On the Hom-Lie CoDer pairs

TL;DR

This paper explores Hom-Lie algebra and coalgebra structures, introducing Hom-Lie coderivation pairs, analyzing their duality with derivation pairs, and connecting them with other algebraic pairs using key operators.

Contribution

It introduces the concept of Hom-Lie coderivation pairs and investigates their duality with derivation pairs, expanding the understanding of Hom-Lie algebraic structures.

Findings

Defined Hom-Lie coderivation pairs and their properties

Established duality between Hom-Lie coderivation and derivation pairs

Constructed Hom-pre-Lie coderivation pairs via endomorphism operators

Abstract

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.