Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and...

Lie Superalgebras Generalities Classification Root Systems Representation Theory

Lie Superalgebras and Representation Theory

Johan van de Leur

Johan van de Leur



Lie Superalgebras

A lie superalgebra g is a Z2-graded vector spaceg = g0 ⊕ g1 together with a multiplication [·, ·] thatsatisfies conditions:

I [·, ·] is bilinear and [ga, gb] ⊂ ga+b,

I supersymmetric: [a, b] = −(−)ab[b, a], wherea ∈ ga, b ∈ gb,

I Jacobi identity:

[a, [b, c]] = [[a, b], c] + (−)ab[b, [a, c]].

Johan van de Leur



Example 1, gl(m, n):

(Am,m OO Dn,n

)⊕

(O Bm,n

Cn,m O

)

gl(m, n) =gl(m, n)0 ⊕ gl(m, n)1, where

gl(m, n)0 =glm ⊕ gln,

gl(m, n)1 =Cm ⊗ Cn ⊕ Cn ⊗ Cm

Johan van de Leur



Example 1, gl(m, n):

(Am,m OO Dn,n

)⊕

(O Bm,n

Cn,m O

)gl(m, n) =gl(m, n)0 ⊕ gl(m, n)1, where

gl(m, n)0 =glm ⊕ gln,

gl(m, n)1 =Cm ⊗ Cn ⊕ Cn ⊗ Cm

Johan van de Leur



Example 2, sl(m, n) or A(m, n):

M =

(Am,m Bm,n

Cn,m Dn,n

)∈ gl(m, n)

supertrace: str M =m∑

i=1

aii −n∑

j=1

djj , then the

special linear Lie superalgebra:

sl(m, n) = {M ∈ gl(m, n) | str M = 0}.If m 6= n then sl(m, n) = A(m, n) is simple,

otherwise CI2n is an ideal. A(n, n) = sl(n, n)/CI2nis simple.

Johan van de Leur




M =

(Am,m Bm,n

Cn,m Dn,n

)∈ gl(m, n)


i=1

aii −n∑

j=1

djj

, then the




Johan van de Leur




M =

(Am,m Bm,n

Cn,m Dn,n

)∈ gl(m, n)


i=1

aii −n∑

j=1

djj , then the


sl(m, n) = {M ∈ gl(m, n) | str M = 0}.

If m 6= n then sl(m, n) = A(m, n) is simple,otherwise CI2n is an ideal. A(n, n) = sl(n, n)/CI2nis simple.

Johan van de Leur




M =

(Am,m Bm,n

Cn,m Dn,n

)∈ gl(m, n)


i=1

aii −n∑

j=1

djj , then the



otherwise CI2n is an ideal.

A(n, n) = sl(n, n)/CI2nis simple.

Johan van de Leur




M =

(Am,m Bm,n

Cn,m Dn,n

)∈ gl(m, n)


i=1

aii −n∑

j=1

djj , then the




Johan van de Leur



Example 3, osp(m, n)

Let

B =

iIm O OO O InO −In O

then the orthosymplectic Lie superalgebra:

osp(m, 2n) = {M ∈ gl(m, 2n) |MB + iMBM = 0}

B(m, n) = osp(2m + 1, 2n),D(m, n) = osp(2m, 2n),C (n) = osp(2, 2n − 2).

Johan van de Leur



Example 4, the strange Lie algebra Q(n)

Q(n) = {(

a bb a

)∈ gl(n+1, n+1) |a ∈ gln+1, b ∈ sln+1}

CI2n+2 is an ideal in Q(n) andQ(n) = Q(n)/CI2n+2 is simple.

Johan van de Leur



Example 5, the Cartan type superalgebra W (n)

Grassmann algebra Λ(n) = C[θ1, θ2, . . . θn]

,θi are Grassmann variables,satisfying θiθj = −θjθi and θ2

i = 0.

W (n) = der Λ(n) = {n∑

i=1

Pi∂

∂θi|Pi ∈ Λ(n)}

W (n) is simple for n ≥ 2

W (2) ' sl(2, 1)

Johan van de Leur



Example 5, the Cartan type superalgebra W (n)

Grassmann algebra Λ(n) = C[θ1, θ2, . . . θn] ,θi are Grassmann variables,satisfying θiθj = −θjθi and θ2

i = 0.

W (n) = der Λ(n) = {n∑

i=1

Pi∂

∂θi|Pi ∈ Λ(n)}

W (n) is simple for n ≥ 2

W (2) ' sl(2, 1)

Johan van de Leur



Semi-simple Lie Superalgebras?

