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Let $Vsubseteq A$ 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.
Let $V$ be an $mathbb{N}$-graded, simple, self-contragredient, $C_2$-cofinite vertex operator algebra. We show that if the $S$-transformation of the character of $V$ is a linear combination of characters of $V$-modules, then the category $mathcal{C}$
We show that direct limit completions of vertex tensor categories inherit vertex and braided tensor category structures, under conditions that hold for example for all known Virasoro and affine Lie algebra tensor categories. A consequence is that the
We construct two non-semisimple braided ribbon tensor categories of modules for each singlet vertex operator algebra $mathcal{M}(p)$, $pgeq 2$. The first category consists of all finite-length $mathcal{M}(p)$-modules with atypical composition factors
In the first part of this paper, we give a new look at inclusions of von Neumann algebras with finite-dimensional centers and finite Jones index. The minimal conditional expectation is characterized by means of a canonical state on the relative commu
The first author constructed a $q$-parameterized spherical category $sC$ over $mathbb{C}(q)$ in [Liu15], whose simple objects are labelled by all Young diagrams. In this paper, we compute closed-form expressions for the fusion rule of $sC$, using Lit