Cohomologies, deformations and extensions of n-Hom-Lie algebras

TL;DR

This paper develops the cohomology theory, deformation analysis, and extension methods for n-Hom-Lie algebras, introducing new concepts like derivations, Nijenhuis operators, and generalized derivations to advance understanding of their structure.

Contribution

It provides the first comprehensive cohomology framework, deformation theory, and extension techniques for n-Hom-Lie algebras, including novel operators and derivation concepts.

Findings

Cohomology of n-Hom-Lie algebras established

Introduction of Hom-Nijenhuis operators for trivial deformations

Construction of generalized derivation extensions

Abstract

In this paper, first we give the cohomologies of an $n$-Hom-Lie algebra and introduce the notion of a derivation of an $n$-Hom-Lie algebra. We show that a derivation of an $n$-Hom-Lie algebra is a $1$-cocycle with the coefficient in the adjoint representation. We also give the formula of the dual representation of a representation of an $n$-Hom-Lie algebra. Then, we study $(n-1)$-order deformation of an $n$-Hom-Lie algebra. We introduce the notion of a Hom-Nijenhuis operator, which could generate a trivial $(n-1)$-order deformation of an $n$-Hom-Lie algebra. Finally, we introduce the notion of a generalized derivation of an $n$-Hom-Lie algebra, by which we can construct a new $n$-Hom-Lie algebra, which is called the generalized derivation extension of an $n$-Hom-Lie algebra.