈ X∗

such that w = LafXbgXc ∈ TA, the polynomial

C3〈f, g〉w =L

fY[afbgXc]f −

L

gY[afXbgc]g

is called the external composition of f and g with respect to w.

If fY 6= 1, then for any special normal f -word [afb]f , a, b ∈ X∗, the polynomial

C4〈f〉w = [afXb][afb]f

is called the multiplication composition of f with respect to w, where w = afXbafb.

Immediately, we have that Ci〈−〉w ≺ w, i ∈ 1, 2, 3, 4.

Remarks

1) When Y = ∅, there are no external and multiplication compositions. This is the case

of Shirshov’s compositions over a field.

2) In the cases of C1 and C2, the corresponding w ∈ TA by the property of ALSW’s, but

in the case of C4, w 6∈ TA.

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3) For any fixed f, g, there are finitely many compositions C1〈f, g〉w, C2〈f, g〉w, but

infinitely many C3〈f, g〉w, C4〈f〉w.

Definition 6.8 Given a k-monic subset S ⊂ Liek[Y ](X) and w ∈ [Y ]X∗ (not nec-

essary in TA), an element h ∈ Liek[Y ](X) is called trivial modulo (S,w), denoted by

h ≡ 0 mod(S,w), if h can be presented as a k[Y ]-linear combination of special normal

S-words with leading words less than w, i.e., h =∑

i αiβi[aisibi]si, where αi ∈ k, βi ∈ [Y ],

ai, bi ∈ X∗, si ∈ S, and βiaisibi ≺ w.

In general, for p, q ∈ Liek[Y ](X), we write p ≡ q mod(S,w) if p− q ≡ 0 mod(S,w).

S is a Grobner-Shirshov basis in Liek[Y ](X) if all the possible compositions of ele-

ments in S are trivial modulo S and corresponding w.

If a subset S of Liek[Y ](X) is not a Grobner-Shirshov basis then one can add all

nontrivial compositions of polynomials of S to S. Continuing this process repeatedly,

we finally obtain a Grobner-Shirshov basis Sc that contains S. Such a process is called

Shirshov’s algorithm. Sc is called Grobner-Shirshov complement of S.

Lemma 6.9 Let f be a k-monic polynomial in Liek[Y ](X). If fY = 1 or f = gf ′ where

g ∈ k[Y ] and f ′ ∈ Liek(X), then for any special normal f -word [afb]f , a, b ∈ X∗,

(u)f = [afXb][afb]f has a presentation:

(u)f = [afXb][afb]f =∑

buicf(u)f

αiβibuicf

where αi ∈ k, βi ∈ [Y ].

Proof. Case 1. fY = 1, i.e., f = fX . By Lemma 6.3 and since ≺ is monomial, we have

[afb] = [afb]f −∑

βivi≺afb αiβi[vi], where αi ∈ k, βi ∈ [Y ], vi ∈ ALSW (X). Then

(u)f = [afb][afb]f = [afb]f [afb]f +∑

βivi≺afb

αiβi[afb]f [vi] =∑

βivi≺afb

αiβi[afb]f [vi].

The result follows since vi ≺ afb and each [afb]f [vi] is normal.

Case 2. f = gf ′, i.e., fX = f ′. Then we have

(u)f = [af ′b][afb]f = g([af ′b][af ′b]f ′).

The result follows from Case 1.

The following lemma plays a key role.

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Lemma 6.10 Let S be a k-monic subset of Liek[Y ](X) in which each multiplication com-

position is trivial. Then for any normal s-word bucs = (asb) and w = asb = bucs, where

a, b ∈ X∗, we have

bucs ≡ [asb]s mod(S,w).

Proof. For w = s the lemma is clear.

For w 6= s, since either bucs = (asb) = [a1](a2sb) or bucs = (asb) = (asb1)[b2], there

are two cases to consider.

Let

δ(asb) =

|a1| if (asb) = [a1](a2sb),

s-length of (asb1) if (asb) = (asb1)[b2].

The proof will be proceeding by induction on (w, δ(asb)), where (w′,m′) < (w,m)⇔ w ≺w′ or w = w′, m′ < m (w,w′ ∈ TA,m,m

′ ∈ N).

Case 1. bucs = (asb) = [a1](a2sb), where a1 > a2sXb, a = a1a2 and (a2sb) is normal

s-word. In this case, (w, δ(asb)) = (w, |a1|).Since w = asb = a1a2sb a2sb, by induction, we may assume that (a2sb) = [a2sb]s +∑

αiβi[cisidi]si, where βicisidi ≺ a2sb, a1, a2, ci, di ∈ X∗, si ∈ S, αi ∈ k and βi ∈ [Y ].

Thus,

bucs = (asb) = [a1][a2sb]s +∑

αiβi[a1][cisidi]si.

Consider the term [a1][cisidi]si.

If a1 > cisiXdi, then [a1][cisidi]si

is normal s-word with a1cisidi ≺ w. Note that

βia1cisidi ≺ w, then by induction, βi[a1][cisidi]si≡ 0 mod(S,w).

If a1 < cisiXdi, then [a1][cisidi]si

= −[cisidi]si[a1] and [cisidi]si

[a1] is normal s-word

with βicisidia1 ≺ βia2sba1 ≺ βia1a2sb = w.

If a1 = cisiXdi, then there are two possibilities. For si

Y = 1, by Lemma 6.9 and by

induction on w we have βi[a1][cisidi]si≡ 0 mod(S,w). For si

Y 6= 1, [a1][cisidi]siis the

multiplication composition, then by assumption, it is trivial mod(S,w).

This shows that in any case, βi[a1][cisidi]siis a linear combination of normal s-words

with leading words less than w, i.e., βi[a1][cisidi]si≡ 0 mod(S,w) for all i.

Therefore, we may assume that bucs = (asb) = [a1][a2sb]s and a1 > wX > a2sXb.

If either |a1| = 1 or [a1] = [[a11][a12]] and a12 ≤ a2sXb, then bucs = [a1][a2sb]s is

already a normal s-word, i.e., bucs = [a1][a2sb]s = [a1a2sb]s = [asb]s.

If [a1] = [[a11][a12]] and a12 > a2sXb, then

bucs = [a1][a2sb]s = [[a11][a12]][a2sb]s = [a11][[a12][a2sb]s] + [[a11][a2sb]s][a12].

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Let us consider the second summand [[a11][a2sb]s][a12]. Then by induction on w and

by noting that [a11][a2sb]s is normal, we may assume that [a11][a2sb]s =∑αiβi[cisidi]si

,

where βicisidi a11a2sb, si ∈ S, αi ∈ k, βi ∈ [Y ], ci, di ∈ X∗. Thus,

[[a11][a2sb]s][a12] =∑

αiβi[cisidi]si[a12],

where a11 > a12 > a2sXb, w = a11a12a2sb.

If a12 < cisiXdi, then [cisidi]si

[a12] is normal with w′ = βicisidia12 βia11a2sba12 ≺w. By induction, βi[cisidi]si

[a12] ≡ 0 mod(S,w).

If a12 > cisiXdi, then [cisidi]si

[a12] = −[a12][cisidi]siand [a12][cisidi]si

is normal with

w′ = βia12cisidi βia12a11a2sb ≺ w. Again we can apply the induction.

If a12 = cisiXdi, then as discussed above, it is either the case in Lemma 6.9 or the

multiplication composition and each is trivial mod(S,w).

These show that [[a11][a2sb]s][a12] ≡ 0 mod(S,w).

Hence,

bucs ≡ [a11][[a12][a2sb]s] mod(S,w).

where a11 > a12 > a2sXb.

Noting that [a11][[a12][a2sb]s] is normal and now (w, δ[a11][[a12][a2sb]s]) = (w, |a11|) <(w, |a1|), the result follows by induction.

Case 2. bucs = (asb) = (asb1)[b2] where asXb1 > b2, b = b1b2 and (asb1) is normal

s-word. In this case, (w, δ(asb)) = (w,m) where m is the s-length of (asb1).

By induction on w, we may assume that

bucs = (asb) = [asb1]s[b2] +∑

αiβi[cisidi]si[b2].

where βicisidi ≺ asb1, si ∈ S, αi ∈ k, βi ∈ [Y ], ci, di ∈ X∗.

Consider the term βi[cisidi]si[b2] for each i.

If b2 < cisiXdi, then [cisidi]si

[b2] is normal s-word with βicisidib2 ≺ w.

If b2 > cisiXdi, then [cisidi]si

[b2] = −[b2][cisidi]siand [b2][cisidi]si

is normal s-word

with βib2cisidi ≺ βib2asb1 ≺ βiasb1b2 = w.

If b2 = cisiXdi, then as above, by Lemma 6.9 and induction on w or by assumption,

βi[cisidi]si[b2] ≡ 0 mod(S,w).

These show that for each i, βi[cisidi]si[b2] ≡ 0 mod(S,w).

Therefore, we may assume that bucs = (asb) = [asb1]s[b2], a, b ∈ X∗, where b = b1b2

and asXb1 > b2.

Noting that for [asb1]s = s or [asb1]s = [a1][a2sb1]s with a2sXb1 ≤ b2 or [asb1]s =

[asb11]s[b12] with b12 ≤ b2, bucs is already normal. Now we consider the remained cases.

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Case 2.1. Let [asb1]s = [a1][a2sb1]s with a1 > a1a2sXb1 > a2s

Xb1 > b2. Then we have

bucs = [[a1][a2sb1]s][b2] = [[a1][b2]][a2sb1]s + [a1][[a2sb1]s[b2]].

We consider the term [[a1][b2]][a2sb1]s.

By noting that a1 > b2, we may assume that [a1][b2] =∑

uia1b2αi[ui] where αi ∈

k, ui ∈ ALSW (X). We will prove that [ui][a2sb1]s ≡ 0 mod(S,w).

If ui > a2sXb1, then [ui][a2sb1]s is normal s-word with w′ = uia2sb1 a1b2a2sb1 ≺

w = a1a2sb1b2.

If ui < a2sXb1, then [ui][a2sb1]s = −[a2sb1]s[ui] and [a2sb1]s[ui] is normal s-word with

w′ = a2sb1ui a2sb1a1b2 ≺ w, since a1a2sb1 is an ALSW.

If ui = a2sXb1, then as above, by Lemma 6.9 and induction on w or by assumption,

[ui][a2sb1]s ≡ 0 mod(S,w).

This shows that

bucs ≡ [a1][[a2sb1]s[b2]] mod(S,w).

By noting that a1 > a2sXb1 > b2, the result now follows from the Case 1.

Case 2.2. Let [asb1]s = [asb11]s[b12] with asXb11 > asXb11b12 > b12 > b2. Then we

have

bucs = [[asb11]s[b12]][b2] = [[asb11]s[b2]][b12] + [asb11]s[[b12][b2]].

Let us first deal with [[asb11]s[b2]][b12]. Since asb11b2 < asb11b12, we may apply induc-

tion on w and have that

[[asb11]s[b2]][b12] =∑

αiβi[cisidi]si[b12],

where βicisidi asb11b2, w = asb11b12b2.

If b12 < cisiXdi, then [cisidi]si

[b12] is normal s-word with w′ = βicisidib12 ≺ w.

If b12 > cisiXdi, then [cisidi]si

[b12] = −[b12][cisidi]siand [b12][cisidi]si

is a normal

s-word with w′ = βib12cisidi βib12asb11b2 ≺ asb11b12b2 = w.

If b12 = cisiXdi, then as above, by Lemma 6.9 and induction on w or by assumption,

βi[cisidi]si[b12] ≡ 0 mod(S,w).

These show that

bucs ≡ [asb11]s[[b12][b2]] mod(S,w).

Let [b12][b2] = [b12b2] +∑

ui≺a1b2αi[ui] where αi ∈ k, ui ∈ ALSW (X). By noting

that asXb11 > b12b2, we have [asb11]s[ui] ≡ 0 mod(S,w) for any i. Therefore,

bucs ≡ [asb11]s[b12b2] mod(S,w).

97

Noting that [asb11]s[b12b2] is normal and now (w, δ[asb11]s[b12b2]) < (w, δ[asb1]s[b2]), the

result follows by induction.

The proof is complete.

Lemma 6.11 Let S be a k-monic subset of Liek[Y ](X) in which each multiplication com-

position is trivial. Then any element of the k[Y ]-ideal generated by S can be written as a

k[Y ]-linear combination of normal S-words.

Proof. Note that for any h ∈ Id(S), h can be presented by a k[Y ]-linear combination

of S-words of the form

(u)s = [c1][c2] · · · [ck]s[d1][d2] · · · [dl] (6.1)

with some placement of parentheses, where s ∈ S, cj, dj ∈ ALSW (X), k, l ≥ 0. By

Lemma 6.10 it suffices to prove that (6.1) is a linear combination of normal S-words. We

will prove the result by induction on k + l. It is trivial when k + l = 0, i.e., (u)s = s.

Suppose that the result holds for k + l = n. Now let us consider

(u)s = [cn+1]([c1][c2] · · · [ck]s[d1][d2] · · · [dn−k]) = [cn+1](v)s.

By inductive hypothesis, we may assume without loss of generality that (v)s is a normal

s-word, i.e., (v)s = bvcs = (csd) where csd ∈ TA, c, d ∈ X∗. If cn+1 > csXd, then (u)s is

normal. If cn+1 < csXd then (u)s = −bvcs[cn+1] where bvcs[cn+1] is normal. If cn+1 = csXd

then by Lemma 6.10, (u)s = [cn+1](csd) ≡ [cn+1][csd]s. Now the result follows from the

multiplication composition and Lemma 6.9.

Lemma 6.12 Let S be a k-monic subset of Liek[Y ](X) in which each multiplication com-

position is trivial. Then for any normal S-word basbcs = [a1][a2] · · · [ak]bvcs[b1][b2] · · · [bl]with some placement of parentheses, the three following S-words are linear combinations

of normal S-words with the leading words less than asb:

(i) w1 = basbcs|[ai] 7→[c] where c ≺ ai;

(ii) w2 = basbcs|[bj ] 7→[d] where d ≺ bj;

(iii) w3 = basbcs|bvcs 7→bv′cs where bv′cs ≺ bvcs.

Proof. We first prove (iii). For k + l = 1, for example, basbcs = bvcs[b1], it is easy to

see that the result follows from Lemmas 6.11 and 6.9 since either bv′cs[b1] or [b1]bv′cs is

98

normal or w3 is the multiplication composition. Now the result follows by induction on

k + l.

We now prove (i), and (ii) is similar. For k+ l = 1, basbcs = [a1]bvcs and then w1 =

[c]bvcs. Then either bvcs[c] or [c]bvcs is normal or w1 is equivalent to the multiplication

composition with respect to w = bvcX

s bvcs. Again by Lemmas 6.11 and 6.9, the result

holds. For k + l ≥ 2, it follows from (iii).

Let s1, s2 ∈ Liek[Y ](X) be two k-monic polynomials in Liek[Y ](X). If asX1 bs

X2 c ∈

ALSW (X) for some a, b, c ∈ X∗, then by Lemma 6.2, there exits a bracketing way

[asX1 bs

X2 c]sX

1 ,sX2

such that [asX1 bs

X2 c]sX

1 ,sX2

= asX1 bs

X2 c. Denote

[as1bs2c]s1,s2 = sY2 [asX

1 bsX2 c]sX

1 ,sX2|[sX

1 ] 7→s1,

[as1bs2c]s1,s2 = sY1 [asX

1 bsX2 c]sX

1 ,sX2|[sX

2 ] 7→s2,

[as1bs2c]s1,s2 = [asX1 bs

X2 c]sX

1 ,sX2|[sX

1 ] 7→s1,[sX2 ] 7→s2

.

Thus, the leading words of the above three polynomials are as1bs2c = sY1 s

Y2 as

X1 bs

X2 c.

The following lemma is also essential.

Lemma 6.13 Let S be a Grobner-Shirshov basis in Liek[Y ](X). For any s1, s2 ∈ S, β1, β2 ∈[Y ], a1, a2, b1, b2 ∈ X∗ such that w = β1a1s1b1 = β2a2s2b2 ∈ TA, we have

β1[a1s1b1]s1 ≡ β2[a2s2b2]s2 mod(S,w).

Proof. Let L be the least common multiple of sY1 and sY

2 . Then wY = β1sY1 = β2s

Y2 = Lt

for some t ∈ [Y ], wX = a1sX1 b1 = a2s

X2 b2 and

β1[a1s1b1]s1 − β2[a2s2b2]s2 = t(L

sY1

[a1s1b1]s1 −L

sY2

[a2s2b2]s2).

Consider the first case in which sX2 is a subword of b1, i.e., wX = a1s

X1 as

X2 b2 for some

a ∈ X∗ such that b1 = asX2 b2 and a2 = a1s

X1 a. Then

β1[a1s1b1]s1 − β2[a2s2b2]s2

= t(L

sY1

[a1s1asX2 b2]s1 −

L

sY2

[a1sX1 as2b2]s2)

= tC3〈s1, s2〉w′

if L 6= sY1 s

Y2 , where w′ = LwX . Since S is a Grobner-Shirshov basis, C3〈s1, s2〉 ≡

0 mod(S, LwX). The result follows from w = tLwX = tw′.

99

Suppose that L = sY1 s

Y2 . By noting that 1

sY1[a1s1as2b2]s1,s2 and 1

sY2[a1s1as2b2]s1,s2 are

normal, by Lemma 6.10 we have

[a1s1as2b2]s1,s2 ≡ sY2 [a1s1as

X2 b2]s1 mod(S,w′),

[a1s1as2b2]s1,s2 ≡ sY1 [a1s

X1 as2b2]s2 mod(S,w′).

Thus, by Lemma 6.12, we have

β1[a1s1b1]s1 − β2[a2s2b2]s2

= t(sY2 [a1s1as

X2 b2]s1 − sY

1 [a1sX1 as2b2]s2)

= t((sY2 [a1s1as

X2 b2]s1 − [a1s1as2b2]s1,s2) + ([a1s1as2b2]s1,s2 − [a1s1as2b2]s1,s2)

−([a1s1as2b2]s1,s2 − [a1s1as2b2]s1,s2)− (sY1 [a1s

X1 as2b2]s2 − [a1s1as2b2]s1,s2))

= t((sY1 [a1s1as

X2 b2]s1 − [a1s1as2b2]s1,s2) + [a1(s1 − [s1])as2b2]s1,s2

−[a1s1a(s2 − [s2])b2]s1,s2 − (sY1 [a1s

X1 as2b2]s2 − [a1s1as2b2]s1,s2))

≡ 0 mod(S,w).

