EXTENSIONS OF THE JACOBI IDENTITY 17
243 8 \~^r)z« s \r~zr)Z328
\~^r)'
•[Y(v4,zA) xZil [Y(v3,z3) xZil [Y(v2,z2) xZ21 K(v1,«i)]]]+
+ *3 4
* l " i 7 " J *
4 2
* l ~ ^ " J
Z M
* l"^r~J
, Z3)
X
2 3 1
,
-3r
^ 4 2

^32
\ - 1
f
/^2
~
*41
\ -1
r
/^31

*2
\
+241
* h s r J *1
2 4
*
\~^~)*3281
\n^-)
•[Y{v3,z3) xZ31 [Y(v2,z2) xZ21 [Y(v4,z4) xZil y(«
1
,z
1
)]]]+
+243
*1-^3—)
*4 2
*
l~^~J2 23 6
\~^r)'
? ^ 3 j X2T31
, Z4)
X
2 4 1
+ the expression obtained permuting the indices 3 and 4
in all the previous summands. (1.52)
The existence of the expressions (1.47) (1.52) will be verified via the
next two lemmas. The equality of (1.51) and (1.52) will be called "four-vertex
operator Jacobi identity". However, as in the case of three vertex operators,
to justify it we must verify the existence of the expressions involved.
L e m m a 1.6 (cf. Lemma 1.1). The expressions
n
z
^ (
Z j
^ ~
Z j i
) (1.53)
and
n
*£i4Et'*M~1V (1
-
54)
where Z& is as above and where
s
fY?k=izk,k-i\
is to fee expanded in nonnegative integral powers of Zk,k-i for all k such that
i+lkj
(cf Remark 1.2), exist and are equal.
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