For a Lie algebra one has the following equivalentstatements for semi-simplicity:

I g does not contain nonzero solvable ideals.

I g is the direct sum of simple Lie algebras.

I The Killing form of g is nondegenerate.

I All finite dimensional representations of g arecompletely reducible.

These conditions are not equivalent for Liesuperalgebras.

Johan van de Leur



Graded representations

Let V = V0 ⊕ V1 be a graded vector space, wedefine a graded representation by

ρ : g → End V , such that

ρ(a)Vi = Vi+a a ∈ ga

and

ρ([a, b]) = ρ(a)ρ(b)− (−)abρ(b)ρ(a).

Ex. The adjoint representation: ad a(b) = [a, b]

Johan van de Leur



Killing form:

K(a, b) = str(ad a ad b)

Properties:

Supersymmetric: K(a, b) = (−)abK(b, a),Invariant:K(a,[b,c])=K([a,b],c),Even: K (g0, g1) = 0.

On sl(m, n): K(a, b) = 2(m − n)str(ab)

Johan van de Leur



Killing form:


Properties:

Supersymmetric: K(a, b) = (−)abK(b, a),

Invariant:K(a,[b,c])=K([a,b],c),Even: K (g0, g1) = 0.


Johan van de Leur



Killing form:


Properties:

Supersymmetric: K(a, b) = (−)abK(b, a),Invariant:K(a,[b,c])=K([a,b],c),

Even: K (g0, g1) = 0.


Johan van de Leur



Killing form:


Properties:

Supersymmetric: K(a, b) = (−)abK(b, a),Invariant:K(a,[b,c])=K([a,b],c),Even: K (g0, g1) = 0.


Johan van de Leur



Poincare–Birkhoff–Witt

g = g0 ⊕ g1, witha1, a2, . . . , am basis g0,b1, b2, . . . , bn basis g1,basis universal enveloping Lie superalgebra U(g):

ak1

1 ak2

2 · · · akmm b`1

1 b`2

2 · · · b`nn

0 ≤ ki ∈ Z, `j = 0, 1.

Johan van de Leur



Classification of simple finite dim. Lie superalgebras

There exist 3 types of Lie superalgebras:basic classical, strange classical and of Cartan type.

Classical: The representation of g0 on g1 iscompletely reducible.Basic: If there exists a nondegenerate, invariant,even bilinear form on g.Strange: Classical, but not basic.Cartan type: Defined as certain derivations.

Johan van de Leur




There exist 3 types of Lie superalgebras:basic classical, strange classical and of Cartan type.Classical: The representation of g0 on g1 iscompletely reducible.

Basic: If there exists a nondegenerate, invariant,even bilinear form on g.Strange: Classical, but not basic.Cartan type: Defined as certain derivations.

Johan van de Leur




There exist 3 types of Lie superalgebras:basic classical, strange classical and of Cartan type.Classical: The representation of g0 on g1 iscompletely reducible.Basic: If there exists a nondegenerate, invariant,even bilinear form on g.

Strange: Classical, but not basic.Cartan type: Defined as certain derivations.

Johan van de Leur




There exist 3 types of Lie superalgebras:basic classical, strange classical and of Cartan type.Classical: The representation of g0 on g1 iscompletely reducible.Basic: If there exists a nondegenerate, invariant,even bilinear form on g.Strange: Classical, but not basic.

Cartan type: Defined as certain derivations.

Johan van de Leur




There exist 3 types of Lie superalgebras:basic classical, strange classical and of Cartan type.Classical: The representation of g0 on g1 iscompletely reducible.Basic: If there exists a nondegenerate, invariant,even bilinear form on g.Strange: Classical, but not basic.Cartan type: Defined as certain derivations.

Johan van de Leur



Classical Lie superalgebras:

superalgebra g g0 g1

A(m, n) C⊕ Am ⊕ An glm+1 ⊗ sln+1 ⊕ contragr.A(n, n) An ⊕ An sln+1 ⊗ sln+1 ⊕ contragr.

C (n + 1) Cn ⊕ C sp2n−2 ⊗ C⊕ contragr.

