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

It is well known that Yetter-Drinfeld modules are special cases of Doi-Hopf modules and Doi-Hopf modules are special cases of entwined modules. All these modules are obtained by considering the category of vector spaces. The aim of the present article is to generalize the above results to a monoidal category and to construct nontrivial examples of entwining structures. More precisely, the authors introduce the notions of entwining structures in a monoidal category (Definition 3.1), lax Doi-Koppinen structures in a braided monoidal category (Definition 3.3), lax Hopf algebras in a braided monoidal category (Definition 3.5) and lax Yetter-Drinfeld structures in a symmetric monoidal category (Definition 3.6). If C is a braided monoidal category, the authors show that particular examples of entwining structures in C can be obtained from the lax Doi-Koppinen structures in C (Proposition 3.4). If C is a symmetric monoidal category, they show that particular examples of lax Doi-Koppinen structures in C can be obtained from lax Yetter-Drinfeld structures in C over a lax Hopf algebra in C (Proposition 3.9). The main theorem of the paper is Theorem 4.6.

Publication year:2014

Keywords:Doi-Hopf module, Yetter-Drinfeld module, monoidal category