Ribbon entwining datum

Abstract

Let (C,A,?) be an entwining structure over a field k. In this paper, we introduce the notion of the ribbon entwined datum to generalize the definition of (co)ribbon structures, and give several necessary and sufficient conditions for the category of entwined modules to be a ribbon category. We also discuss the ribbon structures in the Long dimodule category and Yetter-Drinfel?d category for applications.