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 = 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.
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