Weak Hopf Algebras, Smash Products and Applications to Adjoint-Stable Algebras

TL;DR

This paper explores the structure of weak Hopf algebras derived from semisimple quasi-triangular Hopf algebras and their module algebras, revealing new embeddings and equivalences that deepen understanding of their representation categories.

Contribution

It demonstrates that certain smash products form weak Hopf algebras and embeds them into endomorphism-based weak Hopf algebras, linking module categories and clarifying subgroup structures.

Findings

A smash product with a strongly separable quantum commutative algebra forms a weak Hopf algebra.

The module category of the smash product is monoidally equivalent to that of the original Hopf algebra.

Embedding into a weak Hopf algebra can be quasi-triangular under specific conditions.

Abstract

For a semisimple quasi-triangular Hopf algebra $\left( H,R\right) $ over a field $k$ of characteristic zero, and a strongly separable quantum commutative $H$-module algebra $A$ over which the Drinfeld element of $H$ acts trivially, we show that $A\#H$ is a weak Hopf algebra, and it can be embedded into a weak Hopf algebra $\operatorname{End}A^{\ast}\otimes H$. With these structure, $_{A\#H}\operatorname{Mod}$ is the monoidal category introduced by Cohen and Westreich, and $_{\operatorname{End}A^{\ast}\otimes H}\mathcal{M}$ is tensor equivalent to $_{H}\mathcal{M}$. If $A$ is in the M{\"{u}}ger center of $_{H}{\mathcal{M}}$, then the embedding is a quasi-triangular weak Hopf algebra morphism. This explains the presence of a subgroup inclusion in the characterization of irreducible Yetter-Drinfeld modules for a finite group algebra.