Doi–Hopf Modules and Yetter–Drinfeld Modules for Quasi-Hopf Algebras

Abstract

For a quasi-Hopf algebra H, a left H-comodule algebra  and a right H-module coalgebra C we will characterize the category of Doi–Hopf modules ^C ℳ(H) in terms of modules. We will also show that for an H-bicomodule algebra  and an H-bimodule coalgebra C the category of generalized Yetter–Drinfeld modules (H) ^C is isomorphic to a certain category of Doi–Hopf modules. Using this isomorphism we will transport the properties from the category of Doi–Hopf modules to the category of generalized Yetter–Drinfeld modules.

Key Words:

• Doi–Hopf module
• Quasi-Hopf algebra
• Yetter–Drinfeld module

1991 Mathematics Subject Classification:

• 16W30

ACKNOWLEDGMENTS

Research supported by the bilateral project “Hopf Algebras in Algebra, Topology, Geometry, and Physics” of the Flemish and Romanian governments. D. Bulacu was financially supported by SB 2002-0286 and CERES 4-147/12.11.2004. He would like to thank the Free University of Brussels (Belgium) and the University of Almería (Spain) for their warm hospitality.

Notes

Communicated by A. Facchini.