Yetter-Drinfeld Categories for Quasi-Hopf Algebras

Abstract

Abstract. We show that all possible categories of Yetter-Drinfeld modules over a quasi-Hopf algebra H are isomorphic. We prove also that the category H H YD fd of finite dimensional left Yetter-Drinfeld modules is rigid and then we compute explicitly the canonical isomorphisms in H H YD fd . Finally, we show that certain duals of H 0 , the braided Hopf algebra introduced in [6,7], are isomorphic as braided Hopf algebras if H is a finite dimensional triangular quasi-Hopf algebra.

Two-sided Two-cosided Hopf Modules and Yetter–Drinfeld Modules for Quasi-Hopf Algebras

“…, for all m ∈ M. In conclusion, F and G provide just the pair of isomorphisms obtained in [7], and this finishes the proof.…”

“…Finally, we will see that the structure theorem for (H, A)-Hopf modules extends to a category equivalence between two-sided two-cosided (H, A, C)-Hopf modules and right-left (H, A, C)-Yetter-Drinfeld modules. As an application, in Section 5 we will show how the equivalence between C H M H A and C Y D(H) A can be used in order to obtain the isomorphism described in [7] between the category of left-right Yetter-Drinfeld modules H Y D H and the category of left-left Yetter-Drinfeld modules H H Y D. This fact generalizes [17,Corollary 6.5]. Note that the proof in [7] uses the left and right center constructions, lifting the isomorphism to the braided level and making it more transparent.…”

“…Note that all these categories can be defined in a more general context. The aim of this paper is to go on with the study of the categories of twosided two-cosided Hopf modules and Yetter-Drinfeld modules over a quasi-Hopf algebra, study initiated in [2] and continued in [6][7][8]. Given a quasi-Hopf algebra H, an H-bimodule coalgebra C and an H-bicomodule algebra A, we can consider the category of (generalized) two-sided two-cosided Hopf modules C H M H A and the category of (generalized) right-left Yetter-Drinfeld modules C Y D(H) A , as well.…”

“…Moreover, since the (left and right) center construction Z l/r (C) can be applied to any monoidal category C we can also introduce the four YetterDrinfeld categories over H as being the left or right center of the monoidal category H M or M H , respectively, (see [7,14]). Note that all these categories can be defined in a more general context.…”

Involutory Quasi-Hopf Algebras

“…4.2 is satisfied.For H a quasi-Hopf algebra and C = H M fd the isomorphisms λ V,W were computed in[1, Proposition 4.2], namely…”