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.
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