14 CRISTIANO HUSU

expressions, it too exists, and so does (1.28), by the Jacobi identity. Since it

is illuminating to have a direct proof that (1.28) exists, we also present such

an argument based on Lemma 1.1. We use Lemma 1.1 to rewrite (1.28) as

Y(Y(Y(v3,Z32)v2,zn)vllZl)z;H (*±£»)

Z

-H (*+ ** + **) •

.Wffclifcy

(L40)

The coefficient of

z2x~x z^1"3 z3±~k,

i , j , k G Z in (1.40) equals

Y(Y(Y(v3,z32)v2,z2i)vlyzl)(z1 + z21)\z1 + z2l +

z32)j(z21

+

z32)k.

(1.41)

Expanding Y(v3, z32) according to (1.4) and using (1.5), the coefficient of

zl32

(for each / G Z) in (1.41) can be written as a finite linear combination of

expressions

Y(Y{wJz21)vuz1)(z1+z21)mz^

(1.42)

where w G V and m, n are fixed integers. The same argument shows that

these exist. (Cf. [6], Chapter 8). •

Lemma 1.3 gives the existence of (1.25) — (1.29).

Remark 1.4 Using (1*19) the expression (1.29) can be formally written as

a sum of eight terms, each one of which is a product of three 8-functions and

the three vertex operators Y{yi,Zi), i = 1,2,3. The existence of (1.29) does

not imply the existence of each of these eight summands, and, in particular,

the distributive law does not apply since the summands do not all exist. For

instance, the two summands obtained by applying (1*19) to

[Y(v2, z2) x221 Y(v1,z1)}Y(v3, zz)z32H f *

3 1

" *

2 1

) z;i6 (?L1*) (1.43)

\ z32 / \ z13 J

do not exist separately.