Representations and cohomologies of Hom-Lie-Yamaguti algebras with applications

TL;DR

This paper develops the representation and cohomology theory for Hom-Lie-Yamaguti algebras, applying it to study their deformations and extensions, and establishing classification results via cohomology groups.

Contribution

It introduces the cohomology theory for Hom-Lie-Yamaguti algebras and applies it to deformation and extension problems, providing new classification results.

Findings

Deformation of Hom-Lie-Yamaguti algebras corresponds to (2,3)-cocycles.

Abelian extensions are classified by (2,3)-cohomology groups.

A 1-parameter deformation relates to a specific cocycle.

Abstract

The representation and cohomology theory of Hom-Lie-Yamaguti algebras is introduced. As an application, we study deformation and extension of Hom-Lie-Yamaguti algebras. It proved that a 1-parameter infinitesimal deformation of a Hom-Lie-Yamaguti algebra $T$ corresponds to a Hom-Lie-Yamaguti algebra of deformation type and a (2,3)-cocycle of $T$ with coefficients in the adjoint representation. We also prove that abelian extensions of Hom-Lie-Yamaguti algebras are classified by the (2,3)-cohomology group.