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

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.