VERTEX OPERATORS AND INTEGRAL BASES 9

where h € h, and a, /3 € A, and c(a,/9) = ( - l ) * 0 ^ . )

Next extend (•, •) to a bilinear symmetric form on g x g by defining

(•*)lhxh = (••

h,z

a

) = 0

{ZayXb)

(2.1.9)

1 if aft = 1 ,

0 ifa&£{l,«}

where a, b £ A. (In terms of a section e, we have (xa,xp) = e(a,/3)8a+py0 for

a, /? G A.) We next state the following well-known result.

Theorem 2.1.1 ([F-K],[Seg]) fg, [•, -]j is a Lie algebra and the nonsingular form

(•, •) is symmetric and ^-invariant. Furthermore, depending on the lattice type, g is

isomorphic to A\, D\ or E\.

The affine Kac-Moody algebra g associated with (g, (•, •)) is the Lie algebra, which

as a vector space is given by

I = g® c[t,

r1]

e Cc e CM, (2.1.10)

with commutation relations given by

[x 8tm,y® tn] = [x, y] g +(x, y)m6m+n,

0

c

[c,fl = |fcc]= 0 (2.1.H)

[J, x 0 t m ] = -[a* g *m, J] = mx 0 *m

for a;, y E g and ra, n £ Z. The element c is called the central element and the

element d is called the degree operator. We next extend the form (•, •) to a symmetric

g-invariant bilinear form on g x g by setting

(x®tm,y®tn) = (x,y)Srn^0

(c,x®tm) = (a;®tm,c) = 0

(d, x®tm) = (x® tm, i) = 0 (2.1.12)

(c,d) = (d,c) = l

(c, c) = (d, d) = 0.