12 IGOR B. FRENKEL, YI-ZHI HUANG AND JAMES LEPOWSKY

where

L(n)=ujn+1 for n E Z, i.e., Y(u,z) = ^

L(n)z-n-2

(2.2.10)

and

r a n k F E F ; (2.2.11)

L(0)v = nv = (wt v)v for n G Z and v G V(n); (2.2.12)

~ 7 ( V ) ^ W - 1 ) V ) . (2.2.13)

This completes the definition. We may denote the vertex operator algebra

just defined by

(V,r,l,«) . (2.2.14)

In practice we will typically have

rank V G Q, rank V 0. (2.2.15)

Remark 2.2.2. Axioms (2.2.4) and (2.2.5) are together equivalent to a single

axiom - that V be equipped with a linear map

V ® V — V((z))

v1®v2*- Y(vu z)v2. (2.2.16)

Remark 2.2.3. Property (2.1.14) of the ^-function amounts to the case u =

v = 1 of the Jacobi identity.

Remark 2.2.4-

I*1 ^ne

presence of the other axioms, we can replace the creation-

property axiom (2.2.7), whose naturality will become apparent, by the natural

injectivity condition

Y(v,z) = 0 implies v = 0 for v 6 V. (2.2.17)

To see that (2.2.7) and (2.2.17) are equivalent, first note that (2.2.17) follows

immediately from (2.2.7). The following converse argument provides an excellent

illustration of the methods of formal calculus. Assume that all the axioms except

(2.2.7) hold, together with (2.2.17). Using the basic 6-function properties, the