Non-commutative Computer Algebra and its Applications with ... · Non-commutative Computer Algebra...

Non-commutative Computer Algebraand its Applications

with the Computer Algebra SystemSINGULAR:PLURAL

Viktor Levandovskyy

SFB Project F1301 of the Austrian FWFResearch Institute for Symbolic Computation (RISC)

Johannes Kepler UniversityLinz, Austria

13.12.2006, Aachen

Viktor Levandovskyy (RISC) PLURAL 13.12.2006, Aachen 1 / 40

Introduction

What do we call Non-commutative Computer Algebra?Potentially computer–assisted symbolic algebraic manipulationwith constructive (presented by finite data) objects

elements of algebras (polynomials etc.)ideals and submodules of free modulesfactor modulessubalgebrasring homomorphismsmodule homomorphisms

Key instrument: Grobner basesG. Bergman, Teo Mora, Edward Green, Victor Ufnarovski, . . .


Applications of Grobner bases

Grobner Basics are, according to Buchberger, Sturmfels et.al....the most important and fundamental applications of Grobner Bases.

Ideal (resp. module) membership problemIntersection with subrings (elimination of variables)Intersection of ideals (resp. submodules)Quotient and saturation of (two–sided) idealsKernel of a module homomorphismKernel of a ring homomorphismAlgebraic relations between pairwise commuting polynomialsHilbert polynomial of graded ideals and modules

This list is universal for both commutative and non–commutativesituations.


Origins Of Non-commutativity

Let C be some algebra of functions (C∞ etc).For any function f ∈ C, we introduce an operator

F : C → C, F (t) = f · t .

We call f a representative of F . ∀f , g ∈ C we have F ◦G = G ◦ F .

DefinitionA map ∂ : C → C is called a differential if ∂ is C–linear and ∀f , g ∈ C,∂(fg) = ∂(f )g + f∂(g).

In particular, ∂i = ∂∂ti

on C are differentials.

NewsBad news: operators F and ∂i do not commute.Good news: ∂j ◦ ∂i = ∂i ◦ ∂j and there is a relation between F and ∂i .


Non–commutative RelationsLemmaFor any differential ∂ and f ∈ C, ∂ ◦ F = F ◦ ∂ + ∂(f ).

Proof.∀ h ∈ C, we have the following:

(∂ ◦ F )(h) = ∂(f · h) = f · ∂(h) + ∂(f ) · h =

= (F ◦ ∂)(h) + ∂(f ) · (h) = (F ◦ ∂ + ∂(f ))(h).

Example

Let C = K[t1, . . . , tn] and ∂i = ∂∂ti

. Then there is a n–th Weyl algebraK〈t1, . . . , tn, ∂1, . . . , ∂n | {tj ti = ti tj , ∂j∂i = ∂i∂j ,∂k tk = tk∂k + 1} ∪ {∂j ti = ti∂j}i 6=j〉,an algebra of linear differentional operators with polynomialcoefficients.


More Non–commutative Relations

Shift AlgebraFor small 4t ∈ R, we define a shift operator

σt : C → C, σt(f (t)) = f (t +4t).

Then, since σt(f · g) = σt(f ) · σt(g), we define a real shift algebraK(4x)〈x , σx | σxx = xσx +4xσx〉.

The Center of an AlgebraFor a K–algebra A, we define the center of A to be

Z (A) = {a ∈ A | a · b = b · a ∀b ∈ A}.

It is a subalgebra of A, containing constants of K.


q–Calculus and Non–commutative RelationsLet k be a field of char 0 and K = k(q).

q–dilation operatorDq : C → C, Dq(f (x)) = f (qx):

K(q)〈x , Dq | Dq · x = q · x · Dq〉.

Continuous q–difference Operator∆q : C → C, ∆q(f (x)) = f (qx)− f (x):

K(q)〈x ,∆q | ∆q · x = q · x ·∆q + (q − 1) · x〉.

q–differential Operator

∂q : C → C, ∂q(f (x)) = f (qx)−f (x)(q−1)x :

K(q)〈x , ∂q | ∂q · x = q · x · ∂q + 1〉.


