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Splitting of operations for Hom-diassociative and Hom-triassociative Algebras

Abdelkader Hamdouni, Imed Basdouri, Mariem Jendoubi, Ahmed Zahari Abdou Damdji

TL;DR

The paper develops Hom-type analogues of classical splitting algebras by defining Hom-quadri-dendriform algebras as splits of Hom-diassociative algebras and Hom-six-dendriform algebras as splits of Hom-triassociative algebras, all twisted by a map $\alpha$. It builds a unified framework using relative averaging operators, graph constructions, and semi-direct products to connect these new algebras with existing Hom-dendriform and Hom-diassociative structures, and proves embedding and representation results. A key contribution is the explicit low-dimensional classification of Hom-quadri-dendriform algebras (2- and 3-dimensional cases) along with associated averaging-operator data, providing concrete models and parametrized families. These results extend operadic dualities and Koszul-type splittings to the Hom-setting, offering new tools for deformation theory, representation theory, and potential applications in algebraic combinatorics and mathematical physics.

Abstract

Hom-quadri dendriform algebras and Hom-six-dendriform agebras are introduced and studied which is a splitting of a Hom-diassociative and Hom-triassociative algebras, respectively. Moreover we explore the connections be tween these categories of Hom-algebras. Finally We elaborate a classification of Hom-quadri-dendriform algebra in low dimensional.

Splitting of operations for Hom-diassociative and Hom-triassociative Algebras

TL;DR

The paper develops Hom-type analogues of classical splitting algebras by defining Hom-quadri-dendriform algebras as splits of Hom-diassociative algebras and Hom-six-dendriform algebras as splits of Hom-triassociative algebras, all twisted by a map . It builds a unified framework using relative averaging operators, graph constructions, and semi-direct products to connect these new algebras with existing Hom-dendriform and Hom-diassociative structures, and proves embedding and representation results. A key contribution is the explicit low-dimensional classification of Hom-quadri-dendriform algebras (2- and 3-dimensional cases) along with associated averaging-operator data, providing concrete models and parametrized families. These results extend operadic dualities and Koszul-type splittings to the Hom-setting, offering new tools for deformation theory, representation theory, and potential applications in algebraic combinatorics and mathematical physics.

Abstract

Hom-quadri dendriform algebras and Hom-six-dendriform agebras are introduced and studied which is a splitting of a Hom-diassociative and Hom-triassociative algebras, respectively. Moreover we explore the connections be tween these categories of Hom-algebras. Finally We elaborate a classification of Hom-quadri-dendriform algebra in low dimensional.
Paper Structure (7 sections, 17 theorems, 49 equations)

This paper contains 7 sections, 17 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Hom-dendriform algebra and Hom-diassociative algebra
  3. Hom-dendriform algebras
  4. Hom-diassociative algebras
  5. Hom-quadri-dendriform algebra
  6. Hom-Six-dendriform algebras
  7. Classification

Key Result

Proposition 2.8

Let $(D ,\ \dashv,\ \vdash ,\ \alpha)$ be a Hom-diassociative algebra and $R :\ D\longrightarrow D$ be a Rota-Baxter operator of weight $0$ on D i.e R is linear and commutes with both $\alpha$ and Then $(D ,\ \unlhd,\ \unrhd,\ \alpha)$ is also Hom-diassociative algebra with: $x\unlhd y=R(x) \dashv y + x \dashv R(y)$ and $x\unrhd y =R(x) \vdash y + x \vdash R(y) ,\ \forall x,\ y\in D.$ M

Theorems & Definitions (46)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • proof
  • Example 2.9
  • ...and 36 more