Hom-Algebras and Hom-Coalgebras

TL;DR

This paper develops the theory of Hom-coalgebras and related structures, including Hom-bialgebras and Hom-Hopf algebras, with classifications and fundamental properties explored.

Contribution

It introduces and analyzes Hom-coalgebras, Hom-bialgebras, and Hom-Hopf algebras, expanding the framework of Hom-algebra theory with new examples and classifications.

Findings

Defined Hom-coalgebra structures and properties

Constructed Hom-bialgebras and Hom-Hopf algebras with examples

Classified Hom-Lie admissible Hom-coalgebras

Abstract

The aim of this paper is to develop the theory of Hom-coalgebras and related structures. After reviewing some key constructions and examples of quasi-deformations of Lie algebras involving twisted derivations and giving rise to the class of quasi-Lie algebras incorporating Hom-Lie algebras, we describe the notion and some properties of Hom-algebras and provide examples. We introduce Hom-coalgebra structures, leading to the notions of Hom-bialgebra and Hom-Hopf algebras and prove some fundamental properties and give examples. Finally, we define the concept of Hom-Lie admissible Hom-coalgebra and provide their classification based on subgroups of the symmetric group.