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Let $V$ be a vertex operator algebra with a category $\mathcal{C}$ of(generalized) modules that has vertex tensor category structure, and thusbraided tensor category structure, and let $A$ be a vertex operator(super)algebra extension of $V$. We employ tensor categories to study untwisted(also called local) $A$-modules in $\mathcal{C}$, using results ofHuang-Kirillov-Lepowsky showing that $A$ is a (super)algebra object in$\mathcal{C}$ and that generalized $A$-modules in $\mathcal{C}$ correspondexactly to local modules for the corresponding (super)algebra object. Bothcategories, of local modules for a $\mathcal{C}$-algebra and (under suitableconditions) of generalized $A$-modules, have natural braided monoidal categorystructure, given in the first case by Pareigis and Kirillov-Ostrik and in thesecond case by Huang-Lepowsky-Zhang. Our main result is that theHuang-Kirillov-Lepowsky isomorphism of categories between local (super)algebramodules and extended vertex operator (super)algebra modules is also anisomorphism of braided monoidal (super)categories. Using this result, we showthat induction from a suitable subcategory of $V$-modules to $A$-modules is avertex tensor functor. We give two applications. First, we derive Verlindeformulae for regular vertex operator superalgebras and regular$(1/2)\mathbb{Z}$-graded vertex operator algebras by realizing them as(super)algebra objects in the vertex tensor categories of their even and$\mathbb{Z}$-graded components, respectively. Second, we analyze parafermioniccosets $C=\mathrm{Com}(V_L,V)$ where $L$ is a positive definite even latticeand $V$ is regular. If the category of either $V$-modules or $C$-modules isunderstood, then our results classify all inequivalent simple modules for theother algebra and determine their fusion rules and modular charactertransformations. We illustrate both directions with several examples.

Tensor categories for vertex operator superalgebra extensions