Major Directions. Algebras and Systems

Free and path algebras, their factor-algebras

BERGMAN, http://servus.math.su.se/bergman/OPAL, GRB, the group of Edward Green (Blacksburg)NCGB/NCALGEBRA, MATHEMATICA packageGBNP/GROBNER, GAP packageFELIX, http://felix.hgb-leipzig.de/

FunctionalityFull or reduced Grobner Bases w.r.t. some orderings, membershipproblems, basic ideal arithmetics.

Wanted: projective resolutions and cohomology.


Major Directions. Algebras and Systems

Algebras, close to commutative.Noetherian integral domains with PBW bases

MACAULAY2, Weyl algebras, exterior algebrasKAN/SM1, rings of differential and (q–)difference operatorsMAS2, GR–algebras and beyondFELIX, G–algebrasSINGULAR:PLURAL, G– and GR–algebras.


What is PLURAL?

What is PLURAL?PLURAL is the kernel extension of SINGULAR,providing a wide range of symbolic algoritms with non–commutativepolynomial algebras (GR–algebras).

Grobner bases, Grobner basics, non–commutative Grobner basicsmore advanced algorithms for non–commutative algebras,

PLURAL is distributed with SINGULAR (from version 3-0-0 on)freely distributable under GNU Public Licenseavailable for most hardware and software platforms


PreliminariesLet K be a field and R be a commutative ring R = K[x1, . . . , xn].

Mon(R) 3 xα = xα11 xα2

2 . . . xαnn 7→ (α1, α2, . . . , αn) = α ∈ Nn.

Definition1 a total ordering ≺ on Nn is called a well–ordering, if

I ∀F ⊆ Nn there exists a minimal element of F ,in particular ∀ a ∈ Nn, 0 ≺ a

2 an ordering ≺ is called a monomial ordering on R, ifI ∀α, β ∈ Nn α ≺ β ⇒ xα ≺ xβ

I ∀α, β, γ ∈ Nn such that xα ≺ xβ we have xα+γ ≺ xβ+γ .3 Any f ∈ R \ {0} can be written uniquely as f = cxα + f ′, with

c ∈ K∗ and xα′ ≺ xα for any non–zero term c′xα′of f ′. We define

lm(f ) = xα, the leading monomial of flc(f ) = c, the leading coefficient of f


Towards GR–algebras

Suppose we are given the following data1 a field K and a commutative ring R = K[x1, . . . , xn],2 a set C = {cij} ⊂ K∗, 1 ≤ i < j ≤ n3 a set D = {dij} ⊂ R, 1 ≤ i < j ≤ n

Assume, that there exists a monomial well–ordering ≺ on R such that

∀1 ≤ i < j ≤ n, lm(dij) ≺ xixj .

The ConstructionTo the data (R, C, D,≺) we associate an algebra

A = K〈x1, . . . , xn | {xjxi = cijxixj + dij} ∀1 ≤ i < j ≤ n〉


PBW Bases and G–algebras

Define the (i , j , k)–nondegeneracy condition to be the polynomial

NDCijk := cikcjk · dijxk − xkdij + cjk · xjdik − cij · dikxj + djkxi − cijcik · xidjk .

TheoremA = A(R, C, D,≺) has a PBW basis {xα1

1 xα22 . . . xαn

n } if and only if

∀ 1 ≤ i < j < k ≤ n, NDCijk reduces to 0 w.r.t. relations

DefinitionAn algebra A = A(R, C, D,≺), where nondegeneracy conditionsvanish, is called a G–algebra (in n variables).


DefinitionLet A be an associative K–algebra and M be a left A–module.

1 The grade of M is defined to be j(M) = min{i | ExtiA(M, A) 6= 0},or j(M) = ∞, if no such i exists or M = {0}.

2 A satisfies the Auslander condition, if for every fin. gen.A–module M, for all i ≥ 0 and for all submodules N ⊆ ExtiA(M, A)the inequality j(N) ≥ i holds.

3 A is called an Auslander regular algebra, if it is Noetherian withgl. dim(A) < ∞ and the Auslander condition holds.

4 A is called a Cohen–Macaulay algebra, if for every fin. gen.nonzero A–module M, j(M) + GKdim(M) = GKdim(A) < ∞.


We collect the properties in the following Theorem.