B(m, n) Bm ⊕ Cn so2m+1 ⊗ sp2n

D(m, n) Dm ⊕ Cn so2m ⊗ sp2n

F (4) A1 ⊕ B3 sl2 ⊗ spin7

G (3) A1 ⊕ G2 sl2 ⊗ G2

D(2, 1;α) A1 ⊕ A1 ⊕ A1 sl2 ⊗ sl2 ⊗ sl2

P(n) An Λ2sl∗n+1 ⊕ S2sln+1

Q(n) An ad sln+1

Johan van de Leur



Classical Lie superalgebras:

superalgebra g g0 g1

A(m, n) C⊕ Am ⊕ An glm+1 ⊗ sln+1 ⊕ contragr.A(n, n) An ⊕ An sln+1 ⊗ sln+1 ⊕ contragr.

C (n + 1) Cn ⊕ C sp2n−2 ⊗ C⊕ contragr.B(m, n) Bm ⊕ Cn so2m+1 ⊗ sp2n

D(m, n) Dm ⊕ Cn so2m ⊗ sp2n

F (4) A1 ⊕ B3 sl2 ⊗ spin7

G (3) A1 ⊕ G2 sl2 ⊗ G2

D(2, 1;α) A1 ⊕ A1 ⊕ A1 sl2 ⊗ sl2 ⊗ sl2

P(n) An Λ2sl∗n+1 ⊕ S2sln+1

Q(n) An ad sln+1

Johan van de Leur



Cartan type: W (n), H(n), S(n), S(n)

Let ω = (dθ1)2 + · · ·+ (dθn)

2, then

H(n) = {D ∈ W (n) |D(ω) = 0}

and

H(n) = [H(n), H(n)].

S(n) = {∑

i

Pi∂

∂θi|∑

i

∂Pi

∂θi= 0}.

S(n) = {D ∈ W (n) |D((1 + θ1θ2 · · · θn)ξθ1 ∧ · · · ∧ ξθn) = 0}

here ξ is a differential of degree 0.

Johan van de Leur



Cartan type: W (n), H(n), S(n), S(n)

Let ω = (dθ1)2 + · · ·+ (dθn)

2, then

H(n) = {D ∈ W (n) |D(ω) = 0} and

H(n) = [H(n), H(n)].

S(n) = {∑

i

Pi∂

∂θi|∑

i

∂Pi

∂θi= 0}.

S(n) = {D ∈ W (n) |D((1 + θ1θ2 · · · θn)ξθ1 ∧ · · · ∧ ξθn) = 0}

here ξ is a differential of degree 0.

Johan van de Leur



Root systems

Let g be a basic classical Lie superalgebra.

Fix a Cartan subalgebra of h ∈ g0 and make the root spacedecomposition:

g = h⊕⊕

0 6=α∈h∗

gα,

α is called a root if gα ∩ g 6= 0. Let ∆ be the set of roots, then

∆ = ∆0 ∪∆1 disjoint union

Problem: There is not a unique simple root system.Fix a simple rootsystem of ∆0

Johan van de Leur



Root systems

Let g be a basic classical Lie superalgebra.Fix a Cartan subalgebra of h ∈ g0 and make the root spacedecomposition:

g = h⊕⊕

0 6=α∈h∗

gα,




Johan van de Leur



Root systems


g = h⊕⊕

0 6=α∈h∗

gα,

α is called a root if gα ∩ g 6= 0. Let ∆ be the set of roots

, then



Johan van de Leur



Root systems


g = h⊕⊕

0 6=α∈h∗

gα,




Johan van de Leur



Root systems


g = h⊕⊕

0 6=α∈h∗

gα,



Problem: There is not a unique simple root system.

Fix a simple rootsystem of ∆0

Johan van de Leur



Root systems


g = h⊕⊕

0 6=α∈h∗

gα,




Johan van de Leur



Weyl group

If α ∈ ∆ is non-isotropic, i.e. (α, α) 6= 0, then one can definereflections:

rα(β) = β − 2(β, α)

(α, α)α

and the Weyl group W is the group generated by all suchreflections.

Then: W is the Weyl group of g0.

Johan van de Leur



Weyl group

If α ∈ ∆ is non-isotropic, i.e. (α, α) 6= 0, then one can definereflections:

rα(β) = β − 2(β, α)

(α, α)α

and the Weyl group W is the group generated by all suchreflections.Then: W is the Weyl group of g0.

Johan van de Leur



Example A(m − 1, n − 1)

g0 = slm ⊕ sln(⊕C)

Then

∆0 = {εi − εj , δk − δ` | 1 ≤ i , j ≤ m, 1 ≤ k, ` ≤ n}

with bilinear form

(εi , εj) = δij , (δk , δ`) = −δk`, (εi , δk) = 0.

