VERTEX OPERATOR ALGEBRAS AND MODULES
5
appeared symbolically in (1.11) and are discussed further below. This Jacobi
identity has a natural ^-symmetr y which might be thought of as a blending of
the separate symmetries of the two classical formulas. The restoration of this
symmetry by means of the introduction of the third variable and the elimination
of the test function f(z) in the Jacobi-Cauchy identity (1.11) are advantages of
the formal-variable language that we use throughout the main text.
Besides the main axiom it is appropriate to introduce a few additional axioms
which further specify the notion of vertex operator algebra. These axioms are
natural from several points of view that we shall not discuss in this Introduction.
(See instead the Introduction in [FLM].) By analogy with (1.3), we assume that
Y(v,z) = 0 implies v = 0 for v G V. (1.14)
We also require the existence of two special elements 1 and u of V such that
y ( l , z ) = 1 (the identity operator) (1.15)
Y(
W
,z ) = £ L ( n ) z - » - 2 , (1.16)
n£7L
where the L(n) are the generators of the Virasoro algebra with a normalized
central element acting as multiplication by a scalar called the rank of the vertex
operator algebra. We also assume that
±Y(v,z) = Y(L(-l)v,z) (1.17)
and that V is Z-graded, with the grading truncated from below and with the ho-
mogeneous subspaces finite-dimensional and equal to the corresponding eigen-
spaces of L(0). In the presence of the other axioms, the injectivity (1.14) is
equivalent to the creation property, which states that Y(v,z)l is a power series
in z, and that its constant term is v. Appropriate related axioms define modules
(representation spaces) and intertwining operators. One important difference
between the vertex operator algebra case and the Lie algebra case (recall (1.7)
and (1.8)) is that in the present case, the tensor product of modules for a vertex
operator algebra does not in fact carry a module structure, but intertwining op-
erators are well defined nevertheless. This completes our sketch of the definition
of the concepts of vertex operator algebra and the related structures.
The three operators L{—1), L(0) and L(l) (the latter playing the role of L{—1)
in the contragredient module; see below) are particularly important in the struc-
ture of vertex operator algebras. They span a subalgebra isomorphic to 5t(2),
and in a suitable sense, the corresponding group acts by projective transforma-
tions on the variable z. One can weaken the vertex operator algebra axioms,
postulating instead of the full Virasoro algebra only this three-dimensional sub-
algebra, with appropriate properties. We call such objects quasi-vertex operator
algebras. There are a few other natural ways to modify the axiom system, as the
results in this paper, and in [B] and [FLM], make clear. However, the present
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