Theorem (Properties of G–algebras)Let A be a G–algebra in n variables. Then

A is left and right Noetherian,A is an integral domain,the Gel’fand–Kirillov dimension GKdim(A) = n + GKdim(K),the global homological dimension gl. dim(A) ≤ n,the Krull dimension Kr.dim(A) ≤ n,A is Auslander-regular and a Cohen-Macaulay algebra.

We say that a GR–algebra A = A/TA is a factor of a G–algebra in nvariables A by a proper two–sided ideal TA.


Examples of GR–algebras

Mora, Apel, Kandri–Rody and Weispfenning, . . .

algebras of solvable type, skew polynomial ringsuniv. enveloping algebras of fin. dim. Lie algebrasquasi–commutative algebras, rings of quantum polynomialspositive (resp. negative) parts of quantized enveloping algebrassome iterated Ore extensions, some nonstandard quantumdeformationsmany quantum groupsWeyl, Clifford, exterior algebrasWitten’s deformation of U(sl2), Smith algebrasalgebras, associated to (q–)differential, (q–)shift, (q–)differenceand other linear operators


Left, right and twosided structures

It suffices to have implementedleft Grobner basesfunctionality for opposite algebras Aop

functionality for enveloping algebras Aenv = A⊗K Aop

mapping A → Aop → A

Then1 for a finite set F ⊂ A, RGBA(F ) =

(LGBAop(F op)

)op

2 the two–sided Grobner can be computed, for instance, with thealgorithm by Manuel and Maria Garcia Roman in Aenv .


Grobner Trinity

With essentially the same algorithm, we can compute

1 GB left Grobner basis G of a module M

2 SYZ left Grobner basis of the 1st syzygy module of M

3 LIFT the transformation matrix between two bases G and M

The algorithm for Grobner Trinity must be able to compute ...

with submodules of free modulesI accept monomial module orderings as inputI distinguish preferred module components

within factor algebraswith extra weights for the ordering / module generatorsand to use the information on Hilbert polynomial


Implementation of Grobner Basics

Universal Grobner BasicsIdeal (resp. module) membership problem (NF, REDUCE)Intersection with subrings (elimination of variables) (ELIMINATE)Intersection of ideals (resp. submodules) (INTERSECT)Quotient and saturation of ideals (QUOT)Kernel of a module homomorphism (MODULO)Kernel of a ring homomorphism (NCPREIMAGE.LIB)Algebraic relations between pairwise commuting polynomialsHilbert polynomial of graded ideals and modules


Anomalies With Elimination

Admissible SubalgebrasLet A = K〈x1, . . . , xn | {xjxi = cijxixj + dij}1≤i<j≤n〉 be a G–algebra.Consider a subalgebra Ar , generated by {xr+1, . . . , xn}.We say that such Ar is an admissible subalgebra, ifdij are polynomials in xr+1, . . . , xn for r + 1 ≤ i < j ≤ nand Ar ( A is a G–algebra.

Definition (Elimination ordering)Let A and Ar be as before and B := K〈x1, . . . , xr | . . . 〉 ⊂ AAn ordering ≺ on A is an elimination ordering for x1, . . . , xrif for any f ∈ A, lm(f ) ∈ B implies f ∈ B.


Constructive Elimination Lemma

”Elimination of variables x1, . . . , xr from an ideal I”means the intersection I ∩ Ar with an admissible subalgebra Ar .In contrast to the commutative case:• not every subset of variables determines an admissible subalgebra• there can be no admissible elimination ordering ≺Ar on A

Lemma

Let A be a G–algebra, generated by {x1, . . . , xn} and I ⊂ A be an ideal.Suppose, that the following conditions are satisfied:

{xr+1, . . . , xn} generate an admissible subalgebra B,∃ an admissible elimination ordering ≺B for x1, . . . , xr on A.

Then, if S is a left Grobner basis of I with respect to ≺B, we have S ∩Bis a left Grobner basis of I ∩ B.


