More properties of Yetter-Drinfeld category over dual quasi-Hopf algebras

TL;DR

This paper explores the structure of Yetter-Drinfeld modules over dual quasi-Hopf algebras, establishing their monoidal and braided properties, proving rigidity for finite-dimensional modules, and examining the braided cocommutativity of a specific object.

Contribution

It introduces all categories of Yetter-Drinfeld modules over dual quasi-Hopf algebras, explicitly describes their monoidal and braided structures, and proves the rigidity of finite-dimensional modules.

Findings

Categories of Yetter-Drinfeld modules are explicitly constructed.

The monoidal and braided structures of these categories are detailed.

The category of finite-dimensional modules is shown to be rigid.

Abstract

Let $H$ be a dual quasi-Hopf algebra. In this paper we will firstly introduce all possible categories of Yetter-Drinfeld modules over $H$, and give explicitly the monoidal and braided structure of them. Then we prove that the category $^H_H\mathcal{YD}^{fd}$ of finite-dimensional left-left Yetter-Drinfeld modules is rigid. Finally we will study the braided cocommunitivity of $H_0$ in $^H_H\mathcal{YD}$.