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On compatible Lie and pre-Lie Yamaguti algebras

Asif Sania, Basdouri Imed, Sadraoui Mohamed Amin

Abstract

This study aims to generalize the notion of compatible Lie algebras to the compatible Lie Yamaguti algebras. Along with describing the representation of the compatible Lie Yamaguti algebra in detail, we also introduce the Maurer-Cartan characterization and cohomology of Lie Yamaguti algebras. As a result of the obtained cohomology, we studied its deformation. We define Rota-Baxter operators on compatible Lie Yamaguti algebras as well as on compatible pre-Lie Yamaguti algebras. Using Rota-Baxter operators, we examine how compatible Lie (and compatible pre-Lie) Yamaguti algebras are related.

On compatible Lie and pre-Lie Yamaguti algebras

Abstract

This study aims to generalize the notion of compatible Lie algebras to the compatible Lie Yamaguti algebras. Along with describing the representation of the compatible Lie Yamaguti algebra in detail, we also introduce the Maurer-Cartan characterization and cohomology of Lie Yamaguti algebras. As a result of the obtained cohomology, we studied its deformation. We define Rota-Baxter operators on compatible Lie Yamaguti algebras as well as on compatible pre-Lie Yamaguti algebras. Using Rota-Baxter operators, we examine how compatible Lie (and compatible pre-Lie) Yamaguti algebras are related.
Paper Structure (6 sections, 16 theorems, 97 equations)

This paper contains 6 sections, 16 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Compatible Lie Yamaguti algebras
  4. Maurer-Cartan characterization and cohomology of compatible Lie Yamaguti algebras
  5. Infinitesimal deformations of compatible Lie Yamaguti algebras
  6. Compatible Pre-Lie Yamaguti algebra

Key Result

Proposition 2.2

Let $(L, [\cdot, \cdot], [\cdot, \cdot, \cdot])$ be a Lie Yamaguti algebra and $V$ a vector space. Let $\rho : L \rightarrow gl(V)$ and $\mu : \otimes^2L \rightarrow gl(V)$ are linear maps. Then $(V;\rho, \mu)$ is a representation of $(L, [\cdot, \cdot], [\cdot, \cdot, \cdot])$ if and only if there for all $x, y, z \in L$; $u, v, w \in V$. Such a Lie Yamaguti algebra $(L \oplus V, [\cdot, \cdot]_

Theorems & Definitions (59)

  • Definition 2.1
  • Example 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.1
  • Proposition 2.2
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.3
  • Definition 3.1
  • ...and 49 more