Representations and cohomology of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>n</mml:mi></mml:math>-ary multiplicative Hom–Nambu–Lie algebras

Ammar, Faouzi; Mabrouk, Sami; Makhlouf, Abdenacer

doi:10.1016/j.geomphys.2011.04.022

Cited by 98 publications

(64 citation statements)

References 31 publications

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“…for all X = x 1 ∧ · · · ∧ x n−1 and Y = y 1 ∧ · · · ∧ y n−1 . It is proved in [2] that (∧ n−1 g, [·, ·] F , α) is a Hom-Leibniz algebra.…”

Section: Preliminariesmentioning

confidence: 99%

“…Then several aspects about n-Hom-Lie algebras are studied. For example, the cohomologies adapted to central extensions and deformations are studied in [2]; 2-cocycles that used to studied abelian extensions are studied in [5]; Construction of 3-Hom-Lie algebras from Hom-Lie algebras are studied in [3], and extensions of 3-Hom-Lie algebras are studied in [14]. However, the systematic study of the cohomology of an n-Hom-Lie algebra is still lost.…”

Section: Introductionmentioning

confidence: 99%

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

Song

Tang

2019

Journal of Geometry and Physics

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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.

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“…for all X = x 1 ∧ · · · ∧ x n−1 and Y = y 1 ∧ · · · ∧ y n−1 . It is proved in [2] that (∧ n−1 g, [·, ·] F , α) is a Hom-Leibniz algebra.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Song

Tang

2019

Journal of Geometry and Physics

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“…. , x n ], σ ∈ S n , then the structure is called Hom-Lie n-algebra (or n-ary Hom-Nambu algebras in [1,3], or n-Hom-Lie algebras [23]). Let x = (x 1 , .…”

Section: Basic Definitionsmentioning

confidence: 99%

Universal α-central extensions of Hom-Leibniz n-algebras

Casas

Rego

2017

Linear and Multilinear Algebra

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We construct homology with trivial coefficients of Hom-Leibniz n-algebras. We introduce and characterize universal (α)-central extensions of Hom-Leibniz n-algebras. In particular, we show their interplay with the zeroth and first homology with trivial coefficients. When n = 2 we recover the corresponding results on universal central extensions of Hom-Leibniz algebras. The notion of non-abelian tensor product of Hom-Leibniz n-algebras is introduced and we establish its relationship with the universal central extensions. We develop a generalization of the concept and properties of unicentral Leibniz algebras to the setting of Hom-Leibniz n-algebras.

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“…We also wish to mention that Z 3 -graded generalizations of supersymmetry, Z 3 -graded algebras, ternary structures and related algebraic models for classifications of elementary particles and unification problems for interactions, quantum gravity and non-commutative gauge theories [2,31,32,33,34] also provide interesting examples related to Homassociative algebras, graded Hom-Lie algebras, twisted differential calculi and n-ary Hom-algebra structures. It would be a project of grate interest to extend and apply all the constructions and results in the present paper in the relevant contexts of the articles [2,6,4,10,9,8,31,32,33,37,39,45].An important direction with many fundamental open problems in the theory of (color) quasi-Lie algebras and in particular (color) quasi-hom-Lie algebras and (color) hom-Lie algebras is the development of comprehensive fundamental theory, explicit constructions, examples and algorithms for enveloping algebraic structures, expanding the corresponding more developed fundamental theory and constructions for enveloping algebras of Lie algebras, Lie superalgebras and general color Lie algebras [11, 29, 49,…”

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confidence: 94%

Enveloping algebras of color hom-Lie algebras

Armakan¹,

Silvestrov²,

Farhangdoost³

2019

Turk J Math

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In this paper the universal enveloping algebra of color hom-Lie algebras is studied. A construction of the free involutive hom-associative color algebra on a hom-module is described and applied to obtain the universal enveloping algebra of an involutive hom-Lie color algebra. Finally, the construction is applied to obtain the well-known Poincaré-Birkhoff-Witt theorem for Lie algebras to the enveloping algebra of an involutive color hom-Lie algebra. of abstract quasi-Lie algebras and subclasses of quasi-Hom-Lie algebras and Hom-Lie algebras as well as their general colored (graded) counterparts in [25,36,39,58,37]. These generalized Lie algebra structures with (graded) twisted skew-symmetry and twisted Jacobi conditions by linear maps are tailored to encompass within the same algebraic framework such quasi-deformations and discretizations of Lie algebras of vector fields using σ-derivations, describing general descritizations and deformations of derivations with twisted Leibniz rule, and the well-known generalizations of Lie algebras such as color Lie algebras which are the natural generalizations of Lie algebras and Lie superalgebras.Quasi-Lie algebras are non-associative algebras for which the skew-symmetry and the Jacobi identity are twisted by several deforming twisting maps and also the Jacobi identity in quasi-Lie and quasi-Hom-Lie algebras in general contains six twisted triple bracket terms. Hom-Lie algebras is a special class of quasi-Lie algebras with the bilinear product satisfying the non-twisted skew-symmetry property as in Lie algebras, whereas the Jacobi identity contains three terms twisted by a single linear map, reducing to the Jacobi identity for ordinary Lie algebras when the linear twisting map is the identity map. Subsequently, hom-Lie admissible algebras have been considered in [45] where also the hom-associative algebras have been introduced and shown to be hom-Lie admissible natural generalizations of associative algebras corresponding to hom-Lie algebras. In [45], moreover several other interesting classes of hom-Lie admissible algebras generalising some non-associative algebras, as well as examples of finite-dimentional hom-Lie algebras have been described. Since these pioneering works [25,36,39,37,40,45], hom-algebra structures have become a popular area with increasing number of publications in various directions.Hom-Lie algebras, hom-Lie superalgebras and hom-Lie color algebras are important special classes of color (Γ-graded) quasi-Lie algebras introduced first by Larsson and Silvestrov in [39,37]. Hom-Lie algebras and hom-Lie superalgebras have been studied further in different aspects by Makhlouf, Silvestrov, Sheng, Ammar, Yau and other authors [62,61,60,45,48,46,47,68,12,44,69,64,65,66,38,51,52,57,59], and hom-Lie color algebras have been considered for example in [68,14,13,1]. In [4], the constructions of Hom-Lie and quasi-hom Lie algebras based on twisted discretizations of vector fields [25] and Hom-Lie admissible algebras have been extended to Hom-Lie superalgebras, a subclass...

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Representations and cohomology of n-ary multiplicative Hom–Nambu–Lie algebras

Cited by 98 publications

References 31 publications

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

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

Universal α-central extensions of Hom-Leibniz n-algebras

Enveloping algebras of color hom-Lie algebras

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