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
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