===== PDFPAGE 259 =====
8.6. Commutators of General Vertex Operators 205 
a Lie algebra whose precise structure we determine. We also relax the 
integrality assumption in (8.4.34). 
Theorem 8.6.1: Let a, b E e and assume that 
(a, 6) E Z and c(a, 6) = (-l)('*'). (8.6.1) 
Let u', u' E S(& and set 
u = u' 0 i(a) = u' * i(a) E VL 
u = U' 0 i(b) = U' * i(b) E VL. 
(8.6.2) 
Then 
Mu, 211, UU, 2211 
= Res, z;'Y(Y(u, z,,)u, z2)e-Zo(a'azl)((z 1 / z~)'B(zI/z~)). (8.6.3) 
If L is integral and (8.6.1) holds for all a, b E L (in which case L is also 
even, c(a, a) being equal to I), then (8.6.3), with the factor (z1/z2)' 
omitted, holds for all u, u E V,. 
Proof: Let k, 12 1 and let 
a,, .. ., a,, bl, . . ., bl E e 
subject to the conditions 
a = a, ak, b = 6, b/ 
Set 
(8.6.4) 
B = exp( c 
\j= I nz I n , XI11 
Using Remark 8.3.9 we see that the coefficients in the formal power series 
A and B span S(@ 0 r(a) and S(&) 0 r(b), respectively, as k, I, ai and bj 
vary subject to the constraint (8.6.4). In fact, the coefficients in 
spans(&) as k and a; vary (let the 0; range through a fixed basis of Q, 
===== PDFPAGE 260 =====
206 General Theory of Untwisted Vertex Operators 
with repetitions as necessary according to Remark 8.3.9); now multiply by 
where 
ak = (a] *'* ak-l)-la. 
Hence it suffices to prove the Theorem with u and u replaced by A and B, 
respectively. 
Recalling (8.4.20), (8.4.22) and Remark 8.4.1 we have 
= SY(a1, W1) .** Y(Uk, Wk)Sl(l) 
= Y(U1, W1) .*. Y(ak, wk)l(l) n (Wi - Wj)-('i*'j), (8.6.6) 
1 sicjs k 
where (wi - wj)-('i9'j) is understood as the binomial expansion in non- 
negative integral powers of wj, so that 
n (wi - wj)-('i*'j) 
Isicjsk 
(Even though A involves only nonnegative integral powers of the wi, it is 
sometimes helpful to express it using arbitrary powers.) Note that without 
the constraint (8.6.4), the expressions (8.6.6) would have been more com- 
plicated. There are analogous expressions for B. In particular, 
B = SY(b1,Xi) **. Y(b,,X1)8l(I). (8.6.8) 
From (8.5.5) and (8.3.3) we see (as in the proof of Theorem 8.4.2) that 
===== PDFPAGE 261 =====
8.6. Commutators of General Vertex Operators 207 
(8.6.9) 
Remark 8.6.2: Note that we have a formal generalization of Remark 
8.5.2: For all a,, . . . , ak E I?, 
- - Y(a1, 21 + w1) *** Y(ak, zl + wk), (8.6.10) 
from (8.6.6), (8.6.9) and a formal application of (8.4.25) (for several 
factors). [To recover (8.5.14), set wk = 0.1 The hypothesis (8.6.1) is not 
used here. We shall interpret (8.6.10) rigorously at the end of this chapter. 