Second, if sX2 is a subword of sX

1 , i.e., sX1 = asX

2 b for some a, b ∈ X∗, then [a2s2b2]s2 =

[a1as2bb1]s2 . Let w′ = LsX1 . Thus, by noting that [a1[as2b]s2b1] is normal and by Lemmas

6.10 and 6.12,

β1[a1s1b1]s1 − β2[a2s2b2]s2

= t(L

sY1

[a1s1b1]s1 −L

sY2

[a1as2bb1]s2)

= t(L

sY1

[a1s1b1]s1 −L

sY2

[a1s1b1]s1|s1 7→[as2b]s2)− L

sY2

([a1as2bb1]s2 − [a1s1b1]s1|s1 7→[as2b]s2)

= t[a1(L

sY1

s1 −L

sY2

[as2b]s2)b1]−L

sY2

([a1as2bb1]s2 − [aX1 [as2b]s2b1])

= t[a1C1〈s1, s2〉w′b1]−L

sY2

([a1as2bb1]s2 − [a1[as2b]s2b1])

≡ 0 mod(S,w).

One more case is possible: A proper suffix of sX1 is a proper prefix of sX

2 , i.e., sX1 = ab

and sX2 = bc for some a, b, c ∈ X∗ and b 6= 1. Then abc is an ALSW. Let w′ = Labc. Then

by Lemmas 6.10 and 6.12, we have

β1[a1s1b1]s1 − β2[a2s2b2]s2

= t(L

sY1

[a1s1cb2]s1 −L

sY2

[a1as2b2]s2)

= tL

sY1

([a1s1cb2]s1 − [a1[s1c]s1b2])− tL

sY2

([a1as2b2]s2 − [a1[as2]s2b2])

+t([a1C2〈s1, s2〉w′b2]

≡ 0 mod(S,w).

100

The proof is complete.

Theorem 6.14 (CD-lemma for Lie algebras over a commutative algebra) Let S ⊂ Liek[Y ](X)

be a nonempty set of k-monic polynomials and Id(S) be the k[Y ]-ideal of Liek[Y ](X) gen-

erated by S. Then the following statements are equivalent.

(i) S is a Grobner-Shirshov basis in Liek[Y ](X).

(ii) f ∈ Id(S)⇒ f = βasb ∈ TA for some s ∈ S, β ∈ [Y ] and a, b ∈ X∗.

(iii) Irr(S) = [u] | [u] ∈ TN , u 6= βasb, for any s ∈ S, β ∈ [Y ], a, b ∈ X∗ is a k-linear

basis for Liek[Y ](X|S) = Liek[Y ](X)/Id(S).

Proof. (i) ⇒ (ii). Let S be a Grobner-Shirshov basis and 0 6= f ∈ Id(S). Then by

Lemma 6.11 f has an expression f =∑αiβi[aisibi]si

, where αi ∈ k, βi ∈ [Y ], ai, bi ∈X∗, si ∈ S. Denote wi = βi[aisibi]si

, i = 1, 2, . . . . Then wi = βiaisibi. We may assume

without loss of generality that

w1 = w2 = · · · = wl wl+1 wl+2 · · ·

for some l ≥ 1.

The claim of the theorem is obvious if l = 1.

Now suppose that l > 1. Then β1a1s1b1 = w1 = w2 = β2a2s2b2. By Lemma 6.13,

α1β1[a1s1b1]s1 + α2β2[a2s2b2]s2

= (α1 + α2)β1[a1s1b1]s1 + α2(β2[a2s2b2]s2 − β1[a1s1b1]s1)

≡ (α1 + α2)β1[a1s1b1]s1 mod(S,w1).

Therefore, if α1 + α2 6= 0 or l > 2, then the result follows from the induction on l.

For the case α1 + α2 = 0 and l = 2, we use the induction on w1. Now the result follows.

(ii)⇒ (iii). For any f ∈ Liek[Y ](X), we have

f =∑

βi[aisibi]sif

αiβi[aisibi]si+∑

[uj ]f

α′j[uj],

where αi, α′j ∈ k, βi ∈ [Y ], [uj] ∈ Irr(S) and si ∈ S. Therefore, the set Irr(S) generates

the algebra Liek[Y ](X)/Id(S).

On the other hand, suppose that h =∑αi[ui] = 0 in Liek[Y ](X)/Id(S), where

αi ∈ k, [ui] ∈ Irr(S). This means that h ∈ Id(S). Then all αi must be equal to zero.

Otherwise, h = uj for some j which contradicts (ii).

101

(iii)⇒ (i). For any f, g ∈ S, we have

Cτ (f, g)w =∑

βi[aisibi]si≺w

αiβi[aisibi]si+∑

[uj ]≺w

α′j[uj].

For τ = 1, 2, 3, 4, since Cτ (f, g)w ∈ Id(S) and by (iii), we have

Cτ (f, g)w =∑

βi[aisibi]si≺w

αiβi[aisibi]si.

Therefore, S is a Grobner-Shirshov basis.

§6.3 Applications

In this section, all algebras (Lie or associative) are understood to be taken over an

associative and commutative k-algebra K with identity and all associative algebras are

assumed to have identity.

Let L be an arbitrary Lie K-algebra which is presented by generators X and defining

relations S, L = LieK(X|S). Let K have a presentation by generators Y and defining

relations R, K = k[Y |R]. Let Y and X be deg-lex orderings on [Y ] and X∗ respectively.

Let RX = rx|r ∈ R, x ∈ X. Then as k[Y ]-algebras,

L = Liek[Y |R](X|S) ∼= Liek[Y ](X|S,RX).

The Poincare-Birkhoff-Witt theorem cannot be generalized to Lie algebras over an

arbitrary ring. This implies that not any Lie algebra over a commutative algebra has

a faithful representation in an associative algebra over the same commutative algebra.

Following P.M. Cohn, a Lie algebra with the PBW property is said to be “special0 . The

first non-special example was given by A.I. Shirshov ([196], see also [204]), and he also

formulated that if no nonzero element of K annihilates an element of the center of LK ,

then a faithful representation always exists. Another classical non-special example was

given by P. Cartier [67]. In the same paper, he proved that each Lie algebra over Dedekind

domain is special. In both examples the Lie algebras are taken over commutative algebras

over GF(2). Shirshov and Cartier used ad hoc methods to prove that some elements of

corresponding Lie algebras are not zero though they are zero in the universal enveloping

algebras. P.M. Cohn [93] proved that any Lie algebra over Kk, where chk = 0, is special.

Also he claimed that he gave an example of non-special Lie algebra over a truncated

polynomial algebra over a filed of characteristic p > 0. But he did not give a proof.

Here we find Grobner-Shirshov bases of Shirshov and Cartier’s Lie algebras and

then use Theorem 6.14 to get the results and we give proof for P.M. Cohn’s example of

102

characteristics 2, 3 and 5. We present an algorithm that one can check for any p, whether

Cohn’s conjecture is valid.

Note that if L = LieK(X|S), then the universal enveloping algebra of L is UK(L) =

K〈X|S(−)〉 where S(−) is just S but substituting all [u, v] by uv − vu.

Example 6.15 (Shirshov [196]) Let the field k = GF (2) and K = k[Y |R], where

Y = yi, i = 0, 1, 2, 3, R = y0yi = yi (i = 0, 1, 2, 3), yiyj = 0 (i, j 6= 0).

Let L = LieK(X|S1, S2), where X = xi, 1 ≤ i ≤ 13, S1 consists of the following relations

[x2, x1] = x11, [x3, x1] = x13, [x3, x2] = x12,

[x5, x3] = [x6, x2] = [x8, x1] = x10,

[xi, xj] = 0 (for any other i > j),

and S2 consists of the following relations

y0xi = xi (i = 1, 2, . . . , 13),

x4 = y1x1, x5 = y2x1, x5 = y1x2, x6 = y3x1, x6 = y1x3,

x7 = y2x2, x8 = y3x2, x8 = y2x3, x9 = y3x3,

y3x11 = x10, y1x12 = x10, y2x13 = x10,

y1xk = 0 (k = 4, 5, . . . , 11, 13), y2xt = 0 (t = 4, 5, . . . , 12), y3xl = 0 (l = 4, 5, . . . , 10, 12, 13).

Then L is not special.

Proof. L = LieK(X|S1, S2) = Liek[Y ](X|S1, S2, RX). We order Y and X by yi >

yj if i > j and xi > xj if i > j respectively. It is easy to see that for the ordering on [Y ]X∗ as before, S = S1 ∪ S2 ∪ RX ∪ y1x2 = y2x1, y1x3 = y3x1, y2x3 = y3x2 is a

Grobner-Shirshov basis in Liek[Y ](X). Since x10 ∈ Irr(S) and Irr(S) is a k-linear basis

of L by Theorem 6.14, x10 6= 0 in L.

On the other hand, the universal enveloping algebra of L has a presentation:

UK(L) = K〈X|S(−)1 , S2〉 ∼= k[Y ]〈X|S(−)

1 , S2, RX〉,

where S(−)1 is just S1 but substituting all [uv] by uv − vu.

But the Grobner-Shirshov complement (see Mikhalev-Zolotyhk [165]) of S(−)1 ∪ S2 ∪

RX in k[Y ]〈X〉 is

Sc = S(−)1 ∪ S2 ∪RX ∪ y1x2 = y2x1, y1x3 = y3x1, y2x3 = y3x2, x10 = 0.

Thus, L is not special.

103

Example 6.16 (Cartier [67]) Let k = GF (2), K = k[y1, y2, y3|y2i = 0, i = 1, 2, 3] and

L = LieK(X|S), where X = xij, 1 ≤ i ≤ j ≤ 3 and

S = [xii, xjj] = xji (i > j), [xij, xkl] = 0 (otherwise), y3x33 = y2x22 + y1x11.

Then L is not special.

Proof. Let Y = y1, y2, y3. Then

L = LieK(X|S) ∼= Liek[Y ](X|S, y2i xkl = 0 (∀i, k, l)).

Let yi > yj if i > j and xij > xkl if (i, j) >lex (k, l) respectively. It is easy to see

that for the ordering on [Y ]X∗ as before, S ′ = S ∪ y2i xkl = 0 (∀i, k, l) ∪ S1 is a

Grobner-Shirshov basis in Liek[Y ](X), where S1 consists of the following relations

y3x23 = y1x12, y3x13 = y2x12, y2x23 = y1x13, y3y2x22 = y3y1x11,

y3y1x12 = 0, y3y2x12 = 0, y3y2y1x11 = 0, y2y1x13 = 0.

The universal enveloping algebra of L has a presentation:

UK(L) = K〈X|S(−)〉 ∼= k[Y ]〈X|S(−), y2i xkl = 0 (∀i, k, l)〉.

In UK(L), we have

0 = y23x

233 = (y2x22 + y1x11)

2 = y22x

222 + y2

1x211 + y2y1[x22, x11] = y2y1x12.

On the other hand, since y2y1x12 ∈ Irr(S ′), y2y1x12 6= 0 in L. Thus, L is not special.

Cohn conjecture (Cohn [93]) Let K = k[y1, y2, y3|ypi = 0, i = 1, 2, 3] be the algebra

of truncated polynomials over a field k of characteristic p > 0. Let

Lp = LieK(x1, x2, x3 | y3x3 = y2x2 + y1x1).

Then Lp is not special. We call Lp the Cohn’s Lie algebra.

Remark In UK(Lp) we have

0 = (y3x3)p = (y2x2)

p + Λp(y2x2, y1x1) + (y1x1)p = Λp(y2x2, y1x1),

where Λp is a Jacobson-Zassenhaus Lie polynomial. P.M. Cohn conjectured that Λp(y2x2, y1x1) 6=0 in Lp.

Theorem 6.17 Cohn’s Lie algebras L2, L3 and L5 are not special.

104

Proof. Let Y = y1, y2, y3, X = x1, x2, x3 and S = y3x3 = y2x2 + y1x1, ypi xj =

0, 1 ≤ i, j ≤ 3. Then Lp∼= Liek[Y ](X|S) and UK(Lp) ∼= k[Y ]〈X|S〉. Suppose that Sc is

a Grobner-Shirshov complement of S in Liek[Y ](X). Let SX

p ⊂ Lp be the set of all the

elements of Sc whose X-degrees do not exceed p.

First, we consider p = 2 and prove the element Λ2 = [y2x2, y1x1] = y2y1[x2x1] 6= 0 in

L2.

Then by Shirshov’s algorithm we have that SX2 consists of the following relations

y3x3 = y2x2 + y1x1, y2i xj = 0 (1 ≤ i, j ≤ 3), y3y2x2 = y3y1x1, y3y2y1x1 = 0,

y2[x3x2] = y1[x3x1], y3y1[x2x1] = 0, y2y1[x3x1] = 0.

Thus, Λ2 is in the k-linear basis Irr(Sc) of L2.

Now, by the above remark, L2 is not special.

Second, we consider p = 3 and prove the element Λ3 = y22y1[x2x2x1]+y2y

21[x2x1x1] 6= 0

in L3.

Then again by Shirshov’s algorithm, SX3 consists of the following relations

y3x3 = y2x2 + y1x1, y3i xj = 0 (1 ≤ i, j ≤ 3), y2

3y2x2 = y23y1x1, y

23y

22y1x1 = 0,

y2[x3x2] = −y1[x3x1], y23y1[x2x1] = 0, y2

2y1[x3x1] = 0,

y3y22[x2x2x1] = y3y2y1[x2x1x1], y3y

22y1[x2x1x1] = 0, y3y2y1[x2x2x1] = y3y

21[x2x1x1].

Thus, y22y1[x2x2x1], y2y

21[x2x1x1] ∈ Irr(Sc), which implies Λ3 6= 0 in L3.

Third, let p = 5. Again by Shirshov’s algorithm, SX5 consists of the following

relations

1) y3x3 = y2x2 + y1x1,

2) y5i xj = 0, 1 ≤ i, j ≤ 3,

3) y43y2x2 = −y4

3y1x1,

4) y43y

42y1x1 = 0,

5) y2[x3x2] = −y1[x3x1],

6) y43y1[x2x1] = 0,

7) y42y1[x3x1] = 0,

8) y33y

22[x2x2x1] = y3

3y2y1[x2x1x1],

9) y33y

42y1[x2x1x1] = 0,

10) y33y2y1[x2x2x1] = y3

3y21[x2x1x1],

105

11) y1[x3x2x3x1] = 0,

12) y1[x3x1x2x1] = 0,

13) y1[x3x2x2x1] = −y1[x3x2x1x2],

14) y2[x3x1x2x1] = 0,

15) y23y

32[x2x2x2x1] = 2y2

3y22y1[x2x2x1x1]− y2

3y2y21[x2x1x1x1],

16) y33y

32y

21[x2x1x1x1] = 0,

17) y23y

22y1[x2x2x2x1] = 2y2

3y2y21[x2x2x1x1]− y2

3y31[x2x1x1x1],

18) y23y

42y

21[x2x1x1x1] = 0,

19) y23y

42y1[x2x2x1x1] =

1

2y2

3y32y

21[x2x1x1x1],

20) y33y

21[x2x2x1x2x1] = 0,

21) y33y2y1[x2x1x2x1x1] = 0,

22) y33y

21[x2x1x2x1x1] = 0,

23) y33y

22[x2x1x2x1x1] = 0,

24) y23y

22y1[x2x2x1x2x1] = −y2

3y2y21[x2x1x2x1x1],

25) y23y2y

21[x2x2x1x2x1] = −y2

3y31[x2x1x2x1x1],

26) y23y

42y

21[x2x1x2x1x1] = 0,

27) y3y42[x2x2x2x2x1] = 3y3y

32y1[x2x2x2x1x1]− y3y

32y1[x2x2x1x2x1]− 3y3y

22y

21[x2x2x1x1x1]

−2y3y22y

21[x2x1x2x1x1] + y3y2y

31[x2x1x1x1x1],

28) y3y32y1[x2x2x2x2x1] = 3y3y

22y

21[x2x2x2x1x1]− y3y

22y

21[x2x2x1x2x1]− 3y3y2y

31[x2x2x1x1x1]

−2y3y2y31[x2x1x2x1x1] + y3y

41[x2x1x1x1x1],

29) y3y42y

31[x2x1x1x1x1] = 0,

30) y23y

32y

31[x2x1x1x1x1] = 0,

31) y3y42y

21[x2x2x1x1x1] = −2

3y3y

42y

21[x2x1x2x1x1] +

1

3y3y

32y

31[x2x1x1x1x1],

32) y3y42y1[x2x2x2x1x1] =

1

3y3y

42y1[x2x2x1x2x1] + y3y

32y

21[x2x2x1x1x1]

+2

3y3y

32y

21[x2x1x2x1x1]−

1

3y3y

22y

31[x2x1x1x1x1],

33) y32y

21[x3x3x1x3x1] = 0,

34) y32y

21[x3x1x3x1x1] = 0,

35) y33y

22y

31[x2x1x1x1x1] = 0,

36) y23y

32y

21[x2x2x1x1x1] = −2

3y2

3y32y

21[x2x1x2x1x1] +

2

3y2

3y22y

31[x2x1x1x1x1].

Thus, Λ5(y2x2, y1x1) = y42y1[x2x2x2x2x1] ∈ Irr(Sc), which implies Λ5 6= 0 in L5.

106

Remark Note that the Jacobson-Zassenhaus Lie polynomial Λp(y2x2, y1x1) is of X-degree

p. Then Λp(y2x2, y1x1) ∈ Irr(Sc) if and only if Λp(y2x2, y1x1) ∈ Irr(SXp). Since the defin-

ing relation of Lp is homogenous on X, SXp is a finite set. By Shirshov’s algorithm, one

can compute SXp for Lp.

Now we give some examples which are special Lie algebras.

Lemma 6.18 Suppose that f and g are two polynomials in Liek[Y ](X) such that f is

k[Y ]-monic and g = rx, where r ∈ k[Y ] and x ∈ X, is k-monic. Then each inclusion

composition of f and g is trivial modulo f ∪ rX.

