Partially Dualized Quasi-Hopf Algebras Reconstructed from Dual Tensor Categories to Finite-Dimensional Hopf Algebras

TL;DR

This paper reconstructs a quasi-Hopf algebra from the dual tensor category of a finite-dimensional Hopf algebra's representation category, revealing new categorical and algebraic structures with applications to classical group theory results.

Contribution

It introduces a method to reconstruct and analyze quasi-Hopf algebras from dual tensor categories related to Hopf algebras, extending categorical Morita equivalence.

Findings

Reconstruction of quasi-Hopf algebra from dual tensor category

Categorical Morita equivalence to original Hopf algebra

Applications to bismash products and bosonizations

Abstract

Let $\mathsf{Rep}(H)$ be the category of finite-dimensional representations of a finite-dimensional Hopf algebra $H$. Andruskiewitsch and Mombelli proved in 2007 that each indecomposable exact $\mathsf{Rep}(H)$-module category has form $\mathsf{Rep}(B)$ for some indecomposable exact left $H$-comodule algebra $B$. This paper reconstructs and determines a quasi-Hopf algebra structure from the dual tensor category of $\mathsf{Rep}(H)$ with respect to $\mathsf{Rep}(B)$, when $B$ is a left coideal subalgebra of $H$. Consequently, it is categorically Morita equivalent to $H$, and some other elementary properties are also studied. As applications, our construction could be applied to imply some classical results on bismash products of matched pair of groups and bosonizations of dually paired Hopf algebras.