Classification of Simple Real Lie Algebras

Zoltan Halasi

Institute of Mathematics, University of Debrecen

Second BIOMICS Summer Workshop18-20 June 2014, University of St Andrews, Scotland

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Introduction Classification of Simple Real Lie Algebras June 19, 2014 2 / 31

Modellingsystems

Continuousyy Discrete &&DifferentialEquations

Symmetries

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Finiteautomatons

��Semigroups

��(Real) Lie

groupsoo

?BIOMICS // Finite Simple

Groups of Lie type

Introduction Classification of Simple Real Lie Algebras June 19, 2014 3 / 31

Connections between Lie algebras and Lie groups overvarious fields

simple complexLie algebras

��Chevalley basis

((Topic of this talk

wwsimple realLie algebras

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Chevalley algebrasover finite fields

��simple

Lie groupsFinite simple groups

of Lie type

Introduction Classification of Simple Real Lie Algebras June 19, 2014 4 / 31

Structure of Lie algebras

Let g be a Lie algebra over R or over C.g is semisimple if it does not contain any non-trivial solvableideal, g is simple if it does not contain any non-trivial ideal.Levi-Malcev theorem: g = ro s where r is solvable, while s issemisimple.Any semisimple Lie algebra can be uniquely decomposed intoa direct product of simple Lie algebras.The simple components of this decomposition are orthogonalto each other with respect to the Killing form.

Solvable Lie algebras have simple structure, but it is impossible toclassify all of them. On the other hand, simple Lie algebras aredifficult, but we know all of them!

Simple real Lie algebras and their complexifications Classification of Simple Real Lie Algebras June 19, 2014 5 / 31

Complexification and real forms

Let g0 be a real Lie algebra. Its complexificationg = g0(C) := g0 ⊗R C = g0 ⊕ ig0 as a vector space withextended Lie bracket[x1 + iy1, x2 + iy2] := [x1, y1]− [x2, y2] + i([x1, y2] + [x2, y1])

If g is a complex Lie algebra, then its realification gR is just gviewed as a real Lie algebra.If {v1, . . . , vn} is a basis of g0, then it is also a basis of thecomplex Lie algebra g = g0(C) and {v1, . . . , vn, iv1, . . . , ivn} isa basis of the real Lie algebra gR. So

dimR g0 = dimC g, dimR gR = 2 dimC g.

If g is any complex Lie algebra, then a real form of g is a realsubalgebra g0 ≤ gR s.t. g = g0 ⊕ ig0. Then g ' g0(C).

Simple real Lie algebras and their complexifications Classification of Simple Real Lie Algebras June 19, 2014 6 / 31

The Killing form

If g is a Lie algebra and X ∈ g, then the map ad X : g 7→ g isdefined as

ad X (Y ) := [X ,Y ]

Then ad : g 7→ gl(g) is a Lie algebra homomorphism.The Killing form on a (real or complex) Lie algebra g is asymmetric bilinear function on g and defined as

k(X ,Y ) = Tr(ad X · ad Y ) X ,Y ∈ g

g is semisimple ⇐⇒ its Killing form k is non-degenerate.If g0 is a real form of g then kg|g0 = kg0 .g0 is semisimple ⇐⇒ g = g0(C) is semisimple.

Simple real Lie algebras and their complexifications Classification of Simple Real Lie Algebras June 19, 2014 7 / 31

The simple real Lie algebras

If g is a simple complex algebra thengR is a simple real Lie algebra with gR(C) ' g⊕ g.Any real form g0 of g is a simple real Lie algebra.

Conversely, if g0 is a simple real Lie algebra then one of thefollowing holds.

There is a complex structure on g0 i.e. g0 = gR for somecomplex simple Lie algebra g.g0(C) is a simple complex Lie algebra (with real form g0).

