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 "fourvertex
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,ki\
is to fee expanded in nonnegative integral powers of Zk,ki for all k such that
i+lkj
(cf Remark 1.2), exist and are equal.