On semisimplicity of module categories for finite non-zero index vertex operator subalgebras

TL;DR

This paper establishes conditions under which the semisimplicity of module categories for vertex operator algebras is preserved under conformal inclusions, especially when the larger algebra is strongly rational.

Contribution

It provides new criteria for semisimplicity inheritance in module categories of vertex operator algebras and extends results to superalgebras, linking algebraic properties with categorical dimensions.

Findings

Semisimplicity is inherited under certain conditions involving categorical dimension.

Strong rationality of A implies strong rationality of V given specific module conditions.

Generalization of results to vertex operator superalgebras.

Abstract

Let $V\subseteq A$ be a conformal inclusion of vertex operator algebras and let $\mathcal{C}$ be a category of grading-restricted generalized $V$-modules that admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. We give conditions under which $\mathcal{C}$ inherits semisimplicity from the category of grading-restricted generalized $A$-modules in $\mathcal{C}$, and vice versa. The most important condition is that $A$ be a rigid $V$-module in $\mathcal{C}$ with non-zero categorical dimension, that is, we assume the index of $V$ as a subalgebra of $A$ is finite and non-zero. As a consequence, we show that if $A$ is strongly rational, then $V$ is also strongly rational under the following conditions: $A$ contains $V$ as a $V$-module direct summand, $V$ is $C_2$-cofinite with a rigid tensor category of modules, and $A$ has non-zero categorical dimension as a $V$-module. These results are vertex operator algebra interpretations of theorems proved for general commutative algebras in braided tensor categories. We also generalize these results to the case that $A$ is a vertex operator superalgebra.