Thus, Classifying the simple real Lie algebras ⇐⇒ Finding allnon-isomorphic real forms of the simple complex Lie algebras

The simple complex Lie algebras Classification of Simple Real Lie Algebras June 19, 2014 8 / 31

The structure of a complex simple Lie algebra g

Let g be a simple (or semisimple) Lie algebra over C. Then thefollowing holds

There is a maximal toral subalgebra (Cartan subalgebra)h ≤ g such that ad h, h ∈ h are simultaneously diagonalizable.The root space decomposition of g with respect to h is

g = h⊕⊕α∈Φ

gα

where h is a Cartan subalgebra of g, Φ ⊆ h∗ is a system ofroots and gα is the root space corresponding to α ∈ Φ.dim gα = 1 and [h, x ] = α(h) · x ∀α ∈ Φ, h ∈ t, x ∈ gα.If α ∈ Φ then −α ∈ Φ (there are some other nice properties)

The simple complex Lie algebras Classification of Simple Real Lie Algebras June 19, 2014 9 / 31

There is a Π ⊆ Φ is a fundamental system of roots.Φ = Φ+ ∪ Φ− is the decomposition of Φ to the system ofpositive and negative roots with respect to Π.There is a Chevalley basis of the complex Lie algebra g:

CB = {hα, α ∈ Π, eα, α ∈ Φ}where {hα, α ∈ Π} is a basis of h and gα = Ceα, such thatall the structure constants of the Lie algebra with respect tothis basis are reals (even integers).The canonical generator set for g

{hα, α ∈ Π, eα, e−α α ∈ Π} ⊆ CB

The simple complex Lie algebras Classification of Simple Real Lie Algebras June 19, 2014 10 / 31

The Killing form is non-degenerate on h. With its help one candefine a non-degenerate symmetric bilinear form ( . , . ) on h∗.( . , . ) is positive definite on the real subspace E := 〈Φ〉 of h∗,E is a real Euclidean space with scalar product ( . , . ).For every α, β ∈ Π we have the Cartan integers

〈α, β〉 := 2(α, β)(β, β) ∈ Z

and the Dynkin diagramsvertices: Πedges : For α, β ∈ Π the number of edges between α and β is〈α, β〉 · 〈β, α〉 directed to the shorter root.

The simple complex Lie algebras Classification of Simple Real Lie Algebras June 19, 2014 11 / 31

The Dynkin diagrams of simple Lie algebras

Table: Classical Lie algebras

Type Diagram over C

Al c c c c sll+1(C)

Bl c c c c>2 2 2 1 so2l+1(C)

Cl c c c c<1 1 1 2 spl (C)

Dl c c c ccQQQ

���

so2l (C)

The simple complex Lie algebras Classification of Simple Real Lie Algebras June 19, 2014 12 / 31

Table: Exceptional Lie algebras

Type Diagram

E6 c c c cccE7 c c c ccc cE8 c c c c ccc cF4

c cc c<1 1 2 2

G2c c<1 3

The simple complex Lie algebras Classification of Simple Real Lie Algebras June 19, 2014 13 / 31

Existence and Isomorphism theorems

Existence TheoremFor every Dynkin diagram there is a simple complex Lie algebrawith this Dynkin diagram.

Uniqueness TheoremLet (g, h,Φ,Π) and (g′, h′,Π′ ⊆ Φ′) be two simple complex Liealgebras with Cartan subalgebras and associated root systems andfundamental systems.If ϕ : Π 7→ Π′ is a bijection respecting the Cartan integers (i.e. it isan isomorphism of the Dynkin diagrams) then ϕ induces anisomorphism π : h 7→ h′ (via the Killing form).Moreover, choosing non-zero xα ∈ gα, x ′α ∈ g′α for each α ∈ Π, πextends uniquely to an isomorphism π : g 7→ g′ such thatπ(xα) = x ′α for each xα.

Characterizing real forms Classification of Simple Real Lie Algebras June 19, 2014 14 / 31

Split real forms

Let g be a simple complex Lie algebra with real form g0.The conjugation σ : g 7→ g with respect to g0 is

σ(x + iy) = x − iy ∀ x , y ∈ g0.

Theng0 = gσ = {x ∈ g |σ(x) = x}.

A split real form g0 of g can be defined by choosing aChevalley basis CB = {hα, α ∈ Π, eα, α ∈ Φ} of g andtaking the real subspace generated by it. Thus,

g0 =∑α∈Π

Rhα +∑α∈Φ

Reα

Characterizing real forms Classification of Simple Real Lie Algebras June 19, 2014 15 / 31

Compact real formu0 is a compact real form of g if its Killing form is negativedefinite. (⇐⇒ it is the Lie algebra of a compact Lie group.)For the canonical generator set {hα, eα, e−α |α ∈ Π} the map

