Yetter-Drinfeld Categories for Quasi-Hopf Algebras

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 of finite dimensional left Yetter-Drinfeld modules is rigid, and then we compute explicitly the canonical isomorphisms in . Finally, we show that certain duals of H ₀, the braided Hopf algebra (introduced in Bulacu and Nauwelaerts, 2002; Bulacu et al., 2000) are isomorphic as braided Hopf algebras if H is a finite dimensional triangular quasi-Hopf algebra.

Communicated by M. Takeuchi.

Key Words:

• Braided monoidal category
• Quasi-Hopf algebra
• Yetter-Drinfeld module

1991 Mathematics Subject Classification:

• 16W30

ACKNOWLEDGMENTS

The authors wish to thank the referee for his suggestion to use the centre construction to simplify the proofs of the results in Sections 2 and 3.

Research supported by the bilateral project “Hopf Algebras in Algebra, Topology, Geometry, and Physics” of the Flemish and Romanian governments. This article was finished while the first author was visiting the Vrije Universiteit Brussel, and he would like to thank VUB for its warm hospitality. The third author was also partially supported by the programmes SCOPES and EURROMMAT.