Proof. Suppose that f = [axb] for some a, b ∈ X∗, f = f + f ′ and g = rx + r′x. Then

w = raxb and

C1〈f, g〉w = rf − [a[rx]b]rx

= rf ′ − r′[axb]

= rf ′ − r′f

≡ 0 mod(f ∪ rX,w).

Now we give other applications.

Theorem 6.19 Suppose that S is a finite homogeneous subset of Liek(X). Then the word

problem of LieK(X|S) is solvable for any finitely generated commutative k-algebra K.

Proof. Let Sc be a Grobner-Shirshov complement of S in Liek(X). Clearly, Sc consists

of homogeneous elements in Liek(X) since the compositions of homogeneous elements are

homogeneous. Since K is finitely generated commutative k-algebra, we may assume that

K = k[Y |R] with R a finite Grobner-Shirshov basis in k[Y ]. By Lemma 6.18, Sc ∪ RXis a Grobner-Shirshov basis in Liek[Y ](X). For a given f ∈ LieK(X), it is obvious that

after a finite number of steps one can write down all the elements of Sc whose X-degrees

do not exceed the degree of fX . Denote the set of such elements by SfX . Then SfX is a

finite set. By Theorem 6.14, the result follows.

Theorem 6.20 Every finitely or countably generated Lie K-algebra can be embedded into

a two-generated Lie K-algebra, where K is an arbitrary commutative k-algebra.

107

Proof. Let K = k[Y |R] and L = LieK(X|S) where X = xi, i ∈ I and I is a subset

of the set of nature numbers. Without loss of generality, we may assume that with the

ordering on [Y ]X∗ as before, S ∪RX is a Grobner-Shirshov basis in Liek[Y ](X).

Consider the algebra L′ = Liek[Y ](X, a, b|S ′) where S ′ = S∪RX∪Ra, b∪[aabiab]−xi, i ∈ I.

Clearly, L′ is a Lie K-algebra generated by a, b. Thus, in order to prove the theorem,

by using our Theorem 6.14, it suffices to show that with the ordering on [Y ](X∪a, b)∗

as before, where a b xi, xi ∈ X, S ′ is a Grobner-Shirshov basis in Liek[Y ](X, a, b).

It is clear that all the possible compositions of multiplication, intersection and inclu-

sion are trivial. We only check the external compositions of some f ∈ S and ra ∈ Ra:Let w = Lu1f

Xu2au3 where L = L(fY , r) and u1fXu2au3 ∈ ALSW (X, a, b). Then

C3〈f, ra〉w

=L

fY1

[u1fu2au3]f −L

r[u1f

Xu2(ra)u3]

= (L

fY1

[u1fu2au3]f − rL

r[u1f

Xu2au3]fX )− (L

r[u1f

Xu2(ra)u3]− rL

r[u1f

Xu2au3]fX )

= ([u1(L

fY1

f)u2au3]f − [u1(rL

rfX)u2au3]fX )− rL

r([u1f

Xu2au3]− [u1fXu2au3]fX )

≡ [u1C3〈f, rx〉w′u2au3] mod(S ′, w)

for some x occurring in fX and w′ = LfX . Since S ∪ RX is a Grobner-Shirshov basis in

Liek[Y ](X), C3〈f, rx〉w′ ≡ 0 mod(S∪RX,w′). Thus by Lemma 6.12, [u1C3〈f, rx〉w′u2au3] ≡0 mod(S ′, w).

108

Chapter 7 Grobner-Shirshov Bases for Pre-Lie

Algebras

§7.1 Composition-Diamond lemma for pre-Lie algebras

Definition 7.1 An algebra A over a field k is called a pre-Lie (or right-symmetric) algebra

if A satisfies (x, y, z) = (x, z, y) for any x, y, z ∈ A, where (x, y, z) = (xy)z − x(yz).

Let X = xi|i ∈ I be a set, X∗ the set of all associative words u in X, X∗∗ the set

of all non-associative words (u) in X, and |(u)| the length of the word (u).

Let I be a well-ordered set. We order X∗∗ by the induction on the lengths of the

words (u) and (v) in X∗∗:

(i) If |((u)(v))| = 2, then (u) = xi > (v) = xj if and only if i > j.

(ii) If |((u)(v))| > 2, then (u) > (v) if and only if one of the following cases holds:

(a) |(u)| > |(v)|.

(b) If |(u)| = |(v)|, (u) = ((u1)(u2)) and (v) = ((v1)(v2)), then (u1) > (v1) or

((u1) = (v1) and (u2) > (v2)).

It is clear that the ordering < on X∗∗ is well-ordered. This ordering is called deg-lex

(degree-lexicographical) ordering and we use this ordering throughout this paper.

We now cite the definition of good words (cf. [193]) by induction on length:

1) xi is a good word for any xi ∈ X.

Suppose that we define good words of length < n.

2) non-associative word ((v)(w)) is called a good word if

(a) both (v) and (w) are good words,

(b) if (v) = ((v1)(v2)), then (v2) ≤ (w).

We denote (u) by [u], if (u) is a good word.

Let W be the set of all good words in the alphabet X and pre-Lie(X) the free pre-Lie

algebra over a field k generated by X. Then W forms a linear basis of the free pre-Lie

algebra pre-Lie(X), see [193].

109

Every nonzero element f in pre-Lie(X) can be uniquely represented as

f = λ1[w1] + λ2[w2] + · · ·+ λn[wn]

where [wi] ∈ W , 0 6= λi ∈ k for all i and [w1] > [w2] > · · · > [wn]. Denote by f the

leading word [w1] of f . f is called monic if the coefficient of f is 1.

For any (w), (w1) ∈ X∗∗, denote by

(w)R(w1) = ((w)(w1)).

The following results are actually proved in [137].

Lemma 7.2 In X∗∗, every good word [w] ∈ W can be uniquely represented as

[w] = xiR[w1]R[w2] · · ·R[wn]

where [wj] ∈ W for all j and [w1] ≤ [w2] ≤ · · · ≤ [wn].

Lemma 7.3 Let [u] and [v] be arbitrary good words and assume that [u] =

xiR[u1]R[u2] · · ·R[un]. Then, in pre-Lie(X),

[u][v] = xiR[u1] · · ·R[us]R[v]R[us+1] · · ·R[un]

where [u1] ≤ · · · ≤ [us] ≤ [v] < [us+1] ≤ · · · ≤ [un] and s 6 n.

By Lemma 7.3, we have

Corollary 7.4 Let [u], [v] ∈ W and [u] = xiR[u1] · · ·R[um−1]R[um] = [u′][um], where

[u′] = xiR[u1] · · ·R[um−1]. If [um] > [v], then, in pre-Lie(X), [u][v] = ([u′][v])[um] and

[u′]([um][v]), [u′]([v][um]) < [u][v].

Lemma 7.5 Let [u], [v], and [w] be arbitrary good words. If [u] < [v] then [w][u] < [w][v]

and [u][w] < [v][w]. It follows that if f, g ∈ pre-Lie(X), then fg = f g.

Definition 7.6 Let S ⊂ kX∗∗ be a set of monic polynomials, s ∈ S and (u) ∈ X∗∗, where

kX∗∗ is the free non-associative algebra over a field k generated by X. We view S as a

new set of letters with S ∩X = ∅. We define S-word (u)s by induction:

(i) (u)s = s is an S-word of S-length 1.

110

(ii) If (u)s is an S-word with S-length k and [v] is a good word with length l, then

(u)s[v] and [v](u)s

are S-words with S-length k + l.

Definition 7.7 An S-word (u)s is called a normal S-word if (u)s = (asb) is a good word.

We denote (u)s by [u]s if (u)s is a normal S-word. From Lemma 7.5, it follows that

[u]s = [u]s.

Definition 7.8 Let f, g ∈ pre-Lie(X) be monic polynomials, [w] ∈ W , and a, b ∈ X∗.

Then there are two kinds of compositions.

(i) If f = [agb], then

(f, g)f = f − [agb]

is called composition of inclusion.

(ii) If (f [w]) is not good, then

f · [w]

is called composition of right multiplication.

Let S ⊂ pre-Lie(X) be a given nonempty subset. The composition of inclusion (f, g)f

is said to be trivial modulo S if

(f, g)f =∑

i

αi[aisibi]

where each αi ∈ k, ai, bi ∈ X∗, si ∈ S, [aisibi] is normal S-word and [aisibi] < f . If this

is the case, then we write

(f, g)f ≡ 0 mod(S, f).

In general, for any normal word [w] and p, q ∈ pre-Lie(X), we write

p ≡ q mod(S, [w])

which means that p − q =∑αi[aisibi], where each αi ∈ k, ai, bi ∈ X∗, si ∈ S and

[aisibi] < [w].

The composition of right multiplication f · [w] is said to be trivial modulo S if

f · [w] =∑

i

αi[aisibi]

111

where each αi ∈ k, ai, bi ∈ X∗, si ∈ S, [aisibi] is normal S-word and [aisibi] ≤ f · [w]. If

this is the case, then we write

f · [w] ≡ 0 mod(S).

Definition 7.9 Let S ⊂ pre-Lie(X) be a nonempty set of monic polynomials and the

ordering < on X∗∗ be defined as before. Then the set S is called a Grobner-Shirshov basis

in pre-Lie(X) if any composition in S is trivial modulo S.

Lemma 7.10 Let S ⊂ pre-Lie(X) be a set of monic polynomials and (u)s an S-word. If

any right multiplication composition in S is trivial module S, then (u)s has a representa-

tion:

(u)s =∑

i

αi[ui]si

where each αi ∈ k, [ui]siis a normal S-word and [ui]si

≤ (u)s.

Proof. We use induction on (u)s. If (u)s = s, then (u)s = s and the result holds.

Assume that (u)s > s. Then (u)s = (v)s[w] or (u)s = [w](v)s. We consider only the case

(u)s = (v)s[w]. The other case can be similarly proved.

By induction, we may assume that (u)s = [v]s[w]. If [v]s = s, then the result holds

clearly because each right multiplication composition in S is trivial modulo S. Suppose

that [v]s = [v1]s[v2] or [v]s = [v1][v2]s. We consider only the case [v]s = [v1]s[v2]. The

other case can be similarly proved. If [v2] ≤ [w], then (u)s = [v]s[w] is a normal S-word

and we get the result. If [v2] > [w], then

(u)s = ([v1]s[v2])[w] = ([v1]s[w])[v2] + [v1]s([v2][w])− [v1]s([w][v2]).

By induction, [v1]s[w] =∑j

βj[vj]sj, where βj ∈ k, [vj]sj

is a normal S-word, and [vj]sj≤

[v1]s[w]. If [vj]sj= [v1]s[w], then [vj]sj

[v2] is a normal S-word since [vj]sj[v2] = [vj]sj

[v2] =

([v1]s[w])[v2] = ([v1]s[v2])[w] = (u)s by Corollary 7.4. If [vj]sj< [v1]s[w], then [vj]sj

[v2] <

([v1]s[w])[v2] = ([v1]s[w])[v2] = (u)s. By Corollary 7.4 again,

[v1]s([v2][w]), [v1]s([w][v2]) < ([v1]s[w])[v2] = (u)s.

Now the result follows from the induction.

Lemma 7.11 Let [a1s1b1], [a2s2b2] be normal S-words. If S is a Grobner-Shirshov basis

in pre-Lie(X) and [w] = [a1s1b1] = [a2s2b2], then

[a1s1b1] ≡ [a2s2b2] mod(S, [w]).

112

Proof. We have w = a1s1b1 = a2s2b2 as associative words. There are two cases to

consider.

Case 1. Suppose that s1 and s2 are disjoint, say, |a2| ≥ |a1| + |s1|. Then, we can

assume that

a2 = a1s1c and b1 = cs2b2

for some c ∈ X∗. Thus, [w] = [a1s1cs2b2]. Now,

[a1s1b1]− [a2s2b2] = [a1s1cs2b2]− [a1s1cs2b2]

= [a1s1cs2b2]− (a1s1cs2b2) + (a1s1cs2b2)− [a1s1cs2b2]

= (a1s1c(s2 − s2)b2) + (a1(s1 − s1)cs2b2).

Since [s2 − s2] < s2 and [s1 − s1] < s1, and by Lemmas 7.5 and 7.10, we conclude that

[a1s1b1]− [a2s2b2] =∑

i

αi[uisivi]

for some αi ∈ k, normal S-words [uisivi] such that [uis1vi] < [w]. Hence,

[a1s1b1] ≡ [a2s2b2] mod(S, [w]).

Case 2. Suppose that s1 contains s2 as a subword. We assume that

s1 = [as2b], a2 = a1a and b2 = bb1, that is, [w] = [a1[as2b]b1]

for the normal S-word [as2b]. We have

[a1s1b1]− [a2s2b2] = [a1s1b1]− [a1[as2b]b1]

= (a1(s1 − [as2b])b1)

= (a1(s1, s2)s1b1).

Since S is a Grobner-Shirshov basis, (s1, s2)s1 =∑i

αi[cisidi] for some αi ∈ k, normal

S-words [cisidi] with each [cisidi] < s1. By Lemmas 7.5 and 7.10, we have

[a1s1b1]− [a2s2b2] = (a1(s1, s2)s1b1)

=∑

i

αi(a1[cisidi]b1) =∑

j

βj[ajsjbj]

for some βj ∈ k, normal S-words [ajsjbj] with each [aj sjbj] < [w] = [a1s1b1].

Consequently, [a1s1b1] ≡ [a2s2b2] mod(S, [w]).

113

Lemma 7.12 Let S ⊂ pre-Lie(X) be a set of monic polynomials and Irr(S) = [u] ∈W |[u] 6= [asb] a, b ∈ X∗, s ∈ S and [asb] is a normal S-word. Then for any f ∈pre-Lie(X),

f =∑

[ui]≤f

αi[ui] +∑

[aj sjbj ]≤f

βj[ajsjbj],

where each αi, βj ∈ k, [ui] ∈ Irr(S) and [ajsjbj] is a normal S-word.

Proof. Let f =∑i

αi[ui] ∈ pre-Lie(X) where 0 6= αi ∈ k and [u1] > [u2] > · · · . If

[u1] ∈ Irr(S), then let f1 = f − α1[u1]. If [u1] 6∈ Irr(S), then there exist some s ∈ S

and a1, b1 ∈ X∗ such that f = [a1s1b1]. Let f1 = f − α1[a1s1b1]. In both cases, we have

f1 < f . Then the result follows by using induction on f .

Theorem 7.13 (CD-lemma for pre-Lie algebras) Let S ⊂ pre-Lie(X) be a nonempty set

of monic polynomials and the ordering < be defined as before. Let Id(S) be the ideal of

pre-Lie(X) generated by S. Then the following statements are equivalent:

(i) S is a Grobner-Shirshov basis in pre-Lie(X).

(ii) f ∈ Id(S)⇒ f = [asb] for some s ∈ S and a, b ∈ X∗, where [asb] is a normal S-word.

(ii)′ f ∈ Id(S)⇒ f = α1[a1s1b1]+α2[a2s2b2]+ · · · , where αi ∈ k, [a1s1b1] > [a2s2b2] > · · ·and each [aisibi] is a normal S-word.

(iii) Irr(S) = [u] ∈ W |[u] 6= [asb] a, b ∈ X∗, s ∈ S and [asb] is a normal S-word is a

linear basis of the algebra pre-Lie(X|S) = pre-Lie(X)/Id(S).

Proof. (i)⇒ (ii). Let S be a Grobner-Shirshov basis and 0 6= f ∈ Id(S). We can also

assume, by Lemma 7.10, that

f =n∑

i=1

αi[aisibi]

where each αi ∈ k, ai, bi ∈ X∗, si ∈ S and [aisibi] a normal S-word. Let

[wi] = [aisibi], [w1] = [w2] = · · · = [wl] > [wl+1] ≥ · · ·

We will use the induction on l and [w1] to prove that f = [asb] for some s ∈ S and a, b ∈X∗.

If l = 1, then f = [a1s1b1] = [a1s1b1] and hence the result holds. Assume that l ≥ 2.

Then, by Lemma 7.11, we have

[a1s1b1] ≡ [a2s2b2] mod(S, [w1]).

114

Thus, if α1 + α2 6= 0 or l > 2, then the result holds. For the case that α1 + α2 = 0 and

l = 2, we use the induction on [w1]. Hence, the result follows.

(ii)⇒ (ii)′. Assume that (ii) and 0 6= f ∈ Id(S). Let f = α1f + · · · . Then, by (ii),

f = [a1s1b1]. Therefore,

f1 = f − α1[a1s1b1], f1 < f, f1 ∈ Id(S).

Now, by using induction on f , we have (ii)′.

(ii)′ ⇒ (ii). This part is clear.

(ii) ⇒ (iii). Suppose that∑i

αi[ui] = 0 in pre-Lie(X|S), where αi ∈ k, [ui] ∈

Irr(S). It means that∑i

αi[ui] ∈ Id(S) in pre-Lie(X). Then all αi must be equal to zero.

Otherwise, we have∑i

αi[ui] = [uj] ∈ Irr(S) for some j which contradicts (ii).

Now, for any f ∈ pre-Lie(X), by Lemma 7.12, we have

f =∑

[ui]∈Irr(S), [ui]≤f

αi[ui] +∑

[aj sjbj ]≤f

βj[ajsjbj].

Hence, (iii) follows.

(iii) ⇒ (i). Applying Lemma 7.12 to a composition of elements of S, we get by

(iii) that any composition is trivial because any composition belongs to Id(S). So S is a

Grobner-Shirshov basis.

§7.2 PBW theorem for pre-Lie algebras

For any pre-Lie algebra A, the algebra A(−) with new product [xy] = xy−yx is a Lie

algebra. So, for any Lie algebra L, one may define its universal enveloping pre-Lie algebra

Upre-Lie(L) as follows. If L = Lie(X|S), then Upre-Lie(L) = pre-Lie(X|S(−)) which has

the universal enveloping property: there exists a Lie homomorphism i : L→ Upre-Lie(L)

such that for any pre-Lie algebra A and any Lie homomorphism ε : L → A, there exists

a unique pre-Lie homomorphism f : Lie(X) → L such that the following triangle is

commutative

-

?

