Crossed product Hom-Hopf algebras and lazy 2-cocycle

TL;DR

This paper constructs Hom-crossed product Hom-Hopf algebras, establishes their properties as cleft extensions, and explores conditions for them to be Hom-Hopf algebras, along with studying lazy 2-cocycles.

Contribution

It introduces the construction of Hom-crossed products, characterizes their extension properties, and analyzes lazy 2-cocycles within Hom-Hopf algebras.

Findings

Hom-crossed product construction for Hom-Hopf algebras.

Characterization of Hom-crossed products as cleft extensions.

Conditions for Hom-crossed products to be Hom-Hopf algebras.

Abstract

Let $(H,\a)$ be a Hom-Hopf algebra and $(A,\b)$ be a Hom-algebra. In this paper we will construct the Hom-crossed product $(A\#_\sigma H,\b\o\a)$, and prove that the extension $A\subseteq A\#_\sigma H$ is actually a Hom-type cleft extension and vice versa. Then we will give the necessary and sufficient conditions to make $(A\#_\sigma H,\b\o\a)$ a Hom-Hopf algebra. Finally we will study the lazy 2-cocycle on $(H,\a)$.