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
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