Commutative algebras in Grothendieck–Verdier categories, rigidity, and vertex operator algebras

Let [Formula: see text] be a commutative algebra in a braided monoidal category [Formula: see text]. For example, [Formula: see text] could be a vertex operator algebra (VOA) extension of a VOA [Formula: see text] in a category [Formula: see text] of [Formula: see text]-modules. We first find conditions for the category [Formula: see text] of [Formula: see text]-modules in [Formula: see text] and its subcategory [Formula: see text] of local modules to inherit rigidity from [Formula: see text]. Second and more importantly, we prove a converse result, finding conditions under which [Formula: see text] and [Formula: see text] inherit rigidity from [Formula: see text]. For our first results, we assume that [Formula: see text] is a braided finite tensor category and identify mild conditions under which [Formula: see text] and [Formula: see text] are also rigid. These conditions are based on criteria due to Etingof and Ostrik for [Formula: see text] to be an exact algebra in [Formula: see text]. As an application, we show that if [Formula: see text] is a simple [Formula: see text]-graded VOA containing a strongly rational vertex operator subalgebra [Formula: see text], then [Formula: see text] is also strongly rational, without requiring the dimension of [Formula: see text] in the modular tensor category of [Formula: see text]-modules to be non-zero. We also identify conditions under which the category of [Formula: see text]-modules inherits rigidity from the module category of a [Formula: see text]-cofinite non-rational subalgebra [Formula: see text]. For our converse result, we assume that [Formula: see text] is a Grothendieck–Verdier category, which means that [Formula: see text] admits a weaker duality structure than rigidity. We first show that [Formula: see text] is also a Grothendieck–Verdier category. Using this, we then prove that if [Formula: see text] is rigid, then so is [Formula: see text] under conditions that include a mild non-degeneracy assumption on [Formula: see text], as well as assumptions that every simple object of [Formula: see text] is local and that induction [Formula: see text] commutes with duality. These conditions are motivated by free field-like VOA extensions [Formula: see text] where [Formula: see text] is often an indecomposable [Formula: see text]-module, and thus our result will make it more feasible to prove rigidity for many vertex algebraic braided monoidal categories. In a follow-up work, our results are used to prove rigidity of the category of weight modules for the simple affine VOA of [Formula: see text] at any admissible level, which embeds by Adamović’s inverse quantum Hamiltonian reduction into a rational Virasoro VOA tensored with a half-lattice VOA.