Anomalies With Elimination: Example

Example

Consider the algebra A = K〈a, b | ba = ab + b2〉.It is a G–algebra with respect to any well–ordering, such that b2 ≺ ab,that is b ≺ a. Any elimination ordering for b must satisfy b � a, henceA is not a G–algebra w.r.t. any elimination ordering for b.

The Grobner basis of a two–sided ideal, generated by b2 − ba + ab inK〈a, b〉 w.r.t. an ordering b � a is infinite and equals to

{ban−1b − 1n

(ban − anb) | n ≥ 1}.

Finding an admissible elimination ordering can be done by solving alinear programming problem (ongoing work with J. Lobillo, Granada).


Non-commutative Grobner basicsFor the noncommutative PBW world, we need even more basics:

Gel’fand–Kirillov dimension of a module (GKDIM.LIB)Two–sided Grobner basis of a bimodule (e.g. twostd)Annihilator of finite dimensional modulePreimage of one–sided ideal under algebra morphismFinite dimensional representationsGraded Betti numbers (for graded modules over graded algebras)Left and right kernel of the presentation of a moduleCentral Character Decomposition of a module (NCDECOMP.LIB)

Very ImportantExt and Tor modules for centralizing bimodules (NCHOMOLOG.LIB)Hochschild cohomology for modules


Non-commutative Grobner basics in PLURAL

Unrelated to Grobner Bases, but Essential FunctionsCenter of an algebra and centralizers of polynomialsOperations with opposite and enveloping algebras

PLURAL as a Grobner engineimplementation of all the universal Grobner basics availableslimgb is available for Pluraljanet is available for two–sided inputnon–commutative Grobner basics:

I as kernel functions (twostd, opposite etc)I as libraries (NCDECOMP.LIB, NCTOOLS.LIB, NCPREIMAGE.LIB etc)


D–modules (joint work with J. Morales)

What’s BehindLet R = K[x1, . . . , xn] and f ∈ R. We are interested in

R[f−s] = K[x1, . . . , xn,1f s ] as an R–module for s ∈ N.

On the one hand, R[f−s] ∼= R[y ]/〈yf s − 1〉.On the other hand, R[f−s] is a D–module, where D is the n–th Weylalgebra K〈x1, . . . , xn, ∂1, . . . , ∂n | {∂jxi = xi∂j + δij}〉.The algorithm ANNFS computes a D–module structure on R[f−s], thatis a left ideal I ⊂ D, such that R[f−s] ∼= D/I.

Especially interesting are cases when f is irreducible singular (amongother, a reiffen curve), reducibly singular or when f is a hyperplanearrangement (arrange).


Challenge: Ann F s for different FLet char K = 0 and F ∈ K[x1, . . . , xn].

Problem FormulationCompute the ideal Ann F s ∈ K〈x1, . . . , xn, ∂1, . . . , ∂n | ∂ixj = xj∂i + δij〉(n–th Weyl algebra).Both algorithms available use two complicated eliminations.

Oaku–Takayama approach (ANNFSOT command)Briancon–Maisonobe approach (ANNFSBM command)Bernstein polynomial is computed within both approachesmany convenient auxiliary tools

polynomial singularitiesvery hard: Reiffen curves xp + yq + xyq−1, q ≥ p + 1 ≥ 5generic and non–generic hyperplane arrangementsfurther examples by F. Castro and J.-M. Ucha

Systems: KAN/SM1, RISA-ASIR, MACAULAY2, SINGULAR:PLURAL.Viktor Levandovskyy (RISC) PLURAL 13.12.2006, Aachen 26 / 40

Cohen–Macaulay and Autonomy Degree

Let A be a K–algebra and M be a left A–module.

Autonomy DegreeThe autonomy degree of M, ad(M) is the first natural i ≥ 0,such that ∀0 ≤ j < i , ExtjA(M, A) = 0 and ExtiA(N, A) 6= 0.If A is a Cohen–Macaulay algebra,

ad(M) = j(M) = d(A)− d(M),

hence we need only to compute d(M) to determine the ad(M).


Cohen–Macaulay and Controllability Degree

For an A–module M (a system module), one defines N to be theHomA(M, A) (an adjoint or a dual module).

