  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.
