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)

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