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

DOI10.1007/S11005-022-01523-4zbMath1510.17040arXiv2103.07657MaRDI QIDQ6362769

Robert McRae

Publication date: 13 March 2021

Abstract: Let $VsubseteqA$ be a conformal inclusion of vertex operator algebras and let $mathcalC$ 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 $mathcalC$ inherits semisimplicity from the category of grading-restricted generalized $A$-modules in $mathcalC$, and vice versa. The most important condition is that $A$ be a rigid $V$-module in $mathcalC$ 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.

Mathematics Subject Classification ID

Vertex operators; vertex operator algebras and related structures (17B69) Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, (W)-algebras and other current algebras and their representations (81R10) Fusion categories, modular tensor categories, modular functors (18M20) Braided monoidal categories and ribbon categories (18M15)

This page was built for publication: On semisimplicity of module categories for finite non-zero index vertex operator subalgebras