EXTENSIONS OF THE JACOBI IDENTITY 9

1.2 The Jacobi identity and some consequences

Let V be a vertex operator algebra. Reversing the order of the indices 1 and

2 and renaming the variable ZQ as z2\, we can rewrite the Jacobi identity

(1.8) as

z£t

{p-^11)

Y(v2,z2)Y(vuz1) + z£6 (

f l

f ^ ) Y(v1,z1)Y(v2,z2) =

- z?8 (

f i

^

1

) Y{Y{v2,z21)vuzx), (1.15)

where ^1,^2 £ V, ^i,z

2

a n

d

z2\ a r e

independent commuting formal variables

and

z12 = -*

2 1

. (1.16)

As before (and always) 8{z) is as in (1.9) and the binomials are to be expanded

in nonnegative integral powers of the variable on the right.

We shall extend these notations and conventions to other commuting

formal variables. For instance, it will always be understood that

ZH = ~ZH (1A7)

for i and j distinct indices.

Note that the Jacobi identity expresses a certain combination of products

of two vertex operators (the left-hand side of (1.15)) in terms of an iterate

of vertex operators (the right-hand side). Also note that we may set

(*,•-*,•)" = *£ f o r n e Z , (1.18)

in expressions multiplied by

S(Zi~Zj)

(and analogously for other S-functions),

provided that the relevant algebraic limits exist (where "existence" is under-

stood in the precise sense of [6], Chapter 2).

We shall call the left-hand side of the Jacobi identity (1.15) the (general)

cross-bracket of Y(v2,z2) and Y{yi,Z\). It is denoted as follows (as in [6],

Section 8.8):

[Y(v2,z2) xZ21 Y(vi,z1)] =

z.

S

(zJ-Jl\

Y(v2,z2)Y(Vl,Zl) + z£6 ( * — ^ ) Y{vx,Zl)Y{v2,z2)

\ z21 J \ z12 J

(1.19)