Controllability DegreeThe controllability degree of M is the first natural i > 0, that∀0 < j < i , ExtjA(N, A) = 0 and ExtiA(N, A) 6= 0.Suppose that Ext0A(N, A) = 0, that is HomA(HomA(M, A), A) = 0.Then, if A is a Cohen–Macaulay algebra, we have

cd(M) = j(N) = d(A)− d(N)

In general, we test the condition HomA(HomA(M, A), A) = 0 (it issatisfied by many system modules) and compute d(N).


Application of Preimage Algorithm to D–modules

The algorithm of Oaku and Takayama (1999)The 1st step of the OT algorithm requires to compute the preimage ofthe left ideal

L = 〈{tj − fj ,p∑

j=1

∂fj∂xi

∂t j + ∂i}〉, 1 ≤ j ≤ p, 1 ≤ i ≤ n

in the subalgebra K[{tj · ∂t j}]〈{xi , ∂xi | [∂x i , xi ] = 1}〉 of

K〈{tj , ∂t j} | [∂t j , tj ] = 1〉 ⊗K K〈{xi , ∂xi} | [∂xi , xi ] = 1〉

Moreover, in the preimage, tj · ∂t j will be replaced by −sj − 1.


Symmetric Deformation: Theorem

TheoremLet A,B be GR–algebras and Φ ∈ Mor(A,B).Let IΦ be the (A,A)–bimodule A〈{xi − Φ(xi) | 1 ≤ i ≤ n}〉A ⊂ A⊗K Band fi := Φ(xi). Suppose there exists an elimination ordering for B onA⊗K B, such that

1 ≤ i ≤ n, 1 ≤ j ≤ m, lm(lc(fiyj)yj fi − lc(yj fi)fiyj) ≺ xiyj .

Then1) A⊗φ

K B is a G–algebra (resp. A⊗ΦK B is a GR–algebra).

2) Let J ⊂ B be a left ideal, then

Φ−1(J ) = (IΦ + J ) ∩ A.


Application of Preimage Algorithm to D–modules

Setup with the Symmetric Deformation

A := K〈sj , Xi , Di | DiXi = XiDi + 1〉

B := K〈tj , Dtj , xi , di | dixi = xidi + 1, Dtj tj = tjDtj + 1〉

Consider the map φ : A → B, where sj 7→ −tjDtj − 1, Xi 7→ xi , Di 7→ di .

Hence, Iφ = 〈{Xi − xi , Di − di , tjDtj + sj + 1}〉 ⊂ A⊗φK B =: E .

Due to the structure, we replace E with E ′ = K〈tj , Dtj , xi , di , sj〉 subjectto the relations

{[di , xi ] = 1, [Dtj , tj ] = 1, sj tj = tjsj − tj , sjDtj = Dtjsj + Dtj}.

Respectively, Iφ ⊂ E becomes I′φ = 〈{tjDtj + sj + 1}〉 ⊂ E ′.Any ordering ≺ satisfying {tj , Dtj} � {xi , di , sj} (which is very easy tofind) satisfies the conditions of the Theorem.


Dancing Flamenco

By the Theorem, for any L ⊂ B, φ−1(L) = (Iφ + L) ∩ A. Hence,

Iφ + L = 〈{tj − fj ,p∑

j=1

∂fj∂xi

∂t j + ∂i , tj∂tj + sj + 1}〉 =

= 〈{tj − fj ,p∑

j=1

∂fj∂xi

∂t j + ∂i , fj∂tj + sj}〉

Citing Gago–Vargas, Hartillo and Ucha JSC paper from 2005...

”...As far as we know, the example f = (x2 + y3) · (x3 + y2) isintractable for available computer algebra systems.”→ Demonstration.


