Associative algebras and the representation theory of grading-restricted vertex algebras

Abstract

We introduce an associative algebra [Formula: see text] using infinite matrices with entries in a grading-restricted vertex algebra [Formula: see text] such that the associated graded space [Formula: see text] of a filtration of a lower-bounded generalized [Formula: see text]-module [Formula: see text] is an [Formula: see text]-module satisfying additional properties (called a nondegenerate graded [Formula: see text]-module). We prove that a lower-bounded generalized [Formula: see text]-module [Formula: see text] is irreducible or completely reducible if and only if the nondegenerate graded [Formula: see text]-module [Formula: see text] is irreducible or completely reducible, respectively. We also prove that the set of equivalence classes of the lower-bounded generalized [Formula: see text]-modules is in bijection with the set of the equivalence classes of nondegenerate graded [Formula: see text]-modules. For [Formula: see text], there is a subalgebra [Formula: see text] of [Formula: see text] such that the subspace [Formula: see text] of [Formula: see text] is an [Formula: see text]-module satisfying additional properties (called a nondegenerate graded [Formula: see text]-module). We prove that [Formula: see text] are finite-dimensional when [Formula: see text] is of positive energy (CFT type) and [Formula: see text]-cofinite. We prove that the set of the equivalence classes of lower-bounded generalized [Formula: see text]-modules is in bijection with the set of the equivalence classes of nondegenerate graded [Formula: see text]-modules. In the case that [Formula: see text] is a Möbius vertex algebra and the differences between the real parts of the lowest weights of the irreducible lower-bounded generalized [Formula: see text]-modules are less than or equal to [Formula: see text], we prove that a lower-bounded generalized [Formula: see text]-module [Formula: see text] of finite length is irreducible or completely reducible if and only if the nondegenerate graded [Formula: see text]-module [Formula: see text] is irreducible or completely reducible, respectively.