lie superalgebras and representation theoryleur0102/superalg.pdf · lie superalgebras and...
TRANSCRIPT
![Page 1: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/1.jpg)
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
![Page 2: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/2.jpg)
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
![Page 3: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/3.jpg)
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
![Page 4: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/4.jpg)
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
![Page 5: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/5.jpg)
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
![Page 6: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/6.jpg)
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
![Page 7: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/7.jpg)
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
![Page 8: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/8.jpg)
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
![Page 9: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/9.jpg)
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
![Page 10: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/10.jpg)
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
![Page 11: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/11.jpg)
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
![Page 12: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/12.jpg)
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
![Page 13: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/13.jpg)
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
![Page 14: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/14.jpg)
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
![Page 15: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/15.jpg)
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
![Page 16: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/16.jpg)
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
![Page 17: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/17.jpg)
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
![Page 18: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/18.jpg)
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
![Page 19: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/19.jpg)
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
![Page 20: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/20.jpg)
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
![Page 21: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/21.jpg)
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
![Page 22: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/22.jpg)
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
![Page 23: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/23.jpg)
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
![Page 24: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/24.jpg)
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
![Page 25: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/25.jpg)
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
![Page 26: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/26.jpg)
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
![Page 27: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/27.jpg)
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
![Page 28: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/28.jpg)
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
![Page 29: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/29.jpg)
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
![Page 30: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/30.jpg)
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
![Page 31: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/31.jpg)
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
![Page 32: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/32.jpg)
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
![Page 33: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/33.jpg)
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
![Page 34: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/34.jpg)
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
![Page 35: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/35.jpg)
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
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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
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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
![Page 38: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/38.jpg)
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
![Page 39: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/39.jpg)
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
![Page 40: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/40.jpg)
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
![Page 41: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/41.jpg)
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
![Page 42: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/42.jpg)
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
![Page 43: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/43.jpg)
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
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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
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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
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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
![Page 47: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/47.jpg)
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
![Page 48: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/48.jpg)
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
![Page 49: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/49.jpg)
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
![Page 50: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/50.jpg)
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
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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
![Page 52: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/52.jpg)
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
![Page 53: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/53.jpg)
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
![Page 54: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/54.jpg)
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
![Page 55: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/55.jpg)
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
![Page 56: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/56.jpg)
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
![Page 57: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/57.jpg)
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
![Page 58: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/58.jpg)
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
![Page 59: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/59.jpg)
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
![Page 60: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/60.jpg)
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
![Page 61: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/61.jpg)
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
![Page 62: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/62.jpg)
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
![Page 63: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/63.jpg)
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
![Page 64: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/64.jpg)
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
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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
![Page 66: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/66.jpg)
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
![Page 67: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/67.jpg)
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
![Page 68: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/68.jpg)
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
![Page 69: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/69.jpg)
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
![Page 70: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/70.jpg)
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
![Page 71: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/71.jpg)
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
![Page 72: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/72.jpg)
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
![Page 73: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/73.jpg)
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
![Page 74: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/74.jpg)
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
![Page 75: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/75.jpg)
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
![Page 76: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/76.jpg)
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
![Page 77: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/77.jpg)
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
![Page 78: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/78.jpg)
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
![Page 79: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/79.jpg)
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
![Page 80: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/80.jpg)
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
![Page 81: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/81.jpg)
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
![Page 82: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/82.jpg)
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
![Page 83: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/83.jpg)
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
![Page 84: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/84.jpg)
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
![Page 85: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/85.jpg)
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
![Page 86: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/86.jpg)
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
![Page 87: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/87.jpg)
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
![Page 88: Lie Superalgebras and Representation Theoryleur0102/superalg.pdf · Lie Superalgebras and Representation Theory. Lie Superalgebras Generalities Classification Root Systems Representation](https://reader034.vdocument.in/reader034/viewer/2022042219/5ec5c79ba42c8010311794b6/html5/thumbnails/88.jpg)
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