ω(hα) = −hα, ω(eα) = −e−α, ω(e−α) = −eαon the canonical generators extends uniquely to anautomorphism ω ∈ Aut g with ω2 = id. ω is called a Weylinvolution.Let g0 be the split real form of g and σ the conjugation withrespect to g0. Let τ = ωσ. By choosing

u0 := gτ = {x ∈ g | τ(x) = x}

we get a compact real form of g.By using the same CB as in the split real form

u0 =∑α∈Π

Rihα +∑α∈Φ+

R(eα − e−α) +∑α∈Φ+

Ri(eα + e−α)

Characterizing real forms Classification of Simple Real Lie Algebras June 19, 2014 16 / 31

Cartan involutions

Let g be a complex (semi)simple algebra with compact real formu0 and arbitrary real form g0.

An involution of g0 is a map θ ∈ Aut(g0) with θ2 = id.A Cartan involution of g0 is an involution θ of g0 such that

kθ(u, v) := −k(u, θ(v))

is positive definite.g0 is a compact real form of g ⇐⇒ the conjugation of g withrespect to g0 is a Cartan involution of gR.Every real semisimple Lie algebra g0 has a Cartan involution.Any two Cartan involutions of g0 are conjugate via Int(g0).Thus, any two compact real forms of g are isomorphic.

Characterizing real forms Classification of Simple Real Lie Algebras June 19, 2014 17 / 31

Cartan decompositionA Cartan decomposition of g0 is a vector space direct sumg0 = n0 ⊕ p0 with properties

[n0, n0] ⊆ n0, [n0, p0] ⊆ p0, [p0, p0] ⊆ n0.

the Killing form of g0

kg0 is{

negative definite on n0positive definite on p0

Correspondence between Cartan involutions and Cartandecompositions

θ =⇒{

n0 = {x ∈ g0 | θ(x) = x}p0 = {x ∈ g0 | θ(x) = −x}

θ ={

+id on n0−id on p0

⇐= g = n0 ⊕ p0

If g0 = n0 ⊕ p0 is a Cartan decomposition then n0 and p0 areorthogonal with respect to kg0 and kθ.

Characterizing real forms Classification of Simple Real Lie Algebras June 19, 2014 18 / 31

Cartan subalgebras

Let g0 be a real semisimple algebra with Cartan involution θ andcorresponding Cartan decomposition g0 = n0 ⊕ p0.

h0 ⊆ g0 is a Cartan subalgebra of g0 if h0(C) is a Cartansubalgebra of g = g0(C).Any Cartan subalgebra of g0 is conjugate via Int(g0) to aθ-stable Cartan subalgebra of g0 (i.e. with h0 satisfyingθ(h0) = h0).

From now on, let h0 ⊆ g0 be a θ-stable Cartan subalgebra.We have the decomposition

h0 = t0 ⊕ a0 with t0 = h0 ∩ n0 and a0 = h0 ∩ p0.

dim t0 is called the compact dimension of h0dim a0 is called the noncompact dimension of h0

Characterizing real forms Classification of Simple Real Lie Algebras June 19, 2014 19 / 31

The (θ-stable) h0 is called maximally compact if its compactdimension is maximal.If t0 ⊆ n0 is a maximal abelian subspace then h0 := Zg0(t0) isa θ-stable maximally compact Cartan subalgebra of g0.All the maximally compact θ-stable Cartan subalgebras of g0are conjugate via Aut(g0).

From now on, let h0 be maximally compact.Let α be a root of g = g0(C) with respect to the Cartansubalgebra h = h0 ⊕ ih0. Then α is real valued on a0 ⊕ it0.

α is real if it is real valued on h0 (i.e. it vanishes on t0)α is imaginary if it is purely imaginary valued on h0 (i.e. itvanishes on a0)α is complex otherwise.

h0 is maximally compact ⇐⇒ there are no real roots.

Characterizing real forms Classification of Simple Real Lie Algebras June 19, 2014 20 / 31

Vogan diagrams

Let g = h⊕⊕α∈Φ gα be a root space decomposition of g = g0(C)

with Cartan subalgebra h = h0(C).If α ∈ h∗ is a root, then θα(h) := α(θ−1h) is also a root.Let {x1, . . . , xn} and {xn+1, . . . , xl} be basises of it0 and a0,respectively. Then {x1, . . . , xl} is a basis of it0 ⊕ a0. Forα, β ∈ Φ let

α > β ⇐⇒ ∃t : α(xt) > β(xt) and α(xs) = β(xs) ∀s < t

Then > is an ordering on Φ =⇒ Φ+, Φ−, Π ⊆ Φθ is identical on t0 and there are no real roots =⇒

θ(Φ+) = Φ+ and θ(Π) = Π.

