The character algebra for module categories over Hopf algebras

TL;DR

This paper explicitly computes the character algebra for module categories over finite-dimensional Hopf algebras, providing concrete descriptions and applications to group-theoretical fusion categories.

Contribution

It introduces explicit calculations of the adjoint algebra and class functions for module categories over Hopf algebras, extending previous theoretical frameworks.

Findings

Explicit formulas for the adjoint algebra A_M in Yetter-Drinfeld modules.

Description of class functions CF(M) for module categories.

Application to group algebra and dual group algebra cases.

Abstract

Given a finite dimensional Hopf algebra H and an exact indecomposable module category M over Rep(H), we explicitly compute the adjoint algebra A_M as an object in the category of Yetter-Drinfeld modules over H, and the space of class functions CF(M) associated to M, as introduced by K. Shimizu [ Further results on the structure of (Co)ends in fintite tensor categories, preprint arXiv:1801.02493]. We use our construction to describe these algebras when H is a group algebra and a dual group algebra. This result allows us to compute the adjoint algebra for certain group-theoretical fusion categories.