VERTEX OPERATOR ALGEBRAS AND MODULES

3

The above form of the Jacobi identity is also parallel to the definition of Lie

algebra representation. In fact, by a representation of the Lie algebra V on the

module W one understands a linear map

*('):V®W-+W (1.5)

satisfying the identity

ir(u)ir(y) — 7r(v)7r(u) = 7r(ad(ti)-u) (1-6)

for any u, v G V. To round out the basic notions of Lie algebra and representa-

tions one defines the tensor product of two modules (Wi,7Ti), (W2^2) and then

the notion of intertwining operator from their tensor product to a third module

(W3,7T

3

):

/ ( • ) • : Wi ®W2- W3, (1.7)

which we also put into a form similar to (1.2) and (1.6):

7r3(u)I(*0 - I(v)ir2(u) = I(iri(ti)t;). (1.8)

Note that the module structure on the space W\ 8) W2 does not enter into this

formula. Starting from these definitions one can proceed to study representation

theory.

The theory of vertex operator algebras can be developed in a completely

parallel way. Each of the three definitions and identities has its vertex operator

algebra analogue which contains as an additional ingredient the Cauchy residue

formula, which may be written as:

-Resz=OQ f(z) - Resz=0 f(z) = R e s ,

=

,

0

/ ( z ) . (1.9)

Here we take f(z) to be a rational function of one complex variable with its only

poles at 0, 00 and z$, and we observe that this formula makes perfect sense over

our field F. Let V be a vector space over F and let

acU(-) -:V®V-+V((z)) (1.10)

be a linear map, where V((z)) denotes the algebra of those formal Laurent series

in the formal variable z involving at most finitely many negative powers of z.

Then ad

z

is the generating function of an infinite family of linear maps from

V 8 V to V. The main axiom for a vertex operator algebra is what we call the

Jacobi-Cauchy identity, or, especially in the alternative version given in the main

text of this paper, the Jacobi identity for vertex operator algebras:

- R e s

2 = 0 0

(f(z)a,dz(u)a,dZo(v)) - Res

z = 0

(f(z)adZo(v)adz(u))

= Res,

=

,

0

(/(^)ad,

0

(ad,.,

0

(ii)t;)) , (1.11)

where / is as above and where the residues are defined, as for scalar-valued ratio-

nal functions or formal series, as certain coefficients in appropriate expansions.