Cocycle deformations for Hom-Hopf algebras

Abstract

In this paper we introduce the Hom-analogue of the definition of 2-cocycle for Hopf algebras, called Hom-2-cocycle, and study its properties in order to give a theory of multiplication alteration by Hom-2-cocycles for Hom-Hopf algebras. We show that, just like in the classical setting, if H is a Hom-Hopf algebra with associated endomorphism α and σ is a convolution invertible Hom-2-cocycle, it is possible to define a new product in H to get a new Hom-Hopf algebra if α4=α. Moreover we introduce the notions of matched pair and skew pairing in the Hom case and, by the close connection between Hom-2-cocycles and Hom-skew pairings, we show that a special case of Hom-matched pair can be obtained as a deformation of a Hom-Hopf algebra by a Hom-2-cocycle built by a Hom-skew pairing.