New York Journal of Mathematics
Volume 6 (2000) 325-329


Peter Schauenburg

Doi-Koppinen Hopf Modules Versus Entwined Modules

Published: December 27, 2000
Keywords: Hopf algebra, Hopf module, entwining structure
Subject: 16W30

A Hopf module is an A-module for an algebra A as well as a C-comodule for a coalgebra C, satisfying a suitable compatibility condition between the module and comodule structures. To formulate the compatibility condition one needs some kind of interaction between A and C. The most classical case arises when A=C=:H is a bialgebra. Many subsequent variants of this were unified independently by Doi and Koppinen; in their version an auxiliary bialgebra H, over which A is a comodule algebra and C a module coalgebra, governs the compatibility. Another very general type of interaction between A and C is an entwining map as studied by Brzeziński -- without an auxiliary bialgebra.

Every Doi-Koppinen datum induces an entwining structure, so Brzeziński's notion of an entwined module includes that of a Doi-Koppinen Hopf module. This paper investigates whether the inclusion is proper.

By work of Tambara, every entwining structure can be obtained from a suitable Doi-Koppinen datum whenever the algebra under consideration is finite dimensional.

We show by examples that this need not be true in general.

Author information

Mathematisches Institut der Universität München, Theresienstr. 39, 80333 München, Germany