Applications

Systems and Control Theory (VL, E. Zerz et. al.)I CONTROL.LIB, NCONTROL.LIB, RATCONTROL.LIBI algebraic analysis tools for System and Control TheoryI In progress: non–commutative polynomial algebras

(NCONTROL.LIB)

Algebraic Geometry (W. Decker, C. Lossen and G. Pfister)I SHEAFCOH.LIBI computation of the cohomology of coherent sheavesI In progress: direct image sheaves (F. - O. Schreyer)

D–Module Theory (VL and J. Morales)I DMOD.LIBI Ann F s algorithms: OT (Oaku and Takayama), BM (Briancon and

Maisonobe)


Applications In Progress

Homological algebra in GR–algebras (with G. Pfister)I NCHOMOLOG.LIBI Ext and Tor modules for centralizing bimodulesI Hochschild cohomology for modules

Clifford Algebras (VL, V. Kisil et. al.)I CLIFFORD.LIBI basic algorithms and techniques of the theory of Clifford algebras

Annihilator of a left module (VL)I NCANN.LIBI the original algorithm of VL for Ann(M) for M with dimK M = ∞I the algorithm terminates for holonomic modules, i.e. for a module

M, such that GKdim(M) = 2 ·GKdim(Ann(M))I high complexity, a lot of tricks and improvements needed


Perspectives

Grobner bases for more non–commutative algebras• tensor product of commutative local algebras with certainnon–commutative algebras (e.g. with exterior algebras for thecomputation of direct image sheaves)• different localizations of G–algebras

localization at some ”coordinate” ideal of commutative variables(producing e.g. local Weyl algebras K[x ]〈x〉〈D | Dx = xD + 1〉)

⇒ local orderings and the generalization of standard basisalgorithm, Grobner basics and homological algebralocalization as field of fractions of commutative variables(producing e.g. rational Weyl algebras K(x)〈D | Dx = xD + 1〉),including Ore Algebras (F. Chyzak, B. Salvy)

⇒ global orderings and a generalization Grobner basis algorithm.Grobner basics require distinct theoretical treatment!


Software from RISC Linz

Algorithmic Combinatorics Group, Prof. Peter Paulemost of the software are packages for MATHEMATICA

created by P. Paule, A. Riese, C. Schneider, M. Kauers,K. Wegschaider, S. Gerhold, M. Schorn, F. Caruso, C. Mallinger,B. Zimmermann, C. Koutschan, T. Bayer, C. Weixlbaumer et al.

The Software is freely available for non–commercial usewww.risc.uni-linz.ac.at/research/combinat/software/


Symbolic Summation

Hypergeometric SummationFASTZEIL, Gosper’s and Zeilberger’s algorithmsZEILBERGER, Gosper and Zeilberger alg’s for MAXIMA

MULTISUM, proving hypergeometric multi-sum identities

q–Hypergeometric SummationQZEIL, q–analogues of Gosper and Zeilberger alg’sBIBASIC TELESCOPE, generalized Gosper’s algorithm to bibasichypergeometric summationQMULTISUM, proving q–hypergeometric multi-sum identities

Symbolic Summation in Difference FieldsSIGMA, discovering and proving multi-sum identities


More Software from RISC LinzSequences and Power Series

ENGEL, q–Engel ExpansionGENERATINGFUNCTIONS, manipulations with univariate holonomicfunctions and sequencesRLANGGFUN, inverse Schutzenberger methodology in MAPLE

Partition Analysis, Permutation GroupsOMEGA, Partition AnalysisPERMGROUP, permutation groups, group actions, Polya theory

Difference/Differential EquationsDIFFTOOLS, solving linear difference eq’s with poly coeffsORESYS, uncoupling systems of linear Ore operator equationsRATDIFF, rat. solutions of lin. difference eq’s after van HoeijSUMCRACKER, identities and inequalities, including summations


Thank you for your attention!

Please visit the SINGULAR homepagehttp://www.singular.uni-kl.de/


Criteria for detecting useless critical pairs

Generalized Product CriterionLet A be a G–algebra of Lie type (that is, all cij = 1). Let f , g ∈ A.Suppose that lm(f ) and lm(g) have no common factors, thenspoly(f , g) →{f ,g} [g, f ], where [g, f ] := gf − fg is the Lie bracket.

Chain CriterionIf (fi , fj), (fi , fk ) and (fj , fk ) are in the set of pairs P andxαj | lcm(xαi , xαk ), then we can delete (fi , fk ) from P.

The Chain Criterion can be proved with the Schreyer’s construction ofthe first syzygy module of a given module, which generalizes to thecase of G–algebras.


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