Hom-Algebras and Hom-Coalgebras

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.