A

Upre-Lie(L)L i

ε∃! f

In this section, we give a Grobner-Shirshov basis for such an algebra.

115

Theorem 7.14 Let (L , [, ]) be a Lie algebra with a well-ordered basis ei| i ∈ I. Let

[ei, ej] =∑m

αmij em

where αmij ∈ k. We denote

∑m

αmij em by eiej. Let

Upre-Lie(L ) = pre-Lie(eiI | eiej − ejei = eiej, i, j ∈ I)

be the universal enveloping pre-Lie algebra of L . Let

S = fij = eiej − ejei − eiej, i, j ∈ I and i > j.

Then

(i) S is a Grobner-Shirshov basis in pre-Lie(X) where X = eiI .

(ii) (Segal PBW theorem) L can be embedded into the universal enveloping pre-Lie

algebra Upre-Lie(L ) as a vector space.

Proof. (i). It is clear that fij = eiej (i > j). So, there exists a unique kind of

composition fijek (i > j > k). Then, in pre-Lie(X), we have

fijek − fikej + fjkei − eifjk + ejfik − ekfij −∑m

αmjkfim −

∑m

αmij fkm −

∑m

αmikfmj

=(eiej)ek − (ejei)ek − eiejek − (eiek)ej + (ekei)ej + eiekej

+ fjkei − eifjk + ejfik − ekfij −∑m

αmjkfim −

∑m

αmij fkm −

∑m

αmikfmj

=(eiek)ej + ei(ejek)− ei(ekej)− (ejek)ei − ej(eiek) + ej(ekei)− eiejek − (eiek)ej + (ekej)ei

+ ek(eiej)− ek(ejei) + eiekej + fjkei − eifjk + ejfik − ekfij

−∑m

αmjkfim −

∑m

αmij fkm −

∑m

αmikfmj

=− (ejek)ei + (ekej)ei + ei(ejek)− ei(ekej) + ej(ekei)− ej(eiek)− ek(ejei) + ek(eiej)

− eiejek + eiekej + fjkei − eifjk + ejfik − ekfij −∑m

αmjkfim −

∑m

αmij fkm −

∑m

αmikfmj

=− ejekei + eiejek − ejeiek+ ekeiej − eiejek + eiekej

−∑m

αmjkfim −

∑m

αmij fkm −

∑m

αmikfmj

=− ekeiej − eiejek − ejekei

=0 (by Jacobi identity).

By invoking fmn = −fnm, we have fijek ≡ 0 mod(S). Therefore, S is a Grobner-

Shirshov basis for Upre-Lie(L ).

(ii). It follows from Theorem 7.13.

116

Chapter 8 Grobner-Shirshov Bases for Dialgebras

§8.1 Preliminaries

Definition 8.1 Let k be a field. A k-linear space D equipped with two bilinear multipli-

cations ` and a is called a dialgebra, if both ` and a are associative and

a a (b ` c) = a a b a c

(a a b) ` c = a ` b ` c

a ` (b a c) = (a ` b) a c

for any a, b, c ∈ D.

Definition 8.2 Let D be a dialgebra, B ⊂ D. Let us define diwords of D in the set B by

induction:

(i) b = (b), b ∈ B is a diword in B of length |b| = 1.

(ii) (u) is called a diword in B of length |(u)| = n, if (u) = ((v) a (w)) or (u) = ((v) `(w)), where (v), (w) are diwords in B of length k, l respectively and k + l = n.

Proposition 8.3 Let D be a dialgebra and B ⊂ D. Any diword of D in the set B is

equal to a diword in B of the form

(u) = b−m ` · · · ` b−1 ` b0 a b1 a · · · a bn (8.1)

where bi ∈ B, −m ≤ i ≤ n, m ≥ 0, n ≥ 0. Any bracketing of the right side of (8.1) gives

the same result.

Definition 8.4 Let X be a set. A free dialgebra D(X) generated by X over k is defined

in a usual way by the following commutative diagram:

-

?

D

D(X)X i

∀ϕ∃!ϕ∗ (di-homomorphism)

where D is any dialgebra.

117

In [147], a construction of a free dialgebra is given.

Proposition 8.5 Let D(X) be a free dialgebra over k generated by X. Any diword in

D(X) is equal to the unique diword of the form

[u] = x−m ` · · · ` x−1 ` x0 a x1 a · · · a xn , x−m · · ·x−1x0x1 · · ·xn (8.2)

where xi ∈ X, m ≥ 0, n ≥ 0, and x0 is called the center of the normal diword [u]. We

call [u] a normal diword (in X) with the associative word u, u ∈ X∗. Clearly, if [u] = [v],

then u = v. In (8.2). Let [u], [v] be two normal diwords. Then [u] ` [v] is the normal

diword [uv] with the center at the center of [v]. Accordingly, [u] a [v] is the normal diword

[uv] with the center at the center of [u].

Example 8.6

(x−1 ` x0 a x1) ` (y−1 ` y0 a y1) = x−1 ` x0 ` x1 ` y−1 ` y0 a y1,

(x−1 ` x0 a x1) a (y−1 ` y0 a y1) = x−1 ` x0 a x1 a y−1 a y0 a y1.

§8.2 Composition-Diamond lemma for dialgebras

Let X be a well-ordered set, D(X) the free dialgebra over k, X∗ the free monoid

generated by X and [X∗] the set of normal diwords in X. Let us define the deg-lex

ordering on [X∗] in the following way: for any [u], [v] ∈ [X∗],

[u] < [v]⇐⇒ wt([u]) < wt([v]) lexicographicaly,

where

wt([u]) = (n+m+ 1,m, x−m, · · · , x0, . . . , xn)

if [u] = x−m · · ·x−1x0x1 · · ·xn.

Throughout this chapter, we will use this ordering.

It is easy to see that the ordering < is satisfied the following properties:

[u] < [v] =⇒ x ` [u] < x ` [v], [u] a x < [v] a x, for any x ∈ X.

Any polynomial f ∈ D(X) has the form

f =∑

[u]∈[X∗]

f([u])[u] = α[f ] +∑

αi[ui],

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where [f ], [ui] are normal diwords in X, [f ] > [ui], α, αi, f([u]) ∈ k, α 6= 0. We call [f ]

the leading term of f . Denote suppf by the set [u]|f([u]) 6= 0 and deg(f) by |[f ]|. f is

called monic if α = 1. f is called left (right) normed if f =∑αiuixi (f =

∑αixiui),

where each αi ∈ k, xi ∈ X and ui ∈ X∗.

If [u], [v] are both left normed or both right normed, then it is clear that for any

[w] ∈ [X∗],

[u] < [v] =⇒ [u] ` [w] < [v] ` [w], [w] ` [u] < [w] ` [v],

[u] a [w] < [v] a [w], [w] a [u] < [w] a [v].

Let S ⊂ D(X). By an S-diword g we will mean a diword in X ∪ S with only one

occurrence of s ∈ S. If this is the case and g = (asb) for some a, b ∈ X∗, s ∈ S, we also

call g an s-diword.

From Proposition 8.3 it follows that any s-diword is equal to

[asb] = x−m ` · · · ` x−1 ` x0 a x1 a · · · a xn|xk 7→s (8.3)

where −m ≤ k ≤ n, s ∈ S, xi ∈ X, −m ≤ i ≤ n. To be more precise, [asb] = [asb] if

k = 0; [asb] = [asb1x0b2] if k < 0 and [asb] = [a1x0a2sb] if k > 0. If the center of the

s-diword [asb] is in a, then we denote it by [asb] = [a1x0a2sb]. Similarly, [asb] = [asb1x0b2]

(of course, either ai or bi may be empty).

Definition 8.7 The s-diword (8.3) is called a normal s-diword if one of the following

conditions holds:

(i) k = 0,

(ii) k < 0 and s is left normed,

(iii) k > 0 and s is right normed.

We call a normal s-diword [asb] a left (right) normed s-diword if both s and [asb] are

left (right) normed. In particulary, s is a left (right) normed s-diword if s is left (right)

normed polynomial.

The following lemma follows from the above properties of the ordering <.

Lemma 8.8 For a normal s-diword [asb], the leading term of [asb] is equal to [a[s]b], that

is, [asb] = [a[s]b]. More specifically, if

[asb] = x−m ` · · · ` x−1 ` x0 a x1 a · · · a xn|xk 7→s,

119

then corresponding to k = 0, k < 0, k > 0, respectively, we have

x−m ` · · · ` x−1 ` s a x1 a · · · a xn = x−m ` · · · ` x−1 ` [s] a x1 a · · · a xn,

x−m ` · · · ` s ` · · · ` x0 a · · · a xn = x−m ` · · · ` [s] ` · · · ` x0 a · · · a xn,

x−m ` · · · ` x0 a · · · a s a · · · a xn = x−m ` · · · ` x0 a · · · a [s] a · · · a xn.

Now, we define compositions of polynomials in D(X).

Definition 8.9 Let the ordering < be as before and f, g ∈ D(X) with f, g monic.

1) Composition of left (right) multiplication.

Let f be not a right normed polynomial and x ∈ X. Then x a f is called the

composition of left multiplication. Clearly, x a f is a right normed polynomial (or 0).

Let f be not a left normed polynomial and x ∈ X. Then f ` x is called the

composition of right multiplication. Clearly, f ` x is a left normed polynomial (or 0).

2) Composition of inclusion.

Let

[w] = [f ] = [a[g]b],

where [agb] is a normal g-diword. Then

(f, g)[w] = f − [agb]

is called the composition of inclusion. The transformation f 7→ f − [agb] is called the

elimination of leading diword (ELW) of g in f , and [w] is called the ambiguity of f

and g.

3) Composition of intersection.

Let

[w] = [[f ]b] = [a[g]], |f |+ |g| > |w|,

where [fb] is a normal f -diword and [ag] a normal g-diword. Then

(f, g)[w] = [fb]− [ag]

is called the composition of intersection, and [w] is called the ambiguity of f and g.

120

Remark In the Definition 8.9, for the case of 2) or 3), we have (f, g)[w] < [w]. For the

case of 1), deg(x a f) ≤ deg(f) + 1 and deg(f ` x) ≤ deg(f) + 1.

Definition 8.10 Let the ordering < be as before, S ⊂ D(X) a monic set and f, g ∈ S.

1) Let x a f be a composition of left multiplication. Then x a f is called trivial modulo

S, denoted by x a f ≡ 0 mod(S), if

x a f =∑

αi[aisibi],

where each αi ∈ k, ai, bi ∈ X∗, si ∈ S, [aisibi] right normed si-diword and |[ai[si]bi]| ≤deg(x a f).

Let f ` x be a composition of right multiplication. Then f ` x is called trivial

modulo S, denoted by f ` x ≡ 0 mod(S), if

f ` x =∑

αi[aisibi],

where each αi ∈ k, ai, bi ∈ X∗, si ∈ S, [aisibi] left normed si-diword and |[ai[si]bi]| ≤deg(f ` x).

2) Composition (f, g)[w] of inclusion (intersection) is called trivial modulo (S, [w]), de-

noted by (f, g)[w] ≡ 0 mod(S, [w]), if

(f, g)[w] =∑

αi[aisibi],

where each αi ∈ k, ai, bi ∈ X∗, si ∈ S, [aisibi] normal si-diword, [ai[si]bi] < [w] and

each [aisibi] is right (left) normed si-diword whenever either both f and [agb] or both

[fb] and [ag] are right (left) normed S-diwords.

We call the set S a Grobner-Shirshov basis in D(X) if any composition of polynomials

in S is trivial modulo S (and [w]).

The following lemmas play key role in the proof of Theorem 8.14.

Lemma 8.11 Let S ⊂ D(X) and [asb] an s-diword, s ∈ S. Assume that each composition

of right and left multiplication is trivial modulo S. Then, [asb] has a presentation:

[asb] =∑

αi[aisibi],

where each αi ∈ k, si ∈ S, ai, bi ∈ X∗ and each [aisibi] is normal si-diword.

121

Proof. Following Proposition 8.3, we assume that

[asb] = x−m ` · · · ` x−1 ` x0 a x1 a · · · a xn|xk 7→s.

There are three cases to consider.

Case 1. k = 0. Then [asb] is a normal s-diword.

Case 2. k < 0. Then [asb] = a ` (s ` xk+1) ` b, k < −1 or [asb] = a ` (s ` x0) a b.If s is left normed then [asb] is a normal s-diword. If s is not left normed then for the

composition s ` xk+1 (k < 0) of right multiplication, we have

s ` xk+1 =∑

αi[aisibi],

where each αi ∈ k, ai, bi ∈ X∗, si ∈ S and [aisibi] is left normed si-diword. Then

[asb] =∑

αi(a ` [aisibi] ` b)

or

[asb] =∑

αi(a ` [aisibi] a b)

is a linear combination of normal si-diwords.

Case 3. k > 0 is similar to the Case 2.

Lemma 8.12 Let S ⊂ D(X) and each composition (f, g)[w] in S of inclusion (inter-

section) trivial modulo (S, [w]). Let [a1s1b1] and [a2s2b2] be normal S-diwords such that

[w] = [a1[s1]b1] = [a2[s2]b2], where s1, s2 ∈ S, a1, a2, b1, b2 ∈ X∗. Then,

[a1s1b1] ≡ [a2s2b2] mod(S, [w]),

i.e., [a1s1b1] − [a2s2b2] =∑αi[aisibi], where each αi ∈ k, ai, bi ∈ X∗, si ∈ S, [aisibi]

normal si-diword and [ai[si]bi] < [w].

Proof. In the following, all letters a, b, c with indexis are words and s1, s2, sj ∈ S.

Because a1s1b1 = a2s2b2 as ordinary words, there are three cases to consider.

Case 1. Subwords s1, s2 have empty intersection. Assume, for example, that b1 =

bs2b2 and a2 = a1s1b. Because any normal S-diword may be bracketing in any way, we

have

[a2s2b2]− [a1s1b1] = (a1s1(b(s2 − [s2])b2))− ((a1(s1 − [s1])b)s2b2).

For any [t] ∈ supp(s2 − [s2]), we prove that (a1s1b[t]b2) is a normal s1-diword. There are

five cases to consider.

1.1 [w] = [a1[s1]b[s2]b2];

122

1.2 [w] = [a1˙[s1]b[s2]b2];

1.3 [w] = [a1[s1]b[s2]b2];

1.4 [w] = [a1[s1]b ˙[s2]b2];

1.5 [w] = [a1[s1]b[s2]b2].

For 1.1, since [a1s1b1] and [a2s2b2] are normal S-diwords, both s1 and s2 are right

normed by the definition, in particular, [t] is right normed. It follows that (a1s1b[t]b2) =

[a1s1b[t]b2] is a normal s1-diword.

For 1.2, it is clear that (a1s1b[t]b2) is a normal s1-diword and [t] is right normed.

For 1.3, 1.4 and 1.5, since [a1s1b1] is normal s1-diword, s1 is left normed by the

definition, which implies that (a1s1b[t]b2) is a normal s1-diword. Moreover, [t] is right

normed, if 1.3, and left normed, if 1.5.

Clearly, for all cases, we have [a1s1b[t]b2] = [a1[s1]b[t]b2] < [a1[s1]b[s2]b2] = [w].

Similarly, for any [t] ∈ supp(s1−[s1]), (a1[t]bs2b2) is a normal s2-diword and [a1[t]b[s2]b2] <

[w].

Case 2. Subwords s1 and s2 have non-empty intersection c. Assume, for example,

that b1 = bb2, a2 = a1a, w1 = s1b = as2 = acb.

There are following five cases to consider:

2.1 [w] = [a1[s1]bb2];

2.2 [w] = [a1[s1]bb2];

2.3 [w] = [a1acbb2];

2.4 [w] = [a1acbb2];

2.5 [w] = [a1acbb2].

Then

[a2s2b2]− [a1s1b1] = (a1([as2]− [s1b])b2) = (a1(s1, s2)[w1]b2),

where [w1] = [acb] = [[s1]b] = [a[s2]] is as follows:

2.1 [w1] is right normed;

2.2 [w1] is left normed;

2.3 [w1] = [acb];

2.4 [w1] = [acb];

2.5 [w1] = [acb].

Since each composition (f, g)[w] in S is trivial modulo (S, [w]), there exist βj ∈k, uj, vj ∈ X∗, sj ∈ S such that [s1b] − [as2] =

∑j βj[ujsjvj], where each [ujsjvj] is

normal S-diword and [uj[sj]vj] < [w1] = [acb]. Therefore,

[a2s2b2]− [a1s1b1] =∑

j

βj(a1[ujsjvj]b2).

123

Now, we prove that each (a1[ujsjvj]b2) is normal sj-diword and (a1[ujsjvj]b2) < [w] =

[a1[[s1]b]b2].

For 2.1, since [a1s1bb2] and [a1as2b2] are normal S-diwords, both [s1b] and [as2] are

right normed S-diwords. Then, by definition, each [ujsjvj] is right normed S-diword, and

so each (a1[ujsjvj]b2) = [a1ujsjvjb2] is normal S-diword.

For 2.2, both [s1b] and [as2] must be left normed S-diwords. Then, by definition,

each [ujsjvj] is left normed S-diword, and so each (a1[ujsjvj]b2) = [a1ujsjvj b2] is normal

S-diword.

For 2.3, 2.4 or 2.5, by noting that (a1[ujsjvj]b2) = ((a1) ` [ujsjvj] a (b2)) and [ujsjvj]

is normal S-diword, (a1[ujsjvj]b2) is also normal S-diword.

Now, for all cases, we have [a1ujsjvjb2] = [a1uj[sj]vjb2] < [w] = [a1[acb]b2].

Case 3. One of the subwords s1 and s2 contains another as a subword. Assume, for

example, that b2 = bb1, a2 = a1a, w1 = s1 = as2b.

Again there are following five cases to consider:

2.1 [w] = [a1a[s2]bb1];

2.2 [w] = [a1a[s2]bb1];

2.3 [w] = [a1a[s2]bb1];

2.4 [w] = [a1a ˙[s2]bb1];

2.5 [w] = [a1a[s2]bb1].

Then

[a1s1b1]− [a2s2b2] = (a1(s1 − as2b)b1) = (a1(s1, s2)[w1]b1).

It is similar to the proof of the Case 2 that we have [a1s1b1] ≡ [a2s2b2] mod(S, [w]).