W = Sm ×Sn, i.e. (permutations of ε’s)×(permutations of δ’s).Odd roots:

∆1 = {εi − δk , δk − εi | 1 ≤ i ≤ m, 1 ≤ k ≤ n}

Johan van de Leur





Then

∆0 = {εi − εj , δk − δ` | 1 ≤ i , j ≤ m, 1 ≤ k, ` ≤ n}

with bilinear form


W = Sm ×Sn, i.e. (permutations of ε’s)×(permutations of δ’s).

Odd roots:

∆1 = {εi − δk , δk − εi | 1 ≤ i ≤ m, 1 ≤ k ≤ n}

Johan van de Leur





Then

∆0 = {εi − εj , δk − δ` | 1 ≤ i , j ≤ m, 1 ≤ k, ` ≤ n}

with bilinear form


W = Sm ×Sn, i.e. (permutations of ε’s)×(permutations of δ’s).Odd roots:

∆1 = {εi − δk , δk − εi | 1 ≤ i ≤ m, 1 ≤ k ≤ n}

Johan van de Leur



Simple root systems for A(m − 1, n − 1):

ε1 − ε2, . . . , εi1−1 − εi1 , εi1 − δ1, δ1 − δ2, . . . , δj1−1 − δj1 , δj1 − εi1+1,

εi1+1 − εi1+2, . . . , εi2−1 − εi2 , εi2 − δj1+1, δj1+1 − δj1+2, . . .

And also one which starts with δ1, so ε’s and δ’s interchanged.

Johan van de Leur



Example A(1, 0), simple root system and Cartan matrix

ε1 − ε2, ε2 − δ

(2 −1−1 0

)

ε1 − δ, δ − ε2

(0 −1−1 0

)

δ − ε1, ε1 − ε2

(0 −1−1 2

)

Johan van de Leur



Example A(1, 0), simple root system and Cartan matrix

ε1 − ε2, ε2 − δ

(2 −1−1 0

)ε1 − δ, δ − ε2

(0 −1−1 0

)δ − ε1, ε1 − ε2

(0 −1−1 2

)

Johan van de Leur



Example D(2, 1; α)

D(2, 1;α) = A1 ⊕ A1 ⊕ A1 ⊕ sl2 ⊗ sl2 ⊗ sl2

Possible Cartan matrices (α 6= 0,−1): 2 −1 0−1 0 −α0 −α 2α

,

0 −1 1 + α−1 0 −α

1 + α −α 0

Johan van de Leur



Irreducible Highest Weight Representations

Problem: Not one unique simple root system.

Hence not one set of positive roots ∆+.Solution, we fix for every basic superalgebra one simple rootsystem.E.g. for A(m − 1, n − 1), we choose

ε1 − ε2, . . . , εm−1 − εm, εm − δ1, δ1 − δ2, . . . , δn−1 − δn

∆+0

= {εi − εj , δi − δj | i < j}∆+

1= {εi − δj}

Johan van de Leur




Problem: Not one unique simple root system.Hence not one set of positive roots ∆+.

Solution, we fix for every basic superalgebra one simple rootsystem.E.g. for A(m − 1, n − 1), we choose


∆+0

= {εi − εj , δi − δj | i < j}∆+

1= {εi − δj}

Johan van de Leur




Problem: Not one unique simple root system.Hence not one set of positive roots ∆+.Solution, we fix for every basic superalgebra one simple rootsystem.

E.g. for A(m − 1, n − 1), we choose


∆+0

= {εi − εj , δi − δj | i < j}∆+

1= {εi − δj}

Johan van de Leur




Problem: Not one unique simple root system.Hence not one set of positive roots ∆+.Solution, we fix for every basic superalgebra one simple rootsystem.E.g. for A(m − 1, n − 1), we choose


∆+0

= {εi − εj , δi − δj | i < j}∆+

1= {εi − δj}

Johan van de Leur






∆+0

= {εi − εj , δi − δj | i < j}

∆+1

= {εi − δj}

Johan van de Leur






∆+0

= {εi − εj , δi − δj | i < j}∆+

1= {εi − δj}

Johan van de Leur



Let Λ ∈ h∗ and let V (Λ) be the irreducible highest weight modulefor g with respect to the triangular decomposition

g =⊕

α∈∆+

g−α ⊕ h⊕⊕

α∈∆+

gα

ThenV (λ) =

⊕µ∈h∗

Vµ,

whereVµ = {v ∈ V (Λ) | hv = µ(h)v for all h ∈ h}.

