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Lie Superalgebras Generalities Classification Root Systems Representation Theory
Lie Superalgebras and Representation Theory
Johan van de Leur
Johan van de Leur
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
Lie Superalgebras Generalities Classification Root Systems Representation Theory
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
Lie Superalgebras and Representation Theory
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