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Structures and comodules of Hom-post Lie coalgebras

Damien Houndedji, Ibrahima Bakayoko

TL;DR

This work defines and analyzes dual structures to Post-Hom-Lie algebras, namely $Post-Hom$-Lie coalgebras and $Hom$-tridendriform coalgebras, establishing core axioms, constructions, and interrelations. It shows how $Hom$-tridendriform coalgebras unify and extend coassociative and dendriform coalgebras, and how their combinations with cobrackets yield post-$Hom$-Lie coalgebras, including links to cocommutative Hom-Poisson coalgebras. The authors develop comodules over these coalgebras, provide twisting techniques (Yau twisting) to generate new comodule and coalgebra structures, and connect the framework to Rota-Baxter theory and Post-Hom-Poisson coalgebras. The results enhance the dual landscape of Hom-type algebraic structures and furnish systematic methods to construct and relate complex coalgebraic objects with potential applications in Hom-type geometry and deformation theory.

Abstract

In this paper, we introduce the notions of Hom-tridendriform coalgebras and Hom-post-Lie coalgebras as the dual notions of Hom-tridendriform algebras and Hom-post-LIe algebras respectively. We give some properties related to them. Then, we study the relationships between them and their connection with post-Hom-Poisson coalgebras. Next, using the Yau stwisting in the modules case, we give some constructions of comodules over post-Hom-Lie coalgebras by twisting either the comodule structures or post-Hom-Lie coalgebra structures.

Structures and comodules of Hom-post Lie coalgebras

TL;DR

This work defines and analyzes dual structures to Post-Hom-Lie algebras, namely -Lie coalgebras and -tridendriform coalgebras, establishing core axioms, constructions, and interrelations. It shows how -tridendriform coalgebras unify and extend coassociative and dendriform coalgebras, and how their combinations with cobrackets yield post--Lie coalgebras, including links to cocommutative Hom-Poisson coalgebras. The authors develop comodules over these coalgebras, provide twisting techniques (Yau twisting) to generate new comodule and coalgebra structures, and connect the framework to Rota-Baxter theory and Post-Hom-Poisson coalgebras. The results enhance the dual landscape of Hom-type algebraic structures and furnish systematic methods to construct and relate complex coalgebraic objects with potential applications in Hom-type geometry and deformation theory.

Abstract

In this paper, we introduce the notions of Hom-tridendriform coalgebras and Hom-post-Lie coalgebras as the dual notions of Hom-tridendriform algebras and Hom-post-LIe algebras respectively. We give some properties related to them. Then, we study the relationships between them and their connection with post-Hom-Poisson coalgebras. Next, using the Yau stwisting in the modules case, we give some constructions of comodules over post-Hom-Lie coalgebras by twisting either the comodule structures or post-Hom-Lie coalgebra structures.
Paper Structure (8 sections, 32 theorems, 105 equations)

This paper contains 8 sections, 32 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hom-tridendriform coalgebras and comodules
  4. Hom-tridendriform coalgebras
  5. Hom-tridendriform comodules
  6. Post-Hom-Lie coalgebra
  7. Connexion with Hom-tridendriform coalgebras
  8. Post-Hom-Lie comodules

Key Result

Lemma 2.2

Let $(A,\Delta)$ be a coassociative coalgebra and $\alpha: A\longrightarrow A$ be a coalgebra endomorphism. Then, the triple $(A,\Delta_\alpha,\alpha)$, where $\Delta_\alpha=\Delta\circ\alpha$, is a comultiplicative Hom-coassociative coalgebra.

Theorems & Definitions (83)

  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Lemma 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • Proposition 2.10
  • ...and 73 more