Related to this we define the formal character

chV (Λ) =∑µ∈h∗

dim(Vµ)eµ

Johan van de Leur



Typical and Atypical Representations

Λ is called dominant integral if

0 ≤ 2(Λ, α)

(α, α)∈ Z, for all α ∈ ∆+

0

Let

ρ0 =1

2

∑α∈∆+

0

α, ρ1 =1

2

∑α∈∆+

1

α, ρ = ρ0 − ρ1

We call the the weight Λ and module V (Λ) typical if

(Λ + ρ, α) 6= 0, for all α ∈ ∆+1,

otherwise it is called atypical.

Johan van de Leur



Theorem (Kac). If Λ is a dominant integral typical weight, then

chV (Λ) =L1

L0

∑w∈W

ε(w)ew(Λ+ρ),

where ε(w) is the signature of w

and

L0 =∏

α∈∆+0

(eα/2 − e−α/2

), L1 =

∏β∈∆+

1

(eβ/2 + e−β/2

)Note that w(L1) = L1 and that

ew(λ+ρ)L1 =w

eλ+ρ0e−ρ1∏

β∈∆+1

(eβ/2 + e−β/2

)=w

eλ+ρ0∏

β∈∆+1

(1 + e−β

)

Johan van de Leur




chV (Λ) =L1

L0

∑w∈W

ε(w)ew(Λ+ρ),

where ε(w) is the signature of w and

L0 =∏

α∈∆+0

(eα/2 − e−α/2

), L1 =

∏β∈∆+

1

(eβ/2 + e−β/2

)

Note that w(L1) = L1 and that

ew(λ+ρ)L1 =w

eλ+ρ0e−ρ1∏

β∈∆+1

(eβ/2 + e−β/2

)=w

eλ+ρ0∏

β∈∆+1

(1 + e−β

)

Johan van de Leur




chV (Λ) =L1

L0

∑w∈W

ε(w)ew(Λ+ρ),

where ε(w) is the signature of w and

L0 =∏

α∈∆+0

(eα/2 − e−α/2

), L1 =

∏β∈∆+

1

(eβ/2 + e−β/2

)Note that w(L1) = L1 and that

ew(λ+ρ)L1 =w

eλ+ρ0e−ρ1∏

β∈∆+1

(eβ/2 + e−β/2

)=w

eλ+ρ0∏

β∈∆+1

(1 + e−β

)Johan van de Leur



Thus

chV (Λ) =1

L0

∑w∈W

ε(w)w

eλ+ρ0∏

β∈∆+1

(1 + e−β

)

For atypical weights of gl(m, n) the formula is more complicated,this formula was conjectured by Van der Jeugt, Hughes, King andThierry-Mieg and a proof was given by Su and Zhang.Important ingredient are the set of atypical roots for the weight Λ,this is the set

ΓΛ = {α ∈ ∆+1|(Λ + ρ, α) = 0}

Johan van de Leur



Thus

chV (Λ) =1

L0

∑w∈W

ε(w)w

eλ+ρ0∏

β∈∆+1

(1 + e−β

)For atypical weights of gl(m, n) the formula is more complicated,

this formula was conjectured by Van der Jeugt, Hughes, King andThierry-Mieg and a proof was given by Su and Zhang.

Important ingredient are the set of atypical roots for the weight Λ,this is the set

ΓΛ = {α ∈ ∆+1|(Λ + ρ, α) = 0}

Johan van de Leur



Thus

chV (Λ) =1

L0

∑w∈W

ε(w)w

eλ+ρ0∏

β∈∆+1

(1 + e−β

)For atypical weights of gl(m, n) the formula is more complicated,

this formula was conjectured by Van der Jeugt, Hughes, King andThierry-Mieg and a proof was given by Su and Zhang.Important ingredient are the set of atypical roots for the weight Λ,this is the set

ΓΛ = {α ∈ ∆+1|(Λ + ρ, α) = 0}

Johan van de Leur



For certain dominant integral atypical weights Λ which are socalled totally disconnected, which is some technical term, theformal character is as follows

chV (Λ) =1

L0

∑w∈W

ε(w)w

eλ+ρ0∏

β∈∆+1\ΓΛ

(1 + e−β

)

E.g. if Λ 6= 0 and |ΓΛ| = 1, then Λ is totally disconnected.

Johan van de Leur



For certain dominant integral atypical weights Λ which are socalled totally disconnected, which is some technical term, theformal character is as follows

chV (Λ) =1

L0

∑w∈W

ε(w)w

eλ+ρ0∏

β∈∆+1\ΓΛ

(1 + e−β

)E.g. if Λ 6= 0 and |ΓΛ| = 1, then Λ is totally disconnected.

Johan van de Leur


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