Chapter 2
Construction of the affine Lie algebras
2.1 Alp (I 1),
D,(1)
( Z 1) and E^ (* = 6,7,8)
In this section we briefly review the construction of the simply-laced (or equal root
length) Lie algebras and their affinizations from a positive definite even lattice L with
a symmetric bilinear form (•,•). We use the "vocabulary" of Chapter 6 of [F-L-M]
to rewrite Chapter 2 of [M]. For a more complete treatment of the above and for
other properties of affine Lie algebras, see [B], [F-K], [F-L-M], [G], [Kac], [L], [L-Wl],
[L-W2], [M], or [Seg].
Let L be an even nondegenerate lattice of rank I (/ 1), with a symmetric bilinear
form (•,•): I x I Z. Suppose further that L is spanned (over Z) by the set
A = {a£L\(a,a) = 2 } , (2.1.1)
where A is a rank / indecomposable root system of type Ai (I 1), Di (I 1) or
Ei (I 6,7,8). Let II = {ai, a 2 , . . . , a/} denote the simple roots of A. Note that II
forms a Z-basis of the root system A, and hence, also of L.
Let (X, —) be a central extension of L by the cyclic group (AC | K2 = 1), i.e.,
1 _ (K |
K
2 = i ) u- l A L - 0 (2.1.2)
and let Co : L x L Z/2Z be the associated commutator map determined by
ab = KC0^^ba (2.1.3)
7
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