θ fixes the imaginary roots while it permutes the complexroots in 2-cycles.

Characterizing real forms Classification of Simple Real Lie Algebras June 19, 2014 21 / 31

If α ∈ Φ is imaginary then gα is θ-stable⇒ gα = (gα ∩ n0)⊕ (gα ∩ p0)⇒ gα ⊆ n0 or gα ⊆ p0.

α is called compact if gα ⊆ n0α is called noncompact if gα ⊆ p0

The Vogan diagram corresponding to (g0, h0,Φ+) is thefollowing

We start with the Dynkin diagram of g0(C)The vertices of the 2-element orbits of θ are connected by anarrow ↔The noncompact imaginary roots are painted.

The “empty” Dynkin diagrams correspond to compact realforms.

Existence and uniqueness Classification of Simple Real Lie Algebras June 19, 2014 22 / 31

Existence and uniqueness theorems

Existence theoremFor any abstract Vogan diagram, there is a real semisimple algebrag0, a Cartan involution θ, a maximally compact θ-stable Cartansubalgebra h0 = t0 ⊕ a0 and a positive system Φ+ such that thetriple (g0, h0,Φ+) defines the given Vogan diagram.

Uniqueness theoremIf (g0, h0,Φ+) and (g′0, h′0,Φ′

+) are two isomorphic triples, thenthey have the same Vogan diagram.

Conversely, if g0 and g′0 are real semisimple algebras having triples(g0, h0,Φ+) and (g′0, h′0,Φ′

+) with the same Vogan diagram theng0 and g′0 are isomorphic.

Existence and uniqueness Classification of Simple Real Lie Algebras June 19, 2014 23 / 31

Unfortunately, different Vogan diagrams can define isomorphic Liealgebras. This redundancy can be resolved by the followingTheorem

Borel and de Siebenthal TheoremIf g0 is a noncomplex simple real Lie algebra with triple(g0, h0,Φ+) then one can choose a fundamental system Π′ ⊆ Φdefining positive system Φ′ such that there is at most one paintedroot in the Vogan diagram defined by (g0, h0,Φ′+).

If the action of θ on the Vogan diagram is the identity, letΠ = {α1, . . . , αl} and let {ω1, . . . , ωl} be the dual basis i.e. whichsatisfies (ωi , αj) = δij . Then the single painted root αi may bechosen s.t. there is no j with (ωi − ωj , ωj) > 0.

Vogan diagrams and associated real Lie algebras Classification of Simple Real Lie Algebras June 19, 2014 24 / 31

c c c csu1,l su2,l−1 sul−1,2 sul,1 Bl

c c c cso2,2l−1 so4,2l−3 so2l−2,3 so2l,1

2 2 2 1

Al

c cc c

cc6

?6?

6?

�� sll+1(R)

Cl

c c c csp1,l−1 sp2,l−2 spl−1,1 spl (R)

1 1 1 2

c cc c

cc6

?6?

6?

##

cc

s sll+1(R)

Dl

c c c ccQQQ

���

so2,2l−2 so4,2l−4 so2l−4,4

so∗2l

so∗2l

c cc c

cc6

?6?

6?

##

cc

c sl(l+1)/2(H) c c c ccQQQ

���

so3,2l−3 so5,2l−5 so2l−3,3

6

?

Table: Vogan diagrams of noncompact classical Lie algebras

Vogan diagrams and associated real Lie algebras Classification of Simple Real Lie Algebras June 19, 2014 25 / 31

E Is c cc cc��HH6? 6? EVII c c c ccc s

E II c c c csc EVIII c c c c ccs cE III c c c ccs EIX c c c c ccc sE IVc c cc cc��HH6? 6? FI

c csc c1 1 2 2

E V c c c csc c FIIc cs c c1 1 2 2

E VI c c c ccs c Gc cs1 3

Table: Vogan diagrams of noncompact exceptional Lie algebras

Vogan diagrams and associated real Lie algebras Classification of Simple Real Lie Algebras June 19, 2014 26 / 31