Let S ⊂ D(X). Denote

Irr(S) = u ∈ [X∗]|u 6= [a[s]b], s ∈ S, a, b ∈ X∗, [asb] is normal s-diword.

Lemma 8.13 Let S ⊂ D(X) and h ∈ D(X). Then h has a representation

h =∑I1

αi[ui] +∑I2

βj[ajsjbj]

where [ui] ∈ Irr(S), i ∈ I1, [ajsjbj] normal sj-diwords, sj ∈ S, j ∈ I2 with [a1[s1]b1] >

[a2[s2]b2] > · · · > [an[sn]bn].

Proof. Let h = α1[h] + · · · . We prove the result by induction on [h].

124

If [h] ∈ Irr(S), then take [u1] = [h] and h1 = h−α1[u1]. Clearly, [h1] < [h] or h1 = 0.

If [h] 6∈ Irr(S), then [h] = [a1[s1]b1] with [a1s1b1] a normal s1-diword. Let h1 =

h− β1[a1s1b1]. Then [h1] < [h] or h1 = 0.

The following theorem is the main result.

Theorem 8.14 (Composition-Diamond lemma for dialgebras) Let S ⊂ D(X) be a monic

set and the ordering < as before, Id(S) is the ideal of D(X) generated by S. Then

(i)⇒ (ii)⇔ (ii)′ ⇔ (iii), where

(i) S is a Grobner-Shirshov basis in D(X).

(ii) f ∈ Id(S)⇒ [f ] = [a[s]b] for some s ∈ S, a, b ∈ X∗ and [asb] a normal S-diword.

(ii)′ f ∈ Id(S)⇒ f = α1[a1s1b1]+α2[a2s2b2]+· · ·+αn[ansnbn] with [a1[s1]b1] > [a2[s2]b2] >

· · · > [an[sn]bn], where [aisibi] is normal si-diword, i = 1, 2, · · · , n.

(iii) The set Irr(S) is a linear basis of the dialgebra D(X|S) = D(X)/Id(S) generated

by X with defining relations S.

Proof. (i) ⇒ (ii). Let S be a Grobner-Shirshov basis and 0 6= f ∈ Id(S). We may

assume, by Lemma 8.11, that

f =n∑

i=1

αi[aisibi],

where each αi ∈ k, ai, bi ∈ X∗, si ∈ S and [aisibi] normal S-diword. Let

[wi] = [ai[si]bi], [w1] = [w2] = · · · = [wl] > [wl+1] ≥ · · · , l ≥ 1.

We will use induction on l and [w1] to prove that [f ] = [a[s]b] for some s ∈ S and a, b ∈ X∗.

If l = 1, then [f ] = [a1s1b1] = [a1[s1]b1] and hence the result holds. Assume that l ≥ 2.

Then, by Lemma 8.12, we have [a1s1b1] ≡ [a2s2b2] mod(S, [w1]).

Thus, if α1 + α2 6= 0 or l > 2, then the result follows from induction on l. For the

case α1 + α2 = 0 and l = 2, we use induction on [w1]. Now, the result follows.

(ii)⇒ (ii)′. Assume (ii) and 0 6= f ∈ Id(S). Let f = α1[f ] +∑

[ui]<[f ] αi[ui]. Then,

by (ii), [f ] = [a1[s1]b1], where [a1s1b1] is a normal S-diword. Therefore,

f1 = f − α1[a1s1b1], [f1] < [f ] or f1 = 0, f1 ∈ Id(S).

Now, by using induction on [f ], we have (ii)′.

(ii)′ ⇒ (ii). This part is clear.

125

(ii)⇒ (iii). Assume (ii). Then by Lemma 8.13, Irr(S) spans D(X|S) as k-space.

Suppose that 0 6=∑αi[ui] ∈ Id(S) where [u1] > [u2] > · · · , [ui] ∈ Irr(S). Then by

(ii), [u1] = [a1[s1]b1] where [a1s1b1] is a normal S-diword, a contradiction.

This shows (iii).

(iii) ⇒ (ii). Assume (iii). Let 0 6= f ∈ Id(S). Since the elements in Irr(S) are

linearly independent in D(X|S), by Lemma 8.13, [f ] = [a[s]b], where [asb] is a normal

S-diword. Thus, (ii) follows.

Remark In general, (iii) 6⇒ (i). For example, it is noted that

Irr(S) = xj a xi1 a · · · a xik | j ∈ I, ip ∈ I − I0, 1 ≤ p ≤ k, i1 ≤ · · · ≤ ik, k ≥ 0

is a linear basis of D(X|S) in Theorem 8.17. Let

S1 = xj ` xi − xi a xj + xi, xj, xt a xi0 , i, j, t ∈ I, i0 ∈ I0.

Then Irr(S1) = Irr(S) is a linear basis of D(X|S). But in the proof of Theorem 8.17,

we know that S1 is not a Grobner-Shirshov basis of D(X|S).

§8.3 Applications

In this section, we give Grobner-Shirshov bases for the universal enveloping dialgebra

of a Leibniz algebra (PBW theorem), the bar extension of a dialgebra, the free product

of two dialgebras, and the Clifford dialgebra. By using Theorem 8.14, we obtain some

normal forms for dialgebras mentioned the above.

Definition 8.15 A k-linear space L equipped with bilinear multiplication [, ] is called a

Leibniz algebra if for any a, b, c ∈ L,

[[a, b], c] = [[a, c], b] + [a, [b, c]]

i.e., the Leibniz identity is valid in L.

It is clear that if (D,a,`) is a dialgebra then D(−) = (D, [, ]) is a Leibniz algebra,

where [a, b] = a a b− b ` a for any a, b ∈ D.

If f is a Leibniz polynomial in variables X, then by f (−) we mean a dialgebra poly-

nomial in X obtained from f by transformation [a, b] 7→ a a b− b ` a.

126

Definition 8.16 Let L be a Leibniz algebra. A dialgebra U(L) together with a Leibniz

homomorphism ε : L → U(L) is called the universal enveloping dialgebra for L, if the

following diagram commute:

-

?

∃!f

D

U(L)L ε

∀δ

where D is a dialgebra, δ is a Leibniz homomorphism and f : U(L) → D is a dialgebra

homomorphism such that fε = δ.

An equivalent definition is as follows: Let L = Lei(X|S) is a Leibniz algebra presented

by generators X and definition relations S. Then U(L) = D(X|S(−)) is the dialgebra with

generators X and definition relations S(−) = s(−)|s ∈ S.

Theorem 8.17 Let L be a Leibniz algebra over a field k with the product , . Let L0 be

the subspace of L generated by the set a, a, a, b + b, a | a, b ∈ L. Let xi|i ∈ I0be a basis of L0 and X = xi|i ∈ I a well-ordered basis of L such that I0 ⊆ I. Let

U(L) = D(X|xi a xj − xj ` xi−xi, xj) be the universal enveloping dialgebra for L and

the ordering < on [X∗] as before. Then

(i) D(X|xi a xj − xj ` xi − xi, xj) = D(X|S), where S consists of the following

polynomials:

(a) fji = xj ` xi − xi a xj + xi, xj (i, j ∈ I)

(b) fji`t = xj ` xi ` xt − xi ` xj ` xt + xi, xj ` xt (i, j, t ∈ I, j > i)

(c) hi0`t = xi0 ` xt (i0 ∈ I0, t ∈ I)

(d) ftaji = xt a xj a xi − xt a xi a xj + xt a xi, xj (i, j, t ∈ I, j > i)

(e) htai0 = xt a xi0 (i0 ∈ I0, t ∈ I)

(ii) S is a Grobner-Shirshov basis in D(X).

(iii) The set

xj a xi1 a · · · a xik | j ∈ I, ip ∈ I − I0, 1 ≤ p ≤ k, i1 ≤ · · · ≤ ik, k ≥ 0

is a linear basis of the universal enveloping algebra U(L). In particular, L is a Leibniz

subalgebra of U(L).

127

Proof. (i) By using the following

fji`t = fji ` xt and fji ` xt + fij ` xt = (xi, xj+ xj, xi) ` xt,

we have (b) and (c) are in Id(fji). By symmetry, (d) and (e) are in Id(fji). This shows

(i).

(ii) We will prove that all compositions in S are trivial modulo S (and [w]). For conve-

nience, we extend linearly the functions fji, fji`t, ftaji, hi0`t and htai0 to fjp,q (fp,qi), fji`p,q

and hp,qai0 , etc respectively. For example, if xp, xq =∑αs

pqxs, then

fjp,q = xj ` xp, xq − xp, xq a xj + xp, xq, xj =∑

αspqfjs,

fji`p,q =∑

αspq(xj ` xi ` xs − xi ` xj ` xs + xi, xj ` xs) = fji ` xp, xq,

hp,qai0 =∑

αspqhsai0 .

By using the Leibniz identity,

a, b, c = a, b, c+ a, c, b, (8.4)

we have

a, b, b = 0 and a, b, c+ c, b = 0

for any a, b, c ∈ L. It means that for any i0 ∈ I0, j ∈ I,

xj, xi0 = 0 (8.5)

and by noting that xi0 , xj = xj, xi0+ xi0 , xj, we have

xi0 , xj ∈ L0. (8.6)

This implies that L0 is an ideal of L. Clearly, L/L0 is a Lie algebra.

The formulas (8.4), (8.5) and (8.6) are useful in the sequel.

In S, all the compositions are as follows.

1) Compositions of left or right multiplication.

All possible compositions in S of left multiplication are ones related to (a), (b) and

(c).

By noting that for any s, i, j, t ∈ I, we have

xs a fji = fsaji (j > i),

xs a fji = −fsaij + xs a (xi, xj+ xj, xi) (j < i),

xs a fii = xs a xi, xi,

xs a fji`t = fsaji a xt (j > i) and

xs a hi0`t = hsai0 a xt,

128

it is clear that all cases are trivial modulo S.

By symmetry, all compositions in S of right multiplication are trivial modulo S.

2) Compositions of inclusion and intersection.

We denote, for example, (a ∧ b) the composition of the polynomials of type (a) and

type (b). It is noted that since (b) and (c) are both left normed, we have to prove that

the corresponding compositions of the cases of (b∧b), (b∧c), (c∧c) and (c∧b) must be a

linear combination of left normed S-diwords in which the leading term of each S-diword

is less than w. Symmetrically, we consider the cases for the right normed (d) and (e).

All possible compositions of inclusion and intersection are as follows.

(a ∧ c) [w] = xi0 ` xi (i0 ∈ I0). We have, by (8.5),

(fi0i, hi0`i)[w] = −xi a xi0 + xi, xi0 = −hiai0 ≡ 0 mod(S, [w]).

(a ∧ d) [w] = xj ` xi a xq a xp (q > p). We have

(fji, fiaqp)[w]

= −xi a xj a xq a xp + xi, xj a xq a xp + xj ` xi a xp a xp − xj ` xi a xp, xq

= −xi a fjaqp + fi,jaqp + fji a xp a xq − fji a xp, xq

≡ 0 mod(S, [w]).

(a ∧ e) [w] = xj ` xi a xi0 (i0 ∈ I0). We have

(fji, hiai0)[w] = −xi a xj a xi0 + xi, xj a xi0 = −xi a hjai0 + hi,jai0 ≡ 0 mod(S, [w]).

(b ∧ a) There are two cases to consider: [w] = xj ` xi ` xt and [w] = xj ` xi ` xt ` xp.

For [w] = xj ` xi ` xt (j > i), by (8.4), we have

(fji`t, fit)[w] = −xi ` xj ` xt + xi, xj ` xt + xj ` xt a xi − xj ` xt, xi

= −xi ` fjt + fi,jt + fjt a xi − fjt,i + fit,j − fit a xj + ftaji

≡ 0 mod(S, [w]).

For [w] = xj ` xi ` xt ` xp (j > i), we have

(fji`t, ftp)[w]

= −xi ` xj ` xt ` xp + xi, xj ` xt ` xp + xj ` xi ` xp a xt − xj ` xi ` xp, xt

= −xi ` xj ` ftp + xi, xj ` ftp + fji`p a xt − fji`p,t

≡ 0 mod(S, [w]).

129

(b ∧ b) There are two cases to consider: [w] = xj ` xi ` xt ` xs ` xp and [w] = xj ` xi `xt ` xp.

For [w] = xj ` xi ` xt ` xs ` xp (j > i, t > s), we have

(fji`t, fts`p)[w]

= −xi ` xj ` xt ` xs ` xp + xi, xj ` xt ` xs ` xp + xj ` xi ` xs ` xt ` xp

−xj ` xi ` xs, xt ` xp

= −xi ` xj ` fts`p + xi, xj ` fts`p + fji`s ` xt ` xp − fji`s,t ` xp

≡ 0 mod(S, [w])

since it is a combination of left normed S-diwords in which the leading term of each

S-diword is less than w.

For [w] = xj ` xi ` xt ` xp (j > i > t), suppose that

xi, xj =∑m∈I1

αmijxm + αt

ijxt +∑n∈I2

αnijxn (m < t < n).

Denote

Bt`i,j`p = xt ` xi, xj ` xp − xi, xj ` xt ` xp − xt, xi, xj ` xp.

Then

Bt`i,j`p =∑m∈I1

αmij ftm`p −

∑n∈I2

αnijfnt`p −

∑q∈I0

βqhq`p

is a linear combination of left normed S-diwords of length 2 or 3, where∑q∈I0

βqxq =∑m∈I1

αmij (xt, xm+ xm, xt) + αt

ijxt, xt.

Denote∑l∈I0

γlxl = −(xj, xt, xi+ xt, xi, xj) + (xi, xt, xj+ xt, xj, xi).

Now, by (8.4), we have

(fji`t, fit`p)[w]

= −xi ` xj ` xt ` xp + xi, xj ` xt ` xp + xj ` xt ` xi ` xp − xj ` xt, xi ` xp

= −xi ` fjt`p −Bt`i,j`p + fjt`i ` xp −Bj`t,i`p +∑l∈I0

γlhl`p

+Bi`t,j`p − fit`j ` xp + xt ` fji`p

≡ 0 mod(S, [w])

since it is a combination of left normed S-diwords in which the leading term of each

S-diword is less than w.

130

(b ∧ c) There are three cases to consider: [w] = xj ` xi0 ` xt (i0 ∈ I0), [w] = xj0 ` xi `xt (j0 ∈ I0) and [w] = xj ` xi ` xt0 ` xn (t0 ∈ I0).

Case 1. [w] = xj ` xi0 ` xt (j > i0, i0 ∈ I0). By (8.6), we can assume that

xi0 , xj =∑

l∈I0γlxl. Then, we have

(fji0`t, hi0`t)[w] = −xi0 ` xj ` xt+xi0 , xj ` xt = −hi0`j ` xt+∑l∈I0

γlhl`t ≡ 0 mod(S, [w]).

Case 2. [w] = xj0 ` xi ` xt (j0 > i, j0 ∈ I0). By (8.5), we have

(fj0i`t, hj0`i)[w] = −xi ` xj0 ` xt + xi, xj0 ` xt = −xi ` hj0`t ≡ 0 mod(S, [w]).

Case 3. [w] = xj ` xi ` xt0 ` xn (j > i, t0 ∈ I0). We have

(fji`t0 , ht0`n)[w] = −xi ` xj ` xt0 ` xn + xi, xj ` xt0 ` xn

= (−xi ` xj + xi, xj) ` ht0`n

≡ 0 mod(S, [w]).

(b ∧ d) [w] = xj ` xi ` xt a xq a xp (j > i, q > p). We have

(fji`t, ftaqp)[w]

= −xi ` xj ` xt a xq a xp + xi, xj ` xt a xq a xp

+xj ` xi ` xt a xp a xq − xj ` xi ` xt a xp, xq

= −xi ` xj ` ftaqp + xi, xj ` ftaqp + fji`t a xp a xq − fji`t a xp, xq

≡ 0 mod(S, [w]).

(b ∧ e) [w] = xj ` xi ` xt a xn0 (j > i, n0 ∈ I0). We have

(fji`t, htan0)[w] = −xi ` xj ` xt a xn0 + xi, xj ` xt a xn0

= (−xi ` xj + xi, xj) ` htan0

≡ 0 mod(S, [w]).

(c ∧ a) There are two cases to consider: [w] = xn0 ` xt (n0 ∈ I0) and [w] = xn0 ` xt `xs (n0 ∈ I0).

For [w] = xn0 ` xt (n0 ∈ I0), we have

(hn0`t, fn0t)[w] = xt a xn0 − xt, xn0 = htan0 ≡ 0 mod(S, [w]).

For [w] = xn0 ` xt ` xs (n0 ∈ I0), we have

(hn0`t, fts)[w] = xn0 ` xs a xt − xn0 ` xs, xt = hn0`s a xt − hn0`s,t ≡ 0 mod(S, [w]).

131

(c ∧ b) [w] = xn0 ` xt ` xs ` xp (t > s, n0 ∈ I0). We have

(hn0`t, fts`p)[w] = xn0 ` xs ` xt ` xp − xn0 ` xs, xt ` xp

= hn0`s ` xt ` xp − hn0`s,t ` xp

≡ 0 mod(S, [w]).

(c ∧ c) [w] = xn0 ` xt0 ` xr (n0, t0 ∈ I0). We have

(hn0`t0 , ht0`r)[w] = 0.

(c ∧ d) [w] = xn0 ` xt a xq a xp (q > p, n0 ∈ I0). We have

(hn0`t, ftaqp)[w] = xn0 ` xt a xp a xq − xn0 ` xt a xp, xq

= hn0`t a (xp a xq − xp, xq)

≡ 0 mod(S, [w]).

(c ∧ e) [w] = xn0 ` xt a xs0 (n0, s0 ∈ I0). We have

(hn0`t, htas0)[w] = 0.

Since (d∧ d), (d∧ e), (e∧ d), (e∧ e) are symmetric with (b∧ b), (b∧ c), (c∧ b), (c∧ c)respectively, they have the similar representations. We omit the details.

So, we show that S is a Grobner-Shirshov basis.

(iii) Clearly, the mentioned set is just the set Irr(S). Now, the results follow from

Theorem 8.14.

Recall a Grobner-Shirshov basis S is called minimal if S has no inclusion composition

in S.

