Cohomologies of n-Lie Algebras with Derivations

Qinxiu Sun,Zhixiang Wu

doi:10.3390/math9192452

Qinxiu Sun, Zhixiang Wu

Open Access

https://doi.org/10.3390/math9192452

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Abstract

The goal of this paper is to study cohomological theory of n-Lie algebras with derivations. We define the representation of an n-LieDer pair and consider its cohomology. Likewise, we verify that a cohomology of an n-LieDer pair could be derived from the cohomology of associated LeibDer pair. Furthermore, we discuss the (n−1)-order deformations and the Nijenhuis operator of n-LieDer pairs. The central extensions of n-LieDer pairs are also investigated in terms of the first cohomology groups with coefficients in the trivial representation.

Highlights

The notion of n-Lie algebras was introduced by Filippov [1] in 1985
Takhtajan [5] systematically developed a foundemental theory of n-Poisson or Nambu–Poisson manifolds, and established a connection between Nambu mechanics and Filippov algebras
We describe the notions of a Nijenhuis operator and an O-operator on n-LieDer pair (g, φg)

Summary

Introduction

The notion of n-Lie algebras was introduced by Filippov [1] in 1985. It is the algebraic structure corresponding to Nambu mechanics [2,3,4]. Lie algebras with derivations are usually called LieDer pairs. An n-LieDer pair is an n-Lie algebra with a derivation, namely, we have the following

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