EXTENSIONS OF THE JACOBI IDENTITY 11

where zn2£n-i,2 ••• ^32

a r e n e w

commuting formal variables and where, in

the spirit of (1.18), we think of Z& as an object which may replace zn — z2\

and thus Zi — z2 in suitable expressions multiplied by appropriate £-functions.

In particular, the notation clarifies which variable plays the role of Zj in

(1.21) when several variables occur. The existence of the expression (1.24)

for n = 3 and n = 4 is proved in this subsection. In the cases n 4,

further manipulations of (1.24) will be performed; the existence of all the

expressions involved will be discussed in this subsection and in the next one.

In particular, it will be clear that the expression (1.24) exists.

The case n — 3 in (1.23) and (1.24) is illuminating. The identity (1.23)

gives the following two identities:

[Y(v3lz3) x231 \Y(v2,z2) x221 y(vi,*i)]] = Y(Y(v3,z3i)Y(v2,Z2i)vuziy

•nW

5

^)

a-25)

j=2

\

Zi

/

and

[Y(v2,z2) x221 [Y(v3,z3) xZ31 Y(vi,zi)]] = Y(Y(v2,z2i)Y(v3,z31)v1,z1)-

.J[z-iSf5iZ^). (1.26)

j=2

\

Z\

/

Multiplying the right-hand side of (1.25) by

\

^32

/

and the right-hand side of (1.26) by

Z

2Z

summing up the expressions obtained, and using the linearity of vertex op-

erators, we get

Y([Y(v3,z31) x232 Y(v2,z21)]v1,z1)f[z^s(^^^) (1.27)

;_o

\

Z\

/