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

Doi-Koppinen Hopf Modules Versus Entwined Modules

Peter Schauenburg

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 Brzezi\'nski --- without an auxiliary bialgebra.

Every Doi-Koppinen datum induces an entwining structure, so Brzezi\'nski'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