Radford $[(m,k),m]$-biproduct Theorem for Generalized Hom-crossed Products

TL;DR

This paper introduces a new approach to constructing Hom-Hopf algebras by generalizing Hom-crossed product structures and establishing a Radford biproduct theorem, expanding the theoretical framework of Hom-Hopf algebra construction.

Contribution

It develops a generalized $(m,k)$-Hom-crossed product framework and proves a Radford biproduct theorem for Hom-Hopf algebras, broadening the methods for their construction.

Findings

Defined the $(m,k)$-Hom-crossed product structure

Established the Radford $[(m,k),m]$-biproduct theorem

Characterized the structure using Hom admissible mapping system

Abstract

In this paper, we mainly provide a new approache to construct Hom-Hopf algebras. For this, we introduce and study the notion of a left $(m,k)$-Hom-crossed product structure as a generalization of $k$-Hom-smash product structure. Then one combines this $(m,k)$-Hom-crossed product structure and a left $m$-Hom-smash coproduct structure to build Radford $[(m,k),m]$-biproduct theorem. Finally, we study Hom admissible mappping system to characterize this Radford $[(m,k),m]$-biproduct structure.