The character algebra for module categories over Hopf algebras
Volume 165 / 2021
Colloquium Mathematicum 165 (2021), 171-197
MSC: 18D20, 16T05.
DOI: 10.4064/cm8170-5-2020
Published online: 17 December 2020
Abstract
Given a finite-dimensional Hopf algebra $H$ and an exact indecomposable module category $\mathcal M $ over $\operatorname{Rep} (H)$, we explicitly compute the adjoint algebra $\mathcal A _\mathcal M $ as an object in the category of Yetter–Drinfeld modules over $H$, and the space of class functions ${\rm CF} (\mathcal M )$ associated to $\mathcal M $, as introduced by K. Shimizu (2020). 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.
