I will introduce an associative algebra A^{\infty}(V) constructed using infinite matrices with entries in a grading-restricted vertex algebra V. The Zhu algebra and its generalizations by Dong-Li-Mason are very special subalgebras of A^{\infty}(V). I will also introduce the new subalgebras A^{N}(V) of A^{\infty}(V), which can be viewed as obtained from finite matrices with entries in V. I will then discuss the relations between lower-bounded generalized V-modules and suitable modules for these associative algebras. This talk is based on the paper arXiv:2009.00262.
Associative algebra and the representation theory of grading-restricted vertex algebras.