EXTENSIONS OF THE JACOBI IDENTITY 15

As we have mentioned, the identity obtained by equating (1.28) with

(1.29) is called the three-operator (or three-vertex operator) Jacobi identity:

zm & ( ) [Y(v3iz3) xz3i \y(v2,Z2) x221

Y(v1,z1))}+

\ z32 J

+z£6 P

2 1

~

Z 3 1

) [Y(v2,z2) x221 [Y(v3,z3) x231 Y(v1,z1)}} =

\ Z23 /

^

221

/ j

= 2

\ ZX J

(1-44)

This achieves the goal of expressing a combination of products of vertex

operators Y(vi,Zi) and suitable ^-functions in terms of the product of an

"iterated" vertex operator and the resulting ^-functions. Moreover, we recall

that both sides of (1.44) equal the expression (1.27).

Remark 1.5 Applying the three-operator Jacobi identity we see that the

identity of Proposition A.3.3 in [6], namely

Y, o-Y(Y(Y(v3,z32)v2,Z2i)vuzl)z^S (— — J •

rG{l,(123),(132)} V Z21 '

•Ii^(Zj-^)=0,

(1.45)

can be written in the form

£ *[Y(v3,z3) x231 [Y(v2,z2) xZ21 Y{vuZl)))z3^ ( *

3 1

~ *

2 1

) = 0, (1.46)

where S3 is the symmetric group on the 3 indices.

The case n = 4 in (1.23) and (1.24) shows all the ideas and subtleties of

the general case, which will be discussed in the next subsection. The identity

(1.23) gives