Lectures on Infinite Dimensional Lie Algebras: Kac-Moody algebras [Lecture notes]

159 Pages·2007·0.69 MB·English

by  Alexander Kleshchev

Lectures on Infinite Dimensional Lie
Algebras
Alexander Kleshchev

Contents
Part one: Kac-Moody Algebras page 1
1 Main Definitions 3
1.1 Some Examples 3
1.1.1 Special Linear Lie Algebras 3
1.1.2 Symplectic Lie Algebras 4
1.1.3 Orthogonal Lie Algebras 7
1.2 Generalized Cartan Matrices 10
1.3 The Lie algebra g˜(A) 13
1.4 The Lie algebra g(A) 16
1.5 Examples 20
2 Invariant bilinear form and generalized Casimir operator 26
2.1 Symmetrizable GCMs 26
2.2 Invariant bilinear form on g 27
2.3 Generalized Casimir operator 32
3 Integrable representations of g and the Weyl group 37
3.1 Integrable modules 37
3.2 Weyl group 39
3.3 Weyl group as a Coxeter group 42
3.4 Geometric properties of Weyl groups 46
4 The Classification of Generalized Cartan Matrices 50
4.1 A trichotomy for indecomposable GCMs 50
4.2 Indecomposable symmetrizable GCMs 58
4.3 The classification of finite and affine GCMs 61
5 Real and Imaginary Roots 68
5.1 Real roots 68
5.2 Real roots for finite and affine types 70
5.3 Imaginary roots 73
iii
iv Contents
6 Affine Algebras 77
6.1 Notation 77
6.2 Standard bilinear form 77
6.3 Roots of affine algebra 80
6.4 Affine Weyl Group 84
6.4.1 A 89
‘
6.4.2 D 89
‘
6.4.3 E 89
8
6.4.4 E 90
7
6.4.5 E 90
6
7 Affine Algebras as Central extensions of Loop Algebras 91
7.1 Loop Algebras 91
7.2 Realization of untwisted algebras 92
7.3 Explicit Construction of Finite Dimensional Lie Algebras 96
8 Twisted Affine Algebras and Automorphisms of Finite Order 99
8.1 Graph Automorphisms 99
8.2 Construction of Twisted Affine Algebras 108
8.3 Finite Order Automorphisms 114
9 Highest weight modules over Kac-Moody algebras 116
9.1 The category O 116
9.2 Formal Characters 118
9.3 Generators and relations 122
10 Weyl-Kac Character formula 127
10.1 Integrable highest weight modules and Weyl group 127
10.2 The character formula 128
10.3 Example: Lˆ(sl ) 132
2
10.4 Complete reducibility 134
10.5 Macdonald’s identities 136
10.6 Specializations of Macdonald’s identities 139
10.7 On converegence of characters 141
11 Irreducible Modules for affine algebras 144
11.1 Weights of irreducible modules 144
11.2 The fundamental modules for sbl2 151
Bibliography 155
Part one
Kac-Moody Algebras

1
Main Definitions
1.1 Some Examples
1.1.1 Special Linear Lie Algebras
Letg=sl =sl (C). Choosethesubalgebrahconsistingofalldiagonal
n n
matrices in g. Then, setting α∨ :=e −e ,
i ii i+1,i+1
α∨,...,α∨
1 n−1
is a basis of h. Next define ε ...,ε ∈h∗ by
1 n
ε :diag(a ,...,a )7→a .
i 1 n i
Then, setting α =ε −ε ,
i i i+1
α ,...,α
1 n−1
is a basis of h∗. Let
a =hα∨,α i.
ij i j
Then the (n−1)×(n−1) matrix A:=(a ) is
ij
 
2 −1 0 0 ... 0 0 0
−1 2 −1 0 ... 0 0 0 
 
 0 −1 2 −1 ... 0 0 0 
 
 . .
 .. 
 