Remark Let the notation be in Theorem 8.17. Let Smin consist of the following polyno-

mials:

(a) fji = xj ` xi − xi a xj + xi, xj (i ∈ I, j ∈ I − I0)

(b) fji`t = xj ` xi ` xt − xi ` xj ` xt + xi, xj ` xt (i, j ∈ I − I0, j > i, t ∈ I)

(c) hi0`t = xi0 ` xt (i0 ∈ I0, t ∈ I)

(d) ftaji = xt a xj a xi − xt a xi a xj + xt a xi, xj (i, j ∈ I − I0, j > i, t ∈ I)

(e) htai0 = xt a xi0 (i0 ∈ I0, t ∈ I)

Then Smin is a minimal Grobner-Shirshov basis for D(X|S).

We have the following corollary.

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Corollary 8.18 Let the notation be as in Theorem 8.17. Then as linear spaces, U(L) is

isomorphic to L⊗U(L/L0), where U(L/L0) is the universal enveloping of the Lie algebra

L/L0.

Proof. Clearly, xj | j ∈ I − I0 is a k-basis of the Lie algebra L/L0. It is well known

that the universal enveloping U(L/L0) of the Lie algebra L/L0 has a k-basis

xi1xi2 . . . xik | i1 ≤ · · · ≤ ik, ip ∈ I − I0, 1 ≤ p ≤ k, k ≥ 0.

By using (iii) in Theorem 8.17, the result follows.

Definition 8.19 Let D be a dialgebra. An element e ∈ D is called a bar unit of D if

e ` x = x a e = x for any x ∈ D.

Theorem 8.20 Each dialgebra has a bar unit extension.

Proof. Let (D,`,a) be an arbitrary dialgebra over a field k and A the ideal of D

generated by the set a a b − a ` b| a, b ∈ D. Let X0 = xi0|i0 ∈ I0 be a k-basis

of A and X = xi|i ∈ I a well-ordered k-basis of D such that I0 ⊆ I. Then D has

a presentation by the multiplication table D = D(X|S), where S = xi ` xj − xi `xj, xi a xj −xi a xj, i, j ∈ I, where xi ` xj and xi a xj are linear combinations

of xt, t ∈ I.Let D1 = D(X ∪ e|S1), where S1 = S ∪ e ` y − y, y a e− y, e a x0, x0 ` e | y ∈

X ∪ e, x0 ∈ X0. Then D1 is a dialgebra with a bar unit e.

Denote

1. fi`j = xi ` xj − xi ` xj,

2. fiaj = xi a xj − xi a xj,

3. ge`y = e ` y − y,

4. gyae = y a e− y,

5. hxi0`e = xi0 ` e,

6. heaxi0= e a xi0 ,

where i, j ∈ I, i0 ∈ I0, y ∈ X ∪ e.We show that xt a xi0 = 0 and xi0 ` xt = 0 for any t ∈ I, i0 ∈ I0.Since xi0 ∈ A, we have xi0 =

∑αi(cifidi), where fi = ai a bi − ai ` bi, αi ∈ k,

ai, bi ∈ D and ci, di ∈ X∗.

133

Since xt a (ci(ai a bi − ai ` bi)di) = 0, we have xt a ciai a bi − ai ` bidi = 0

for each i. Then xt a xi0 = 0.

By symmetry, we have xi0 ` xt = 0.

To prove the theorem, by using our Theorem 8.14, it suffices to prove that with the

ordering on [(X ∪ e)∗] as before, where x < e, x ∈ X, S1 is a Grobner-Shirshov basis

in D(X ∪ e). Now, we show that all compositions in S1 are trivial.

All possible compositions of left and right multiplication are: z a fi`j, z a ge`y,

z a hxi0`e, fiaj ` z, gyae ` z, heaxi0

` z, z ∈ X ∪ e.For z a fi`j, z = xt ∈ X, since (xt a xi) a xj = xt a (xi ` xj), we have xt a xi a

xj = xt a xi ` xj and

xt a fi`j

= xt a xi a xj − xt a xi ` xj

= ftai a xj + ftaiaj − ftai`j + xt a xi a xj − xt a xi ` xj

= ftai a xj + ftaiaj − ftai`j

≡ 0 mod(S1).

For z a fi`j, z = e, let xi a xj − xi ` xj =∑αi0xi0 . Then

e a fi`j = e a xi a xj − e a xi ` xj

= e a (xi a xj − xi a xj) + e a xi a xj − e a xi ` xj

= e a fiaj +∑

αi0heaxi0

≡ 0 mod(S1).

For z a ge`y, we have

z a ge`y = z a e a y − z a y = (z a e− z) a y = gzae a y ≡ 0 mod(S1).

For z a hxi0`e, we have

z a hxi0`e = z a xi0 a e = z a gxi0

ae + z a xi0 .

It is clear that z a xi0 = heaxi0if z = e and z a xi0 = xt a xi0 − xt a xi0 = ftai0 if

z = xt ∈ X, since xt a xi0 = 0. This implies that z a hxi0`e ≡ 0 mod(S1).

Thus we show that all compositions of left multiplication in S1 are trivial modulo S1.

By symmetry, all compositions of right multiplication in S1 are trivial modulo S1.

Now, all possible ambiguities [w] of compositions of intersection in S1 are:

1 ∧ 1, [xixjxt]; 1 ∧ 2, [xixjxt]; 1 ∧ 4, [xixje]; 1 ∧ 5, [xixi0 e].

134

2 ∧ 2, [xixjxt]; 2 ∧ 4, [xixje].

3 ∧ 1, [exixj]; 3 ∧ 2, [exixj]; 3 ∧ 3, [eey]; 3 ∧ 4, [eye]; 3 ∧ 5, [exi0 e]; 3 ∧ 6, [eexi0 ].

4 ∧ 4, [yee]; 4 ∧ 6, [yexi0 ].

5 ∧ 3, [xi0ey]; 5 ∧ 4, [xi0 ee]; 5 ∧ 6, [xi0 exj0 ].

6 ∧ 2, [exi0xj]; 6 ∧ 4, [exi0e].

In the above, all i, j, t ∈ I, i0, j0 ∈ I0 and y ∈ X ∪ e.There is no composition of inclusion in S1.

We will show that all compositions of intersection in S1 are trivial. We check only

the cases of 1 ∧ 2, 1 ∧ 5 and 4 ∧ 6. Others can be similarly proved.

For 1 ∧ 2, [w] = [xixjxt], since (xi ` xj) a xt = xi ` (xj a xt), we have xi ` xj axt = xi ` xj a xt and

(1 ∧ 2)[w] = −xi ` xj a xt + xi ` xj a xt

= −fi`jat + fi`jat − xi ` xj a xt+ xi ` xj a xt

= −fi`jat + fi`jat

≡ 0 mod(S1, [w]).

For 1 ∧ 5, [w] = [xixi0 e], since xi ` xi0 ∈ A, we have xi ` xi0 =∑αj0xj0 and

(1 ∧ 5)[w] = xi ` xi0 ` e =∑

αj0hxj0`e ≡ 0 mod(S1, [w]).

For 4 ∧ 6, [w] = [yexi0 ], we have (4 ∧ 6)[w] = −heaxi0if y = e and (4 ∧ 6)[w] = −ftai0

if y = xt ∈ X since xt a xi0 = 0. Then (4 ∧ 6)[w] ≡ 0 mod(S1, [w]).

Then all the compositions in S1 are trivial.

The proof is complete.

Remark Let the notation be as in the proof of Theorem 8.20. Let D′ = D(X ∪ejJ |S ′)be a dialgebra, where S ′ = S∪ej ` y−y, y a ej−y, ej a x0, x0 ` ej | y ∈ X∪ejJ , x0 ∈X0, j ∈ J. Let J be a well-ordered set. Then with the ordering on [(X ∪ ejJ)∗] as

before, where xi < ej for all i ∈ I, j ∈ J , by a similar proof of Theorem 8.20, S ′ is a

Grobner-Shirshov basis in D(X ∪ ejJ). It follows from Theorem 8.14 that D can be

embedded into the dialgebra D′ while D′ has bar units ejJ .

Definition 8.21 Let D1, D2 be dialgebras over a field k. The dialgebra D1 ∗D2 with two

dialgebra homomorphisms ε1 : D1 → D1 ∗D2, ε2 : D2 → D1 ∗D2 is called the free product

of D1, D2, if the following diagram commute:

135

-

?

@@

@@

@@R

D

∃!f

D1 ∗D2D1 D2ε1 ε2

∀δ1 ∀δ2

where D is a dialgebra, δ1, δ2 are dialgebra homomorphisms and f : D1 ∗ D2 → D is a

dialgebra homomorphism such that fε1 = δ1, fε2 = δ2.

An equivalent definition is as follows: Let Di = D(Xi|Si) be a presentation by genera-

tors and defining relations with X1∩X2 = ∅, i = 1, 2. Then D1∗D2 = D(X1∪X2|S1∪S2).

Let (D1,`,a), (D2,`,a) be two dialgebras over a field k, A1 the ideal of D1 generated

by the set a a b − a ` b| a, b ∈ D1 and A2 the ideal of D2 generated by the set

c a d− c ` d| c, d ∈ D2. Let X0 = xi0|i0 ∈ I0 be a k-basis of A1 and X = xi|i ∈ Ia well-ordered k-basis of D1 such that I0 ⊆ I. Let Y0 = yl0|l0 ∈ J0 be a k-basis of A2

and Y = yl|l ∈ J a well-ordered k-basis of D2 such that J0 ⊆ J . Then D1 and D2 have

multiplication tables:

D1 = D(X|S1), S1 = xi ` xj − xi ` xj, xi a xj − xi a xj, i, j ∈ I,

D2 = D(Y |S2), S2 = yl ` ym − yl ` ym, yl a ym − yl a ym, l,m ∈ J.

The free product D1 ∗D2 of D1 and D2 is

D1 ∗D2 = D(X ∪ Y |S1 ∪ S2).

We order X∪Y by xi < yj for any i ∈ I, j ∈ J . Then we have the following theorem.

Theorem 8.22 (i) S is a Grobner-Shirshov basis of D1 ∗D2 = D(X ∪Y |S1 ∪S2), where

S consists of the following relations:

1. fxi`xj= xi ` xj − xi ` xj, i, j ∈ I,

2. fxiaxj= xi a xj − xi a xj, i, j ∈ I,

3. fyl`ym = yl ` ym − yl ` ym, l,m ∈ J,

4. fylaym = yl a ym − yl a ym, l,m ∈ J,

5. hxi0`yl

= xi0 ` yl, i0 ∈ I0, l ∈ J,

6. hylaxi0= yl a xi0 , i0 ∈ I0, l ∈ J,

7. hyl0`xi

= yl0 ` xi, i ∈ I, l0 ∈ J0,

8. hxiayl0= xi a yl0 , i ∈ I, l0 ∈ J0.

136

(ii) Irr(S), which is a k-linear basis of D1∗D2, consists of all elements z−m · · · z−1z0z1 · · · zn,

where m,n ≥ 0, z0 ∈ X∪Y, zi ∈ (X\X0)∪(Y \Y0),−m ≤ i ≤ n, i 6= 0, neither zj, zj+1 ⊆X nor zj, zj+1 ⊆ Y,−m ≤ j ≤ n− 1.

Proof. By the proof of Theorem 8.20, we have xi a xi0 = 0, xi0 ` xi = 0, yl ayl0 = 0 and yl0 ` yl = 0 for any i ∈ I, i0 ∈ I0, l ∈ J, l0 ∈ J0.

Firstly, we prove that hylaxi0∈ Id(S1 ∪ S2) for any i0 ∈ I0, l ∈ J .

Since yl a (ci(ai a bi−ai ` bi)di) = yl a (ci((ai a bi−ai a bi)− (ai ` bi−ai `bi)di) ∈ Id(S1 ∪S2), we have yl a ciai a bi− ai ` bidi ∈ Id(S1 ∪S2) for all i, l. Then

hylaxi0∈ Id(S1 ∪ S2).

Similarly, we have hxi0`yl, hyl0

`xi, hxiayl0

∈ Id(S1 ∪ S2) for any i ∈ I, i0 ∈ I0, l ∈J, l0 ∈ J0.

Secondly, we will show that all compositions in S are trivial.

All possible compositions of left and right multiplication are: z a fxi`xj, z a

fyl`ym , z a hxi0`yl, z a hyl0

`xi, fxiaxj

` z, fylaym ` z, hylaxi0` z, hxiayl0

` z, where

z ∈ X ∪ Y .

By a similar proof in Theorem 8.20, all compositions of left and right multiplication

mentioned the above are trivial modulo S.

Now, all possible ambiguities [w] of compositions of intersection in S are:

1 ∧ 1, [xixjxt]; 1 ∧ 2, [xixjxt]; 1 ∧ 5, [xixi0 yl]; 1 ∧ 8, [xixjyl0 ].

2 ∧ 2, [xixjxt]; 2 ∧ 8, [xixjyl0 ].

3 ∧ 3, [ylymyt]; 3 ∧ 4, [ylymyt]; 3 ∧ 6, [ylymxi0 ]; 3 ∧ 7, [ymyl0xi].

4 ∧ 4, [ylymyt]; 4 ∧ 6, [ylymxi0 ].

5 ∧ 3, [xi0ylyt]; 5 ∧ 4, [xi0 ylyt]; 5 ∧ 6, [xi0 ylxj0 ]; 5 ∧ 7, [xi0yl0 xt].

6 ∧ 2, [ylxi0xt]; 6 ∧ 8, [ymxi0yl0 ].

7 ∧ 1, [yl0xixj]; 7 ∧ 2, [yl0xixj]; 7 ∧ 5, [yl0xi0 ym]; 7 ∧ 8, [yl0xiym0 ].

8 ∧ 4, [xiyl0yt]; 8 ∧ 6, [xiyl0xi0 ].

There is no composition of inclusion in S.

We will show that all compositions of intersection in S are trivial. We check only the

cases of 1 ∧ 5 and 2 ∧ 8. Others can be similarly proved.

For 1 ∧ 5, [w] = [xixi0 yl], let xi ` xi0 =∑αt0xt0 . Then

(1 ∧ 5)[w] = −xi ` xi0 ` yl = −∑

αt0hxt0`yl≡ 0 mod(S, [w]).

For 2 ∧ 8, [w] = [xixjyl0 ], let xi a xj =∑αtxt. Then

(2 ∧ 8)[w] = −xi a xj a yl0 = −∑

αthxtayl0≡ 0 mod(S, [w]).

137

Then all the compositions in S are trivial. This show (i).

(ii) follows from our Theorem 8.14.

Definition 8.23 Let X = x1, . . . , xn be a set, k a field of characteristic 6= 2 and

(aij)n×n a non-zero symmetric matrix over k. Denote

D(X ∪ e | xi ` xj + xj a xi − 2aije, e ` y − y, y a e− y, xi, xj ∈ X, y ∈ X ∪ e)

by C(n, f). Then C(n, f) is called a Clifford dialgebra.

We order X ∪ e by x1 < · · · < xn < e.

Theorem 8.24 Let the notation be as the above. Then

(i) S is a Grobner-Shirshov basis of Clifford dialgebra C(n, f), where S consists of the

following relations:

1. fxixj= xi ` xj + xj a xi − 2aije,

2. ge`y = e ` y − y,

3. gyae = y a e− y,

4. fyaxixj= y a xi a xj + y a xj a xi − 2aijy, (i > j),

5. fyaxixi= y a xi a xi − aiiy,

6. fxixj`y = xi ` xj ` y + xj ` xi ` y − 2aijy, (i > j),

7. fxixi`y = xi ` xi ` y − aiiy,

8. hxie = xi ` e− e a xi,

where xi, xj ∈ X, y ∈ X ∪ e.

(ii) A k-linear basis of C(n, f) is a set of all elements of the form yxi1 · · ·xik, where

y ∈ X ∪ e, xij ∈ X and i1 < i2 < · · · < ik (k ≥ 0).

Proof. Let S1 = fxixj, ge`y, gyae | xi, xj ∈ X, y ∈ X ∪ e.

Firstly, we will show that fyaxixj, fyaxixi

, fxixj`y, fxixi`y, hxie ∈ Id(S1).

In fact, fyaxixj= y a fxixj

+ 2aijgyae implies fyaxixj, fyaxixi

∈ Id(S1). By symmetry,

we have fxixj`y, fxixi`y ∈ Id(S1).

If there exists t such that ait 6= 0, then

2aithxie = fxixi`xt − xi ` fxi`xt + fxi`xt a xi − fxtaxixi∈ Id(S1).

138

Otherwise, ait = 0 for any t. Since (aij) 6= 0, there exists j 6= i such that ajt 6= 0 for some

t. Then

2ajthxie

= fxixj`xt − xi ` fxj`xt − xj ` fxi`xt + fxi`xt a xj + fxj`xt a xi − fxtaxixj∈ Id(S1).

This shows that hxie ∈ Id(S1).

Secondly, we will show that all compositions in S is trivial.

All possible compositions of left and right multiplication are: z a fxixj, z a ge`y, z a

fxixj`y, z a fxixi`y, z a hxie, fxixj` z, gyae ` z, fyaxixj

` z, fyaxixi` z, hxie ` z, where

z ∈ X ∪ e. We just check the cases of fyaxixj` z and hxie ` z. Others can be similarly

proved.

For fyaxixj` z, we have

fyaxixj` z = y ` xi ` xj ` z + y ` xj ` xi ` z − 2aijy ` z = y ` fxixj`z ≡ 0 mod(S).

For hxie ` z,

hxie ` z = xi ` e ` z − e ` xi ` z = xi ` ge`z − ge`xi` z ≡ 0 mod(S).

Now, all possible ambiguities [w] of compositions of intersection in S are:

1 ∧ 3, [xixje]; 1 ∧ 4, [xixjxmxn] (m > n); 1 ∧ 5, [xixjxnxn].

2 ∧ 1, [exixj]; 2 ∧ 2, [eey]; 2 ∧ 3, [eye]; 2 ∧ 4, [eyxixj] (i > j);

2 ∧ 5, [eyxixi]; 2 ∧ 6, [exixj y] (i > j); 2 ∧ 7, [exixiy]; 2 ∧ 8, [exie].

3 ∧ 3, [yee]; 3 ∧ 4, [yexixj] (i > j); 3 ∧ 5, [yexixi].

