Cohomology and deformations of n-Lie algebra morphisms

Abstract

The study of n-Lie algebras which are natural generalization of Lie algebras is motivated by Nambu Mechanics and recent developments in String Theory and M-branes. The purpose of this paper is to define cohomology complexes and study deformation theory of n-Lie algebra morphisms. We discuss infinitesimal deformations, equivalent deformations and obstructions. Moreover, we provide various examples.Many proprieties on these types of algebras are treated, one cites for example solvability, nilpotency, central extension. A study of (n + 1)-Lie algebras constructed from n-Lie algebras and generalized trace maps was discussed in [2]. In [15], the authors established the relationships between the properties of a n-Lie algebra and its induced (n + 1)-Lie algebra. This aim of this paper is to construct a cohomology complex of n-Lie algebra morphisms. For that, we define a cohomology structure of these algebras with values in a module. The cohomology of n-Lie algebras is a generalization of the Chevalley-Eilenberg Lie algebras cohomology. We refer to [3,13,19] for the cohomology of n-Lie algebras and for representation theory which leads us to describe the deformation cohomology complex (C * (φ, φ), δ) for n-Lie algebra morphisms. Notice that Daletskii and Takhtajan showed, in [5], that the cohomology of n-Lie algebras can be derived from Leibniz algebras cohomology. Furthermore, we are interested in this paper by the deformations of n-Lie algebra morphisms. A cohomology complex adapted to the study of deformations was introduced by Frégier [7] in the case of Lie algebra morphisms.Deformations of n-Lie algebras has been discussed in terms of Chevalley-Eilenberg cohomology in various papers, a survey is available in [16]. In this paper, we introduce and study for n-Lie algebra morphisms the concepts of infinitesimal and equivalent deformations as well as obstructions to extend a given fixed order deformation. We denote by N and N ′ two n-Lie algebras. Equivalence classes of infinitesimal deformations of n-Lie algebras are characterized by the cohomology groups H 2 (N , N ) and by H 1 (N , N ′ ) withe respect to a morphism φ between N and N ′ .The paper is organized as follows. In Section 1, we review some basics about n-Lie algebras and their adjoint representation via a morphism φ. Moreover, we recall the cohomology of n-Lie algebras with values in the algebra. In Section 2, we define the cohomology of n-Lie algebras with values in an adjoint module.In Section 3, we define explicitly a cochain complex with a coboundary operator and the n-cochains module C n (φ, φ) providing a cohomology of n-Lie algebra morphisms. Section 4 focuses on the deformation theory of n-Lie algebra morphisms. Lastly, in Section 5, we deal with some examples. We compute the cohomology and provide examples of deformation of 3-Lie algebra morphisms.

O-operators on Lie triple systems

“…Recently deformations of certain operators, e.g. morphisms and Rota-Baxter operators (Ooperators) were deeply studied, see [1,8,10,21,26]. One needs a cohomology to control deformations and extension problems of a given algebraic structure.…”

Cohomologies of 3-Lie algebras with derivations

“…Representation theory of n-Lie algebras was studied by Kasymov in [20] and cohomology theory of n-Lie algebras was studied by Takhtajan and Gautheron in [18,29]. Deformations of n-Lie algebras were studied in [1,24,29]. See [11,22,32] for more details on extensions of n-Lie algebras.…”

Lie n-algebras and cohomologies of relative Rota-Baxter operators on n-Lie algebras

“…The idea of treating deformation as a tool to study the algebraic structures was introduced by Gerstenhaber in his work of associative algebras ( [24,25]) and then was extended to Lie algebras by Nijenhuis and Richardson ([41,42]). Deformations of 3-Lie algebras and n-Lie algebras are studied in [1,19,36,51]. See the review paper [13,37] for more details.…”

“…See the review paper [13,37] for more details. Recently, people pay more attention on the deformations of morphisms ( [1,21,22,23,38,53]), relative Rota-Baxter operators ( [12,48,49]) and diagram of algebras ( [7,28,39]).…”