The classical Lie algebrasLet In be the identity matrix and

Jn,n =(

0 In−In 0

), Ip,q =

(−Ip 0

0 Iq

), Kn,n =

(0 InIn 0

)Classical Lie algebras over C

sln(C) = {X ∈ gln(C) | Tr X = 0}son(C) = {X ∈ gln(C) |X + X T = 0}spn(C) = {X ∈ gl2n(C) |X T Jn,n + Jn,nX = 0}

Classical Lie algebras over RType Al :

sln(R) = {X ∈ gln(R) | Tr X = 0} split real formsln(H) = {X ∈ gln(H) |Re Tr X = 0}sup,q = {X ∈ gln(C) |X ∗Ip,q + Ip,qX = 0} compact real form

for p = 0

Vogan diagrams and associated real Lie algebras Classification of Simple Real Lie Algebras June 19, 2014 27 / 31

Type Bl and Dl :

sop,q = {X ∈ glp+q(R) |X ∗Ip,q + Ip,qX = 0}so∗2l = {X ∈ sun,n |X T Kn,n + Kn,nX = 0}

Type Cl :

spp,q = {X ∈ glp+q(H) |X ∗Ip,q + Ip,qX = 0}spn(R) = {X ∈ gl2n(R) |X T Jn,n + Jn,nX = 0}

Example: Al Classification of Simple Real Lie Algebras June 19, 2014 28 / 31

Let g = sll+1(C) and {Eij | 1 ≤ i , j ≤ l + 1} be the canonical basisof gll+1(C).

Cartan subalgebra h = {diag(x1, . . . , xl+1) |∑

xi = 0}Let ei ∈ h∗, ei : diag(x1, . . . , xl+1) 7→ xi .

Roots αij = ei − ej ∈ h∗, αij(diag(x1, . . . , xl+1)) = xi − xj

Positive, negative and fundamental roots

Φ+ = {αij |i < j}, Φ− = {αij |i > j}, Π = {αi ,i+1 | 1 ≤ i ≤ l}.

Chevalley basis

{Hi = Eii − Ei+1,i+1 | 1 ≤ i ≤ l} ∪ {Eij | i 6= j}

The scalar product on 〈Φ〉R is defined by

(αi ,i+1, αj,j+1) = αi ,i+1(Hj) =

2 for k = l−1 for |k − l | = 10 else

Example: Al Classification of Simple Real Lie Algebras June 19, 2014 29 / 31

The Cartan matrix Akl containing the Cartan integers〈αij , αkl〉 = 2(αij ,αkl )

(αkl ,αkl )

Al =

2 −1 0 . . . 0 0−1 2 −1 . . . 0 0

......

... . . . ......

0 0 0 . . . 2 −10 0 0 . . . −1 2

Hence we get the Dynkin diagram

c c c ce1−e2 e2−e3 el−1−el el−el+1

Example: Al Classification of Simple Real Lie Algebras June 19, 2014 30 / 31

The split real form of g = sll+1(C) is

sll+1(R) = 〈Hi , 1 ≤ i ≤ l ,Eij , i 6= j〉R

The corresponding conjugation is σ : X 7→ X .The Weyl involution can be given as ω = X 7→ −X T . Hencethe Cartan involution θ = ωσ : X 7→ −X T = −X ∗ defines thecompact real form

gθ = {X ∈ sll+1(C) |X = −X ∗} = sul+1(C)

The corresponding Vogan diagram is just the “empty” Dynkindiagram.

Example: Al Classification of Simple Real Lie Algebras June 19, 2014 31 / 31

For p + q = l + 1, 1 ≤ p, q ≤ l let

sup,q = {X ∈ sll+1(C) |X ∗Ip,q + Ip,qX = 0} =

={

X =(

A BB∗ D

) ∣∣∣∣A = −A∗, D = −D∗}

The complexification of sup,q is sll+1(C), so it is a real form ofsll+1(C).The Cartan decomposition of sup,q corresponding to theCartan involution θ(X ) = −X ∗ is

n0 =(

A 00 D

)p0 =

(0 B

B∗ 0

)The diagonal matrices in sup,q are all in n0, so it is amaximally compact Cartan subalgebra.There are no complex roots ⇒ θ trivially acts on Π.The only simple root whose root space is not in n0 isep − ep+1 ⇒ the only painted root is the p-th.We get he Vogan diagram of type Al with trivialautomorphism and with only the p-th vertex painted.