On Doi-Hopf modules and Yetter-Drinfeld modules in symmetric monoidal categories

Abstract

We study entwining structures on a monoidal category $\Cc$ and their corresponding categories of entwined modules. Examples can be constructed from lax Doi-Koppinen and lax Yetter-Drinfeld structures in $\cal C$. If $\cal C$ is symmetric then lax Yetter-Drinfeld structures appear as special cases of lax Doi-Koppinen structures, at least if we work over a so-called lax Hopf algebra. In this case the corresponding categories of entwined modules are isomorphic, and this generalizes a well-known result of Caenepeel, Militaru and Zhu. In particular, our theory applies to Doi-Koppinen and Yetter-Drinfeld structures in symmetric monoidal categories. We present some examples of entwining structures in monoidal categories coming from actions and coactions of a weak Hopf algebra.