4 ∧ 3, [yxixje] (i > j); 4 ∧ 4, [yxixjxmxn] (i > j,m > n), [yxixjxt] (i > j > t);

4 ∧ 5, [yxixjxtxt] (i > j), [yxixjxj] (i > j).

5 ∧ 3, [yxixie]; 5 ∧ 4, [yxixixmxn] (m > n), [yxixixj] (i > j);

5 ∧ 5, [yxixixmxm], [yxixixi].

6 ∧ 1, [xixjxmxn] (i > j); 6 ∧ 2, [xixjey] (i > j); 6 ∧ 3, [xixj ye] (i > j);

6 ∧ 4, [xixj yxmxn] (i > j,m > n); 6 ∧ 5, [xixj yxmxm] (i > j);

6 ∧ 6, [xixjxmxny] (i > j,m > n), [xixjxty] (i > j > t);

6 ∧ 7, [xixjxmxmy] (i > j), [xixjxj y] (i > j); 6 ∧ 8, [xixjxte] (i > j).

7 ∧ 1, [xixixmxn]; 7 ∧ 2, [xixiey]; 7 ∧ 3, [xixiye]; 7 ∧ 4, [xixiyxmxn] (m > n);

7 ∧ 5, [xixiyxmxm]; 7 ∧ 6, [xixixmxny] (m > n), [xixixty] (i > t);

7 ∧ 7, [xixixmxmy], [xixixiy]; 7 ∧ 8, [xixixj e].

8 ∧ 3, [xiee]; 8 ∧ 4, [xiexmxn] (m > n); 8 ∧ 5, [xiexmxm].

139

All possible ambiguities [w] of compositions of inclusion in S are:

6 ∧ 1, [xixjxt] (i > j); 6 ∧ 8, [xixj e] (i > j).

7 ∧ 1, [xixixj]; 7 ∧ 8, [xixie].

We just check the cases of intersection 1 ∧ 4, 4 ∧ 4, 6 ∧ 4, 6 ∧ 8, 8 ∧ 4 and of inclusion

6 ∧ 1, 6 ∧ 8. Others can be similarly proved.

For 1 ∧ 4, [w] = [xixjxmxn] (m > n), we have

(1 ∧ 4)[w]

= xj a xi a xm a xn − 2aije a xm a xn − xi ` xj a xn a xm + 2amnxi ` xj

= xj a fxiaxmxn − 2aijfeaxmxn − fxixja xn a xm + 2amnfxixj

≡ 0 mod(S, [w]).

For 4 ∧ 4, there are two cases to consider: [w1] = [yxixjxmxn] (i > j,m > n) and

[w2] = [yxixjxt] (i > j > t). We have

(4 ∧ 4)[w1]

= y a xj a xi a xm a xn − 2aijy a xm a xn − y a xi a xj a xn a xm + 2amny a xi a xj

= y a xj a fxiaxmxn − 2aijfyaxmxn − fyaxixja xn a xm + 2amnfyaxixj

≡ 0 mod(S, [w1]) and

(4 ∧ 4)[w2]

= y a xj a xi a xt − 2aijy a xt − y a xi a xt a xj + 2ajty a xi

= y a fxjaxixt − fyaxjxt a xi − fyaxixt a xj + y a fxtaxixj

≡ 0 mod(S, [w2]).

For 6 ∧ 4, [w] = [xixj yxmxn] (i > j,m > n), we have

(6 ∧ 4)[w]

= xj ` xi ` y a xm a xn − 2aijy a xm a xn − xi ` xj ` y a xn a xm + 2amnxi ` xj ` y

= xj ` xi ` fyaxmxn − 2aijfyaxmxn − fxixj`y a xn a xm + 2amnfxixj`y

≡ 0 mod(S, [w]).

For 6 ∧ 8, [w] = [xixjxte] (i > j), we have

(6 ∧ 8)[w] = xj ` xi ` xt ` e− 2aijxt ` e+ xi ` xj ` e a xt

= xj ` xi ` hxte − 2aijhxte + fxixj`e a xt

≡ 0 mod(S, [w]).

140

For 8 ∧ 4, [w] = [xiexmxn] (m > n), we have

(8 ∧ 4)[w] = −e a xi a xm a xn − xi ` e a xn a xm + 2amnxi ` e

= −e a fxiaxmxn − hxie a xn a xm + 2amnhxie

≡ 0 mod(S, [w]).

Now, we check the compositions of inclusion 6 ∧ 1 and 6 ∧ 8.

For 6 ∧ 1, [w] = [xixjxt] (i > j), we have

(6 ∧ 1)[w] = xj ` xi ` xt − 2aijxt − xi ` xt a xj + 2ajtxi ` e

= xj ` fxixt − fxixt a xj + 2ajthxie − fxjxt a xi + fxtaxixj+ 2aithxje

≡ 0 mod(S, [w]).

For 6 ∧ 8, [w] = [xixj e] (i > j), we have

(6 ∧ 8)[w] = xj ` xi ` e− 2aije+ xi ` e a xj

= xj ` hxie + hxie a xj + hxje a xi + feaxixj

≡ 0 mod(S, [w]).

Then all the compositions in S are trivial. We have proved (i).

For (ii), since the mentioned set is just the set Irr(S), by Theorem 8.14 the result

holds.

The proof is complete.

Remark In the Theorem 8.24, if the matrix (aij)n×n = 0, then Clifford dialgebra C(n, f)

has a Grobner-Shirshov basis S ′ which consists of the relations 1–7.

141

Chapter 9 A History and a Survey of

Grobner-Shirshov Bases Theory

A.I. Shirshov (also A.I. Sirsov) in his pioneering work ([202], 1962) posted the fol-

lowing fundamental question:

How to find a linear basis of any Lie algebra presented by generators and defining

relations?

He gave an infinite algorithm for a solution of this problem using a new notion

of composition (“s-polynomial” in a later terminology of Buchberger [61, 62]) of two

Lie polynomials and a new notion of completion of any set of Lie polynomials (to add

consequently “non-trivial” compositions, critical pair/completion (cpc-) algorithm in a

later Knuth-Bendix [135] and Buchberger [64, 65] terminology).

Shirshov’s algorithm is as follows. Let S be a subset of Lie polynomials in Lie(X) of

the free algebra k〈X〉 on X over a field k (the algebra of non-commutative polynomials

on X over k) . Let S ′ be a set by adding all non-trivial Lie compositions (“Lie s-

polynomials”) of elements of S to S (S ⊆ S ′). A problem of triviality of a Lie polynomial

modulo a finite (or recursive) set S is algorithmically solvable using Shirshov Lie reduction

algorithm from his previous paper [198], 1958. Then one gets in general an infinite

sequence of Lie multi-compositions S ⊆ S ′ ⊆ S ′′ ⊆ · · · ⊆ S(n) ⊆ . . . . The union Sc of the

sequence has the property that any Lie composition of elements of Sc is trivial relative

to Sc. It is what is now called a Lie GS basis. Then from a new “Composition-Diamond

lemma∗ for Lie algebras” (Lemma 3 in [202]) it follows that the set Irr(Sc) of all Sc-

irreducible (or Sc-reduced) “basic” Lie monomials [u] in X is a linear basis of the Lie

algebra Lie(X|S) generated by X with defining relations S. Here “basic” Lie monomial

means a Lie monomial from a special linear basis of the free Lie algebra Lie(X) ⊂ k〈X〉,so called Lyndon-Shirshov (LS for short) basis (Shirshov [202] and Chen-Fox-Lyndon [69],

see below). A LS monomial [u] is called Sc-irreducible (or Sc-reduced) if u, the associative

support of [u], does not contain word s for any s ∈ S, where s is the maximum (in deg-

lex ordering) word of s as associative polynomial. To be more precise, Shirshov used

his reduction algorithm at each step S, S ′, S ′′, . . . . Then one has a direct system of sets

S → S ′ → S ′′ → . . . and Sc = lim−→S(n) is what is now called a “minimal GS basis”. As

the result, Shirshov algorithm gives a solution of the above problem for Lie algebras.

Shirshov algorithm dealing with the word problem is an infinite algorithm as Knuth-

∗The name “Composition-Diamond lemma” is a combination name of “Neuman Diamond Lemma”

[167], “Shirshov Composition Lemma” [202] and “Bergman Diamond Lemma” [9].

142

Bendix algorithm [135], 1970 that deals with the identity problem for any variety of

universal algebras.† An initial data for Knuth-Bendix algorithm is defining identities of a

variety. The result of the algorithm, if any, is a “Knuth-Bendix basis” of identities of the

variety (not a GS basis of defining relations, say, a Lie algebra).

Shirshov algorithm gives linear basis and algorithmic decidability of the word problem

for one relator Lie algebras [202], (recursive) linear basis for Lie algebras with (finite)

homogeneous defining relations [202], and linear basis for free product of Lie algebras

with known linear basis [201]. Also he proved the Freiheitssatz (the freeness theorem)

for Lie algebras [202] (for any one-relator Lie algebra Lie(X|f), a subalgebra 〈X\xi〉,where xi is involved in f , is a free Lie algebra). Shirshov problem [202] for decidability

of the word problem for Lie algebras was solved negatively in [19]. More generally, it was

proved [19] that some recursively presented Lie algebras with an unsolvable word problem

can be embedded into finitely presented Lie algebras (with the unsolvable word problem).

It is a weak analogy of Higman embedding theorem for groups [113]. A problem [19]

whether an analogy of Higman embedding theorem is valid for Lie algebras is still open.

For associative algebras the similar problem [19] was solved positively by V.Y. Belyaev

[8]. A simple example of a Lie algebra with the unsolvable word problem was given by

G.P. Kukin [138].

Following Shirshov’s paper [202] a similar algorithm is valid also for associative al-

gebras. It is since he fixed that Lie(X) is the subspace of the Lie polynomials in the

free associative algebra k〈X〉. Then to define a Lie composition 〈f, g〉w of two Lie poly-

nomials relative to an associative word w, he defines first the associative composition

(non-commutative “s-polynomial”) (f, g)w = fb − ag, a, b ∈ X∗. Then he puts some

brackets 〈f, g〉w = [fb]f − [ag]g following his special bracketing lemma in [198]. One may

get Sc for any S ⊂ k〈X〉 in the same way as for Lie polynomials and in the same way

as for Lie algebras (“CD-lemma for associative algebras”) to get that Irr(Sc) is a linear

basis of associative algebra k〈X|S〉 generated by X with defining relations S. All proofs

are in the same manner and much easier than in [202].

Even more, the cases of semigroups and groups presented by generators and defining

relations are just special cases of associative algebras via semigroup and group algebras.

All in all, Shirshov algorithm gives linear bases and normal forms of elements of any Lie

algebra, associative algebra, semigroup or group presented by generators and defining

relations! The algorithm does work in many cases (see below), as opposed to for example

an analogous algorithm for non-associative algebras.

†We use the usual algebraic terminology “the word problem”, “ the identity problem”, see, for example,

O.G. Kharlampovich, M.V. Sapir [134].

143

The Grobner bases theory and Buchberger algorithm were initiated by B. Buchberger

(Thesis [61] 1965, paper [62] 1970) for commutative-associative algebras. Buchberger

algorithm is a finite algorithm for finitely generated commutative algebras. It is one of

the most useful and famous algorithm in modern computer science.

The Shirshov paper [202] was in a spirit of A.G. Kurosh (1908-1972) program to

study non-associative (relatively) free algebras and free products of algebras initiated in

Kurosh paper [139], 1947. In this paper he proved non-associative analogies of Nielsen-

Shreier and Kurosh theorems for groups. It took quite a few years to clarify the situation

for Lie algebras in Shirshov’s papers [195], 1953 and [201], 1962 closely connected with

his GS bases theory. It is important to note that quite unexpectedly the Kurosh program

led to the Shirshov’s theory of GS bases for Lie and associative algebras [202].

A step in the Kurosh program was done by his student A.I. Zhukov in his PhD Thesis

[221], 1950 who algorithmically solved the word problem for non-associative algebras. In a

sense it was a beginning GS bases theory for non-associative algebras. The main difference

with the future Shirshov’s approach is that Zhukov did not use any linear ordering of non-

associative monomials. Instead he chooses any monomial of maximal degree as a “leading”

monomial of a polynomial. Also for non-associative algebras there is no “composition of

intersection” (“s-polynomial”). In this sense it cannot be a model for Lie and associative

algebras.‡

A.I. Shirshov, also a student of Kurosh, defended his Candidate of Sciences Thesis

[194] at Moscow University in 1953. His Thesis together with next followed paper [198],

1958 may be viewed as a background of his later method of GS bases. He proved the

free subalgebra theorem for free Lie algebras (now known as Shirshov-Witt theorem,

see also Witt [214], 1956) using the elimination process rediscovered by Lazard [145],

1960. He used the elimination process later [198], 1958 as a general method to prove

properties of regular (LS) words, including an algorithm of (special) bracketing of a LS

word (with a fix LS subword). The last algorithm is of some importance in his GS

bases theory for Lie algebras (particular in a definition of the compositions of two Lie

polynomials). Also Shirshov proved the free subalgebra theorem for (anti-) commutative

non-associative algebras [197], 1954. He used the paper lately in [203], 1962 for GS bases

theory of (commutative, anti-commutative) non-associative algebras. Last but not the

‡After PhD Thesis, 1950, A.I. Zhukov moved to now Keldysh Institute of Applied Mathemat-

ics (Moscow) and then to an atomic organization that is now known as Chelyabinsk-90 (Ural Re-

gion) doing computational mathematics. S.K. Godunov in “Reminiscence about numerical schemes”,

arxiv.org/pdf/0810.0649, 2008, mentioned his name in according with a creation of the famous Godunov

numerical method. So, A.I. Zhukov was a forerunner of two important computational methods!

144

least Shirshov found a series of bases of a free Lie algebra published 10 years later in the

first issue of the new A.I. Malcev’s Journal “Algebra and Logic” [200], 1962.§

LS basis is a particular case of the Shirshov or Hall-Shirshov series of bases (cf.

Reutenauer [185], where this series was called “Hall series”). In the definition of his series,

Shirshov used the Hall inductive procedure (see Ph. Hall [111], 1933, M. Hall [110], 1950):

a non-associative monomial w = ((u)(v)) is a basic monomial if 1) (u), (v) are basic, 2)

(u) > (v), 3) if (u) = ((u1)(u2)), then (u2) ≤ (v). But instead of an ordering by the

degree function (Hall words), he used any linear ordering of non-associative monomials

such that ((u)(v)) > (v). For example, in his Thesis [194], 1953 he used an ordering by the

“content” of monomials (content of, say, a monomial (u) = ((x2x1)((x2x1)x1)) is a vector

(x2, x2, x1, x1, x1)). Actually content u of (u) may be viewed as a commutative-associative

word that equals to u in the free commutative semigroup. Two contents are compared

lexicographically (a proper prefix of a content is greater than the content).

If one uses lexicographical ordering, (u) (v) if u v lexicographically, then one will

get the LS basis.¶ For example, for alphabets x1, x2, x2 x1 one has Lyndon-Shirshov

basic monomials by induction:

x2, x1, [x2x1], [x2[x2x1]] = [x2x2x1], [[x2x1]x1] = [x2x1x1], [x2[x2x2x1]] = [x2x2x2x1],

[x2[x2x1x1]] = [x2x2x1x1], [[x2x1x1]x1] = [x2x1x1x1], [[x2x1][x2x1x1]] = [x2x1x2x1x1] and

so on.

They are exactly all Shirshov regular (LS) Lie monomials and their associative sup-

ports are exactly all Shirshov regular words with 1-1 correspondence between two sets

given by the Shirshov elimination (bracketing) algorithm of (associative) words.

§It must be pointed out that A.I. Malcev (1909-1967) inspired Shirshov’s works very much. Malcev

was an official opponent (referee) of his (second) Doctor of Sciences Dissertation, MSU, 1958. L.A. Bokut

remembers this event at the Science Council Meeting, Chairman A.N. Kolmogorov, and Malcev’s words

“Shirshov’s Dissertation is a brilliant one!”. Malcev and Shirshov worked together at the now Sobolev

Institute of Mathematics (Novosibirsk) since 1959 until Malcev’ unexpected death at 1967, and have been

friends despite the age difference. Malcev was head of the Algebra and Logic department (by the way,

L.A. Bokut is a member of the departement since 1960) and Shirshov was the first deputy-director of the

Institute (director was S.L. Sobolev). In those years, Malcev was interested in the theory of algorithms

of mathematical logic and algorithmic problems of model theory. Because of that, Shirshov had an

additional motivation to work on algorithmic problems for Lie algebras. Both Maltsev and Kurosh were

delighted with the Shirshov’s results of [202]. Malcev successfully presented the paper for an award of

the Presidium of the SB of the RAS (Sobolev and Malcev were only members of the Presidium from

Institute of mathematics at that time).¶Lyndon-Shirshov basis for the alphabet x1, x2 is different from above Shirshov “content” basis starting

with monomials of degree 7.

145

Let us remind that an elementary step of Shirshov elimination algorithm is to joint

the minimal letter of a word to previous ones by the bracketing and to continue this

process with lexicographical ordering of the new alphabet. For example, let x2 x1.

Then one has induction bracketing:

x2x1x2x1x1x2x1x1x1x1x2x1x1, [x2x1][x2x1x1][x2x1x1x1][x2x1x1],

[x2x1][[x2x1x1][x2x1x1x1]][x2x1x1], [[x2x1][[x2x1x1][x2x1x1x1]]][x2x1x1],

[[[x2x1][[x2x1x1][x2x1x1x1]]][x2x1x1]],

x2x1x1x1x2x1x1x2x1x2x2x1, [x2x1x1x1][x2x1x1][x2x1]x2[x2x1],

[x2x1x1x1][x2x1x1][x2x1][x2[x2x1]], x2x1x1x1 ≺ x2x1x1 ≺ x2x1 ≺ x2x2x1.

By the way, the second series of partial bracketing is an illustration of Shirshov

factorization theorem [198], 1958 that any word is a non-decreasing product of LS words

(it is often called by a mistake, see [10], as “Lyndon theorem”.)

Shirshov’s [198] the special bracketing of any LS word with a fix LS subword is as fol-

lows. Let us give an example of special bracketing of LS word w = x2x2x1x1x2x1x1x1 with

LS-subword u = x2x2x1. The Shirshov standard bracketing is [w] = [x2[[[x2x1]x1][x2x1x1x1]]].

