On $G$-crossed Frobenius $\star$-algebras and fusion rings associated with braided $G$-actions

TL;DR

This paper introduces and classifies $G$-crossed Frobenius $igstar$-algebras, linking them to braided $G$-actions on fusion categories and providing tools like a Verlinde formula for fusion coefficients.

Contribution

It defines $G$-crossed Frobenius $igstar$-algebras and classifies their extensions, connecting them to braided $G$-actions and fusion rings in a novel way.

Findings

Classification of $G$-crossed Frobenius $igstar$-algebras via group cohomology.

Identification of $G$-graded fusion rings as $G$-crossed Frobenius $igstar$-algebras.

Derivation of a Verlinde formula for fusion coefficients.

Abstract

For a finite group $G$, Turaev introduced the notion of a braided $G$-crossed fusion category. The classification of braided $G$-crossed extensions of braided fusion categories was studied by Etingof, Nikshych and Ostrik in terms of certain group cohomological data. In this paper we will define the notion of a $G$-crossed Frobenius $\star$-algebra and give a classification of (strict) $G$-crossed extensions of a commutative Frobenius $\star$-algebra $R$ equipped with a given action of $G$, in terms of the second group cohomology $H^2(G,R^\times)$. Now suppose that $\mathcal{B}$ is a non-degenerate braided fusion category equipped with a braided action of a finite group $G$. We will see that the associated $G$-graded fusion ring is in fact a (strict) $G$-crossed Frobenius $\star$-algebra. We will describe this $G$-crossed fusion ring in terms of the classification of braided $G$-actions by Etingof, Nikshych, Ostrik and derive a Verlinde formula to compute its fusion coefficients.