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

We introduce an associative algebra A(infinity)(V) using infinite matrices with entries in a grading-restricted vertex algebra V such that the associated graded space Gr(W)=coproduct(n is an element of N)Gr(n)(W) of a filtration of a lower-bounded generalized V-module W is an A(infinity)(V)-module satisfying additional properties (called a nondegenerate graded A(infinity)(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(infinity)(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(infinity)(V)-modules. For N is an element of N, there is a subalgebra A(N)(V) of A(infinity)(V) such that the subspace Gr(N)(W)=coproduct(N)(n=0)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 Mobius 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 is an element of 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.