Pivotal and Ribbon Entwining Datums

TL;DR

This paper introduces pivotal and ribbon entwined datums to generalize pivotal and ribbon Hopf algebras, providing conditions for the category of entwined modules to be pivotal or ribbon categories.

Contribution

It defines new notions of pivotal and ribbon entwined datums, extending the theory of Hopf algebras and their module categories.

Findings

Characterizes when the category of entwined modules is pivotal.

Provides conditions for the category to be a ribbon category.

Generalizes existing concepts in Hopf algebra theory.

Abstract

Let $(C,A,\varphi)$ be an entwining structure over $k$. In this paper, we introduce the notions of the pivotal entwined datums and ribbon entwined datums to generalize (co)pivotal Hopf algebras and (co)ribbon Hopf algebras. These notions give necessary and sufficient conditions for the category of entwined modules to be a pivotal category and ribbon category.