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

We introduce an associative algebra A ∞ (V) using infinite matrices with entries in a grading-restricted vertex algebra V such that the associated graded space Gr (W) = ∐ n ∈ ℕ Gr n (W) of a filtration of a lower-bounded generalized V -module W is an A ∞ (V) -module satisfying additional properties (called a nondegenerate graded A ∞ (V) -module). We prove that a lower-bounded generalized V -module W is irreducible or completely reducible if and only if the nondegenerate graded A ∞ (V) -module Gr (W) is irreducible or completely reducible, respectively. We also prove that the set of equivalence classes of the lower-bounded generalized V -modules is in bijection with the set of the equivalence classes of nondegenerate graded A ∞ (V) -modules. For N ∈ ℕ , there is a subalgebra A N (V) of A ∞ (V) such that the subspace Gr N (W) = ∐ n = 0 N Gr n (W) of Gr (W) is an A N (V) -module satisfying additional properties (called a nondegenerate graded A N (V) -module). We prove that A N (V) are finite-dimensional when V is of positive energy (CFT type) and C 2 -cofinite. We prove that the set of the equivalence classes of lower-bounded generalized V -modules is in bijection with the set of the equivalence classes of nondegenerate graded A N (V) -modules. In the case that V is a Möbius vertex algebra and the differences between the real parts of the lowest weights of the irreducible lower-bounded generalized V -modules are less than or equal to N ∈ ℕ , we prove that a lower-bounded generalized V -module W of finite length is irreducible or completely reducible if and only if the nondegenerate graded A N (V) -module Gr N (W) is irreducible or completely reducible, respectively. [ABSTRACT FROM AUTHOR]

Copyright of Communications in Contemporary Mathematics is the property of World Scientific Publishing Company and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)