 
0 0 0 0 ... −1 2 −1
 
0 0 0 0 ... 0 −1 2
This matrix is called the Cartan matrix. Define
X :=e , X :=e (1≤i<j ≤n)
εi−εj ij −εi+εj ji
3
4 Main Definitions
Note that
[h,X ]=α(h)X (h∈h),
α α
and
{α∨,...,α∨ }∪{X |1≤i6=j ≤n}
1 n−1 εi−εj
is a basis of g. Set e =X and f =X for 1≤i<n. It is easy to
i αi i −αi
check that
e ,...,e ,f ,...,f ,α∨,...α∨ (1.1)
1 n−1 1 n−1 1 n−1
generate g and the following relations hold:
[e ,f ]=δ α∨, (1.2)
i j ij i
[α∨,α∨]=0, (1.3)
i j
[α∨,e ]=a e , (1.4)
i j ij j
[α∨,f ]=−a f , (1.5)
i j ij j
(ade )1−aij(e )=0 (i6=j), (1.6)
i j
(adf )1−aij(f )=0 (i6=j). (1.7)
i j
A (special case of a) theorem of Serre claims that g is actually gener-
ated by the elements of (1.1) subject only to these relations. What is
important for us is the fact that the Cartan matrix contains all the in-
formationneededtowritedowntheSerre’spresentationofA. Sincethe
Cartan matrix is all the data we need, it makes sense to find a nicer ge-
ometricwaytopicturethesamedata. SuchpictureiscalledtheDynkin
diagram, and in our case it is:
• • • •...• • • •
α α α
1 2 n−1
Here vertices i and i+1 are connected because a = a = −1,
i,i+1 i+1,i
others are not connected because a = 0 for |i−j| > 1, and we don’t
ij
have to record a since it is always going to be 2.
ii
1.1.2 Symplectic Lie Algebras
Let V be a 2n-dimensional vector space and ϕ : V × V → C be a
non-degenerate symplectic bilinear form on V. Let
g=sp(V,ϕ)={X ∈gl(V)|ϕ(Xv,w)+ϕ(v,Xw)=0for all v,w ∈V}.
1.1 Some Examples 5
An easy check shows that g is a Lie subalgebra of gl(V). It is known
from linear algebra that over C all non-degenerate symplectic forms are
equivalent, i.e. if ϕ0 is another such form then ϕ0(v,w) = ϕ(gv,gw) for
some fixed g ∈GL(V). It follows that
sp(V,ϕ0)=g−1(sp(V,ϕ))g ∼=sp(V,ϕ),
thus we can speak of just sp(V). To think of sp(V) as a Lie algebra of
matrices, choose a symplectic basis e ,...,e ,e ,...,e , that is
1 n −n −1
ϕ(e ,e )=−ϕ(e ,e )=1,
i −i −i i
and all other ϕ(e ,e )=0. Then the Gram matrix is
i j
(cid:18) (cid:19)
0 s
G= ,
−s 0
where
0 0 ... 0 1
0 0 ... 1 0
 . 
s= . . (1.8)
 . 
 
0 1 ... 0 0
1 0 ... 0 0
Itfollowsthatthematricesofsp(V)inthebasisofe ’sarepreciselythe
i
matrices from the Lie algebra
(cid:18) (cid:19)
A B
sp ={ |B =sBts,C =sCts,D =−sAts},
2n C D
so sp(V)∼=sp . Note that sXts is the transpose of X with respect to
2n
the second main diagonal.
Choosethesubalgebrahconsistingofalldiagonalmatricesing. Then,
setting α∨ := e −e −e +e , for 1 ≤ i < n and
i ii i+1,i+1 −i,−i −i−1,−i−1
α∨ =e −e ,
n nn −n,−n
α∨,...,α∨ ,α∨
1 n−1 n
isabasisofh. Next,settingα =ε −ε for1≤i<n,andα :=2ε ,
i i i+1 n n
α ,...,α ,α
1 n−1 n
is a basis of h∗. Let
a =hα∨,α i.
ij i j
6 Main Definitions
Then the Cartan matrix A is the n×n matrix
 
2 −1 0 0 ... 0 0 0
−1 2 −1 0 ... 0 0 0 
 
 0 −1 2 −1 ... 0 0 0 
 
 . .
 .. 
 
 
0 0 0 0 ... −1 2 −2
 
0 0 0 0 ... 0 −1 2
Define
X = e , (1≤i≤n)
2εi i,−i
X = e , (1≤i≤n)
−2εi −i,i
X = e −e (1≤i<j ≤n)
εi−εj ij −j,−i
X = e −e (1≤i<j ≤n)
−εi+εj ji −i,−j
X = e +e (1≤i<j ≤n)
εi+εj i,−j j,−i
X = e +e (1≤i<j ≤n).
−εi−εj −j,i −i,j
Note that
[h,X ]=α(h)X (h∈h),
α α
and
{α∨,...,α∨}∪{X }
1 n α
is a basis of g. Set e =X and f =X for 1≤i≤n. It is easy to
i αi i −αi
check that
e ,...,e ,f ,...,f ,α∨,...α∨ (1.9)
1 n 1 n 1 n
generate g and the relations (1.2-1.7) hold. Again, Serre’s theorem
claims that g is actually generated by the elements of (1.11) subject
only to these relations. The Dynkin diagram in this case is:
• • • •...• • • < •
α α α α
1 2 n−1 n
Theverticesn−1andnareconnectedthewaytheyarebecausea =
n−1,n
−2 and a =−1, and in other places we follow the same rules as in
n,n−1
the case sl.