12

CRISTIANO HUSU

(cf. (1.24)). By the Jacobi identity, (1.27) equals

3

and, on the other hand, using (1.25) and (1.26), it equals

Y(Y(Y(v3,z32)v2,z21)v1,z1)z;116 (*±Z**\ J J z^S (fLILfil)

( 1

.

2 8

)

V Z2\ / ,-_o \ Z\ /

z

-i6f^-^l.) [Y(v3,z3) xZ31 [Y(v2,z2) x221 Y(vuZl)}]+

\ Z32 /

+ z£6 (*

2 1

~*

3 1

) [Y(v2,z2) x221 [Y(v3,z3) x231 Y(vuZl))}. (1.29)

\ Z23 J

The equality of (1.28) and (1.29) will be called "three-vertex operator Jacobi

identity". However, to justify it we must verify the existence of the expres-

sions (1.25) through (1.29) (again cf. [6], Chapter 2). One approach is to use

the next two lemmas.

Lemma 1.1 The expressions

^ ( ^ )

2 f l {

( ^ ) ^ ( - _ _ - )

( 1

,

0 )

and

-lr;fZi+Z2l\ _ ! (ZX + ^21 + ^32\ - \ . (Z2\ + Z32\ /i 01 \

exist and are equal.

Remark 1.2 There is no ambiguity in the second factor of (LSI) since

((*i + 221) + z32)n = (*i + (*2i + z32))n for n e Z. (1.32)

Proof of Lemma 1.1. By Proposition 8.8.5 of [6],

^(^H"'(*£*)•

'-33

and similarly for the other factors of (1.30), so that (1.30) equals

-i£(zi + z2i\ _x (zx + z31\ _icfz21 + z32\

(

.

H