On the Hom-Lie CoDer pairs

The present research paper investigates the intricate fields of Hom-Lie algebra and Hom-Lie coalgebra, providing a complete analysis of their key concepts and important examples. Precisely, the paper introduces the concept of Hom-Lie coderivation pairs and demystifies its duality with Hom-Lie derivation pairs, inspecting pertinent facts such as representation and semi-direct product. Furthermore, the study examines the connection between Hom-Lie, pre-Lie, and Ass-Coder pairs with the use of crucial operators such as commutator, Rota-Baxter operator, and endomorphism operator. Finally, the paper concludes by presenting the construction of Hom-pre-Lie coderivation pairs through a dual to an endomorphism operator.