Vertex operator algebras associated to certain admissible modules for affine Lie algebras of type A

TL;DR

This paper classifies irreducible modules and proves semisimplicity for a specific vertex operator algebra linked to affine Lie algebras of type A at a certain admissible level, enhancing understanding of their module categories.

Contribution

It provides a classification of irreducible modules and establishes semisimplicity for the category of weak modules in a particular vertex operator algebra setting.

Findings

Classification of irreducible modules achieved

Semisimplicity of the module category proven

Enhanced understanding of vertex operator algebra representations

Abstract

Let $L(-{1/2}(l+1),0)$ be the simple vertex operator algebra associated to an affine Lie algebra of type $A_{l}^{(1)}$ with the lowest admissible half-integer level $-{1/2}(l+1)$, for even l. We study the category of weak modules for that vertex operator algebra which are in category $\cal{O}$ as modules for the associated affine Lie algebra. We classify irreducible objects in that category and prove semisimplicity of that category.