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

Creutzig T, Mcrae R, Shimizu K, Yadav H (2025)

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Publication Type: Journal article

Publication year: 2025

Journal

Article Number: 2550091

DOI: 10.1142/S0219199725500919

Abstract

Let A be a commutative algebra in a braided monoidal category C. For example, A could be a vertex operator algebra (VOA) extension of a VOA V in a category C of V-modules. We first find conditions for the category CA of A-modules in C and its subcategory (Formula presented) of local modules to inherit rigidity from C. Second and more importantly, we prove a converse result, finding conditions under which C and CA inherit rigidity from CAloc. For our first results, we assume that C is a braided finite tensor category and identify mild conditions under which CA and (Formula presented) are also rigid. These conditions are based on criteria due to Etingof and Ostrik for A to be an exact algebra in C. As an application, we show that if A is a simple Z≥0-graded VOA containing a strongly rational vertex operator subalgebra V, then A is also strongly rational, without requiring the dimension of A in the modular tensor category of V-modules to be non-zero. We also identify conditions under which the category of A-modules inherits rigidity from the module category of a C2-cofinite non-rational subalgebra V. For our converse result, we assume that C is a Grothendieck–Verdier category, which means that C admits a weaker duality structure than rigidity. We first show that CA is also a Grothendieck–Verdier category. Using this, we then prove that if (Formula presented) is rigid, then so is C under conditions that include a mild non-degeneracy assumption on C, as well as assumptions that every simple object of CA is local and that induction FA : C→CA commutes with duality. These conditions are motivated by free field-like VOA extensions V ↪A where A is often an indecomposable V-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 sl2 at any admissible level, which embeds by Adamović’s inverse quantum Hamiltonian reduction into a rational Virasoro VOA tensored with a half-lattice VOA.

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How to cite

APA:

Creutzig, T., Mcrae, R., Shimizu, K., & Yadav, H. (2025). Commutative algebras in Grothendieck–Verdier categories, rigidity, and vertex operator algebras. Communications in Contemporary Mathematics. https://doi.org/10.1142/S0219199725500919

MLA:

Creutzig, Thomas, et al. "Commutative algebras in Grothendieck–Verdier categories, rigidity, and vertex operator algebras." Communications in Contemporary Mathematics (2025).

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