8

CRISTIANO HUSU

(the Jacobi identity), where

neZ

and where

is to be expanded as a formal power series in the second term in the numer-

ator, Z2, and similarly the other ^-functions expressions;

[L(m), L(n)] = (m - n)L(m + n) + jz(m3 - m)$m+n,0(rankV) (1.10)

for m, n 6 Z, where

X(n) =un+1 for n G Z (i.e., r(w,z) = ^ ( n ) * " 7 1 " 2 ) , (1.11)

and where

rankV G Q; (1.12)

X(0)u = nv = (wtt;)u for n £ Z and t £ V(n); (1-13)

.^K(t,,z) = y(I(-l)t;,z) . (1.14)

The formal Laurent series

Y(v,z)=y£vnz-n-1

neZ

are called vertex operators. Throughout the remainder of the present section,

we will discuss extensions of the Jacobi identity.