The Canonical Isomorphisms in the Yetter-Drinfeld Categories for Dual Quasi-Hopf Algebras

Hopf algebras, as a crucial generalization of groups, have a very symmetric structure and have been playing a prominent role in mathematical physics. In this paper, let H be a dual quasi-Hopf algebra which is a more general Hopf algebra structure. A. Balan firstly introduced the notion of right-right Yetter-Drinfeld modules over H and studied its Galois extension. As a continuation, the aim of this paper is to introduce more properties of Yetter-Drinfeld modules. First, we will describe all the other three kinds of Yetter-Drinfeld modules over H, and the monoidal and braided structure of the categories of Yetter-Drinfeld modules explicitly. Furthermore, we will prove that the category HHYDfd of finite dimensional left-left Yetter-Drinfeld modules is rigid. Then we will compute explicitly the canonical isomorphisms in HHYDfd. Finally, as an application, we will rewrite the isomorphisms in the case of coquasitriangular dual quasi-Hopf algebra.