The Shirshov special bracketing is [w]u = [[[u]x1][x2x1x1x1]]. In general, if w = aub then

the Shirshov standard bracketing gives [w] = [a[uc]d], where b = cd. Now, c = c1 · · · ct,each ci is LS-word, c1 · · · ct in lex-ordering (Shirshov factorization theorem). Then

one must change bracketing of [uc]:

[w]u = [a[. . . [[u][c1]] . . . [ct]]d]

The main property of [w]u is that [w]u is a monic associative polynomial with the maximal

monomial w, i.e., [w]u = w.

Actually, Shirshov [202], 1962 needs to use “double” relative bracketing of a regular

word with two separated LS subwords. Then he used implicitly the following property:

any LS subword of c = c1 · · · ct as above is a subword of some ci, 1 ≤ i ≤ t.

The Shirshov [198], 1958 definition of regular monomials ia as follows: (w) = ((u)(v))

is a regular monomial iff 1) w is a regular word, 2) (u), (v) are regular monomials (then

automatically u v in lex-ordering), 3) if (u) = ((u1)(u2)), then u2 v.

Once again, if one formally omits all Lie brackets in Shirshov’s paper [202], essentially

the same algorithm and essentially the same CD-lemma (with the same but much simpler

proof) give a linear basis of an associative algebra presented by generators and defining

relations. The differences are the following:

—No need to use LS monomials and LS words (since the set X∗ is a linear basis of

the free associative algebra k〈X〉);—The definition of associative compositions for monic polynomials f, g,

146

(f, g)w = fb− ag, w = f b = ag, deg(w) < deg(f) + deg(g), or (f, g)w = f − agb, w =

agb, w, a, b ∈ X∗, are much simpler than the definition of Lie compositions for monic Lie

polynomials f, g,

〈f, g〉w = [fb]f − [ag]g, w = f b = ag, deg(w) < deg(f) + deg(g), or

〈f, g〉w = f − [agb]g, w = agb, w, a, b, f , g ∈ X∗,

where [fb]f , [ag]g, [agb]g are special Shirshov’s bracketing of LS words w with fixed LS

subwords f , g correspondingly.

—A definition of the elimination of a leading word s of an associative monic polyno-

mial s is straightforward (asb→ a(rs)b, if s = s−rs, a, b ∈ X∗). But for Lie polynomials,

it is much more involved and uses the special Shirshov’s bracketing.

—The main idea of the Shirshov’s proof may be formulated as follows. Let S be a

complete set of monic Lie polynomials. If w = a1s1b1 = a2s2b2, where w, , ai, bi ∈ X∗, w

is a LS word, s1, s2 ∈ S, then Lie monomials [a1s1b1]s1 and [a2s2b2]s2 are equal modulo

the less Lie monomials from the Id(S):

[a1s1b1]s1 = [a2s2b2]s2 +∑i>2

αi[aisibi]si, where each si ∈ S, [aisibi]si

= aisibi < w.

Actually Shirshov proved more general result: if (a1s1b1) = a1s1b1, (a2s2b2) = a2s2b2 and

w = a1s1b1 = a2s2b2 then

(a1s1b1) = (a2s2b2) +∑i>2

αi(aisibi), where each si ∈ S, (aisibi) = aisibi < w.

Later we call a Lie polynomial (asb) as Lie normal S-word if (asb) = asb.

Exactly in this place he used the notion of composition and other notions and prop-

erties mentioned above.

For associative algebras (and for commutative-associative algebras) it is much easy

to prove the analogy of this property: If S is a complete monic set in k〈X〉 (k[X]) and

w = a1s1b1 = a2s2b2, ai, bi ∈ X∗, s1, s2 ∈ S, then

a1s1b1 = a2s2b2 +∑i>2

αiaisibi, where each si ∈ S, aisibi < w.

Summarizing we can say with confidence that the work (Shirshov [202]) implicitly

contains the CD-lemma for associative algebras as a simple exercise that does not require

new ideas. L.A. Bokut can confirm that Shirshov clearly understood this and told to him,

that “the case of associative algebras is the same”. Explicitly the lemma was formulated

147

in Bokut [20], 1976 (with reference to the Shirshov’s paper [202]), Bergman [9], 1978 and

Mora [166], 1986.

Let us stress once again that CD-Lemma for associative algebras can be applied to

any semigroups P = sgp〈X|S〉, in particular to any group, through the semigroup algebra

kP over a field k. The last algebra has the same generators and defining relations as P ,

kP = k〈X|S〉. Any composition of “twonomials” u1 − v1, u2 − v2 is a twonomial u − v.As the result, Shirshov’s algorithm applying to a set of semigroups relations S gives rise a

complete set of semigroup relations Sc. The Sc-irreducible words in X is a set of normal

forms of elements of P .

Before we go any farther, let us give some well known examples of presentations of

algebras, groups and semigroup.

—Grassman algebra k〈X|x2i = 0, xixj + xjxi = 0, i > j〉.

—Clifford algebra k〈X|xixj + xjxi = aij, 1 ≤ i, j ≤ n〉, where (aij) is an n × n

symmetric matrix over k.

—Universal enveloping algebra U(L) = k〈X|xixj − xjxi =∑αk

ijxk, i > j〉, where L

is a Lie algebra with a well ordered linear basis X = xi|i ∈ I, [xixj] =∑αk

ijxk, i > j.

The PBW theorem provides a linear basis of U(L): xi1xi2 · · ·xin | i1 ≤ i2 ≤ · · · ≤in, it ∈ I, t = 1, . . . , n, n ≥ 0. A. Kandri-Rody and V. Weispfenning [123] invented

an important class of “algebras of polynomial type” that included Clifford and universal

enveloping algebras. They created a GS bases theory for this class.

—More generally, let L = Lie(X|S) be a Lie algebra presented by generators and

a Lie GS basis S (in deg-lex ordering). Then the universal enveloping algebra U(L) has

presentation U(L) = k〈X|S(−)〉 , where S(−) is S as the set of associative polynomials.

PBW Theorem in a Shirshov form (see Theorem 4.8) says that S(−) is an associative GS

basis and a linear basis of U(L) consists of words u1u2 · · ·un, where ui are S-reduced LS

words, u1 u2 · · · un (in lex-ordering),‖ see [52, 53].

—Free Lie algebras over a commutative algebra. Ph. Hall, M. Hall, A.I. Shirshov,

R. Lyndon provide different linear bases of the algebras (Hall basis, Hall-Shirshov series

of bases, Lyndon-Shirshov basis). See also R-basis [14] and recent papers [31, 34].

—Lie algebras presented by Chevalley generators and defining relations of types

An, Bn,

Cn, Dn, G2, F3, E6, E7, E8. The Serre theorem provides linear bases and multiplication

tables of the algebras (they are finite dimensional simple Lie algebras over a field k). See

also [45, 46, 47].

‖For any Lie polynomial s ∈ Lie(X) ⊂ k〈X〉, the maximal associative word s of s is a LS word and

the regular LS monomial [s] is the maximal LS monomial of s (both in deg-lex ordering).

148

—Finite Coxeter groups. Groups presented by Coxeter generators and defining rela-

tions of types An, Bn, Dn, G2, F3, E6, E7, E8. Coxeter theorem provides normal forms and

Cayley tables (they are finite groups generated by reflections). See also [54].

—Coxeter groups (for any Coxeter matrix). J. Tits [205] algorithmically solved the

word problem for any Coxeter group. See also [13].

—Affine Kac-Moody algebras [118]. Kac-Gabber theorem provides linear bases of

the algebras under “symmetrizability” condition on the Cartan matrix. See also [174,

175, 176].

—Borcherds-Kac-Moody algebras [57, 58, 59, 118].

—Quantum enveloping algebras (V.G. Drinfeld, M. Jimbo). Lusztig theorem [150]

provides linear “canonical bases” of the algebras. Different approaches were given by

C.M. Ringel [186, 187] and V.K. Kharchenko [129, 130, 131, 132, 133]. See also [51, 84].

—Koszul algebras. Quadratic algebras having a basis of “standard” monomials,

called PBW-algebras, are always Koszul (S. Priddy [179]), but not conversely. In another

terminology PBW-algebras are algebras with quadratic GS bases. See [173].

—Elliptic algebras. Elliptic algebras (B. Feigin, A. Odesskii) are associative algebras

presented by n generators and n(n− 1)/2 homogeneous quadratic relations for which the

dimensions of graded components are the same as for polynomial algebra in n variables.

The first example of such algebras was Sklyanin algebra (1982) generated by x1, x2, x3

with 3 defining relations [x3, x2] = x21, [x2, x1] = x2

3, [x1, x3] = x22. See [171].

—Cluster algebras. A cluster algebra (S. Fomin, A. Zelevinski) is a subalgebra of

a field of rational functions Q(x1, . . . , xn) generated by “cluster” elements. Cluster alge-

bras were invented in order to better understand the Lustig canonical bases of quantum

enveloping algebras.

—Leavitt path algebras. GS bases of these algebras are found in A. Alahmedi et al

[2] and applied by the same authors to determination of the structure of Leavitt path

algebras of polynomial growth in [3].

—Hecke (Iwahory-Hecke) algebras. A deep connection of Hecke algebras (of type

An) and braid groups (and invariants of links) were found in V.F.R. Jones [117].

—Artin braid group Bn. Markov-Artin theorem provides normal form and a semi-

direct structure of the group in Burau generators. Other normal forms of Bn were obtained

by Garside, Birman-Ko-Lee and Adjan-Thurston. See also [21, 22].

—Artin-Tits groups. Normal forms are known in the spherical case, see E. Brieskorn,

K. Saito [60].

—Novikov-Boon type of groups [56, 95, 119, 120, 121, 122, 169] with the unsolvable

word or conjugacy problem. They are groups with “standard bases” (“standard normal

149

forms” or “standard GS bases”), see [15, 16, 75].

—Adjan-Rabin constructions of groups with unsolvable isomorphism and Markov

properties [1, 182]. See also [24].

—Markov-Post semigroups with unsolvable word problem. See also [157].

—Markov construction of semigroups with unsolvable isomorphism and Markov prop-

erties.

—Plactic monoids. A theorem due to Richardson, Schensted, Knuth provides normal

form of elements of the monoids. See also [73].

—Groups of quotients of multiplicative semigroups of some power series rings with

“topological quadratic defining relations” (of the type k〈〈x, y, z, t|xy = zt〉〉) that are

embeddable into groups but not embeddable, in general, into division algebras (an answer

to Malcev problem). “Relative standard normal forms” of the groups were found in [17, 18]

between line of what was later called “relative GS basis” [55].

Up to date, GS bases method invented in particular for the following classes of linear

universal algebras as well as for operads, categories and semirings. Unless otherwise

stated, all linear algebras are considered over a field k. Following the terminology of

Higgins-Kurosh, by an ((differential) associative) Ω-algebra we mean a linear space (an

(differential) associative algebra) with a set of multi-linear operations Ω:

—Associative algebras, Shirshov [202], Bokut [20], Bergman [9],

—Associative algebras over a commutative algebra, A.A. Mikhalev, Zolotykh [165],

—Associative Γ-algebras, where Γ is a group, Bokut, Shum [55],

—Lie algebras, Shirshov [202],

—Lie algebras over a commutative algebra, Bokut, Chen, Chen [28],

—Lie p-algebras over k, chk = p, A.A. Mikhalev [161],

—Lie superalgebras, A.A. Mikhalev [160, 162],

—Metabelian Lie algebras, Chen, Chen [72],

—Quiver (path) algebras, Farkas, Feustel, Green [98],

—Tensor products of associative algebras, Bokut, Chen, Chen [27],

—Associative differential algebras, Chen, Chen, Li [74],

—Associative (n−)conformal algebras over k, chk = 0, Bokut, Fong, Ke [42], Bokut,

Chen, Zhang [40],

—Dialgebras, Bokut, Chen, Liu [35],

—Pre-Lie (Vinberg-Koszul-Gerstenhaber, right symmetric) algebras, Bokut, Chen,

Li [32],

—Associative Rota-Baxter algebras over k, chk = 0, Bokut, Chen, Deng [29],

—L-algebras, Bokut, Chen, Huang [30],

150

—Associative Ω-algebras, Bokut, Chen, Qiu [38],

—Associative differential Ω-algebras, Qiu, Chen [180]

—Ω-algebras, Bokut, Chen, Huang [30],

—Differential Rota-Baxter commutative associative algebras,

—Semirings, Bokut, Chen, Mo [37],

—Modules over an associative algebra, Kang, Lee [124, 125], Chibrikov [88],

—Small categories, Bokut, Chen, Li [33],

—Non-associative algebras, Shirshov [203],

—Non-associative algebras over a commutative algebra, Chen, Jing Li, Zeng [79],

—Commutative non-associative algebras, Shirshov [203]

—Anti-commutative non-associative algebras, Shirshov [203],

—Symmetric operads, Dotsenko, Khoroshkin [96].

At the heart of the GS method for a class of linear algebras, there is a CD-lemma

for a free object of the the class. For above cases, free objects are correspondingly a

free associative algebra k〈X〉, a “double free” associative k[Y ]-algebra k[Y ]〈X〉, a free Lie

algebra Lie(X), a “double free” Lie k[Y ]-algebra Liek[Y ](X). For tensor product of two

associative algebras, one needs to use tensor product of two free algebras, k〈X〉 ⊗ k〈Y 〉.Any semiring can be viewed as a “double semigroup” with two associative “products”

· and . So, CD-lemma for semirings is CD-lemma for a “semiring algebra” kRig(X)

of the free semiring Rig(X). CD-lemma for modules is CD-lemma for a “double free”

module Modk〈Y 〉(X), a free module over a free associative algebra. CD-lemma for small

categories is CD-lemma for a “free partial k-algebra” kC〈X〉 generated by an oriented

graph X (a sequence z1z2 · · · zn, where zi ∈ X, is a partial word in X, iff it is a path;

a partial polynomial is a linear combination of partial words with the same sources and

targets).

All CD-lemmas are essentially the same wording. Let V be a class of linear universal

algebras, V(X) a free algebra from V, N(X) a well-ordered term basis of V(X). A subset

S ∈ V(X) is called a GS basis, if any “composition” of elements of S is “trivial” (goes

to zero by “elimination” of leading terms s, s ∈ S). Then the following conditions are

equivalent:

(i) S is a GS basis.

(ii) If f ∈ Id(S), then the leading term f contains a sub-term s for some s ∈ S.

(iii) A set of S-irreducible terms is a linear basis of the V-algebra V〈X|S〉, generated

by X with defining relations S.

151

In some cases ((n−) conformal algebras, dialgebras), conditions (i) and (ii) are not

equivalent. To be more precise in that cases (i)⇒ (ii)⇔ (iii).

Typical compositions are compositions of intersection and inclusion. Shirshov [202,

203] avoided the inclusion composition. Instead, he suggested that a GS basis must be

reduced (leading words do not contain each other as subwords). In some cases, one needs

to define new compositions, for example, the composition of left (right) multiplication.

Also, sometimes one needs to combine all these compositions. We present here a new

approach to the definition of a composition based on the concept of a least common

multiple lcm(u, v) of two terms u, v.

In some cases (Lie algebras, (n−) conformal algebras) the “leading” term f of a

polynomial f ∈ V(X) does not belong to V(X). For Lie algebras, one has f ∈ k〈X〉, for

((n)-) conformal algebras f belongs to an associative Ω-semigroup.

For almost all CD-lemmas one needs to use a new notion of “normal S-term”. A

term (asb) in X, Ω, s ∈ S, with only one occurrence of s is called a normal S-term if

(asb) = (a(s)b). For any S ⊂ k〈X〉, each S-word (i.e., S-term) is a normal S-word. For

any S ⊂ Lie(X), each Lie S-monomial (Lie S-term) is a linear combination of normal Lie

S-terms (Shirshov [202]).

One of the two key lemmas asserts that if S is complete under compositions of

multiplications then any elements of the ideal generated by S is a linear combination of

normal S-terms. Another key lemma says that if S is a GS basis and the leading words

of two normal S-terms are the same, then these terms are the same modulo lower normal

S-terms. As we mentioned above these results were proved in Shirshov [202] for Lie(X)

(there are no compositions of multiplications for Lie and associative algebras).

In conclusion, we give some information about the works of Shirshov, see more on

this in the book [204]. A.I. Shirshov (1921-1981) was a famous Russian mathematician.

His name associated with notions and results on GS bases, the CD-lemma, the Shirshov-

Witt theorem, the Lazard-Shirshov elimination, the Shirshov height theorem, Lyndon-

Shirshov words, Lyndon-Shirshov basis (in free Lie algebras), Hall-Shirshov series of bases,

the Cohn-Shirshov theorem for Jordan algebras , the Shirshov theorem on the Kurosh

problem, the Shirshov factorization theorem. Shirshov’s ideas were used by his students

Efim Zelmanov for a solution of the restricted Burnside problem and Aleksander Kemer

for a solution of the Specht problem.

152

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Notation

CD-lemma: Composition-Diamond lemma.

GS basis: Grobner-Shirshov basis.

LS word (basis): Lyndon-Shirshov word (basis).

ALSW(X): the set of all associative Lyndon-Shirshov words in X.

NLSW(X): the set of all non-associative Lyndon-Shirshov words in X.

PBW theorem: Poincare-Birkhoff-Witt theorem.

X∗: the free monoid generated by the set X.

[X]: the free commutative monoid generated by the set X.

X∗∗: the set of all non-associative words (u) in X.

gp〈X|S〉: the group generated by X with defining relations S.

sgp〈X|S〉: the semigroup generated by X with defining relations S.

k: a field.

K: a commutative algebra over k with unit.

k〈X〉: the free associative algebra over k generated by X.

k〈X|S〉: the associative algebra over k with generators X and defining relations S.

Sc: Grobner-Shirshov completion of S.

Id(S): the ideal generated by the set S.

s: the leading term of the polynomial s.

Irr(S): the set of all monomials which do not contain any subwords s, s ∈ S.

k[X]: the polynomial algebra over k generated by X.

Lie(X): the free Lie algebra over k generated by X.

LieK(X): the free Lie algebra over a commutative algebra K generated by X.

169