Certain associative algebras similar to $U(sl_{2})$ and Zhu's algebra $A(V_{L})$

TL;DR

This paper computes Zhu's algebra for vertex operator algebras linked to positive-definite even lattices, revealing their structure as quotients of Smith's algebra and generalizations thereof.

Contribution

It introduces a new connection between Zhu's algebra for lattice vertex operator algebras and a generalized Smith's algebra, expanding understanding of their algebraic structure.

Findings

Zhu's algebra for rank one lattice VOAs is a finite-dimensional semiprimitive quotient of Smith's algebra.

Zhu's algebra for higher rank lattices is related to a generalized Smith's algebra.

The paper provides explicit calculations of Zhu's algebra for these VOAs.

Abstract

It is proved that Zhu's algebra for vertex operator algebra associated to a positive-definite even lattice of rank one is a finite-dimensional semiprimitive quotient algebra of certain associative algebra introduced by Smith. Zhu's algebra for vertex operator algebra associated to any positive-definite even lattice is also calculated and is related to a generalization of Smith's algebra.