Algebras graded by discrete Doi-Hopf data and the Drinfeld double of a Hopf group-coalgebra

TL;DR

This paper explores the structure of Doi-Hopf modules over Hopf group-coalgebras, introduces a grading framework, and constructs the Drinfeld double as a quasitriangular graded Hopf algebra, enriching the understanding of algebraic symmetries.

Contribution

It introduces modules graded by discrete Doi-Hopf data and constructs the Drinfeld double for Hopf group-coalgebras, connecting category equivalences and algebraic structures.

Findings

Category of Doi-Hopf modules is isomorphic to graded modules over a constructed algebra

Constructed the Drinfeld double as a quasitriangular graded Hopf algebra

Established a grading framework for modules over Hopf group-coalgebras

Abstract

We study Doi-Hopf data and Doi-Hopf modules for Hopf group-coalgebras. We introduce modules graded by a discrete Doi-Hopf datum; to a Doi-Hopf datum over a Hopf group coalgebra, we associate an algebra graded by the underlying discrete Doi-Hopf datum, using a smash product type construction. The category of Doi-Hopf modules is then isomorphic to the category of graded modules over this algebra. This is applied to the category of Yetter-Drinfeld modules over a Hopf group coalgebra, leading to the construction of the Drinfeld double. It is shown that this Drinfeld double is a quasitriangular graded Hopf algebra.