Algebras Graded by Discrete Doi–Hopf Data and the Drinfeld Double of a Hopf Group-Coalgebra

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 \({\mathbb{G}}\)-graded Hopf algebra.