Constructions and Cohomology of Hom–Lie Color Algebras

Abdaoui, K.; Ammar, Faouzi; Makhlouf, Abdenacer

doi:10.1080/00927872.2014.910797

Cited by 43 publications

(38 citation statements)

References 35 publications

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“…Although one can see that extensions of a given Abelian hom-Lie color algebra is characterized by elements of its second cohomology group, we concentrate on some geometric aspects in this research. The cohomology has been studied for Lie superalgebras and color Lie algebras [20,49,57,58] and hom-Lie color algebras [1].…”

Section: Extensions Of Hom-lie Color Algebrasmentioning

confidence: 99%

“…Hom-Lie algebras, hom-Lie superalgebras and Hom-Lie color algebras are important special classes of color (graded) quasi-Lie algebras introduced first by Larsson and Silvestrov [35,37]. Hom-Lie algebras and hom-Lie superalgebras have been studied in different aspects by Makhlouf, Silvestrov, Sheng, Ammar, Yau and other authors [4, 13, 36, 43-47, 50-52, 54, 59-66], and Hom-Lie color algebras have been considered for example in [1,14,15,65]. We wish to mention specially [6], where the constructions of Hom-Lie and quasi-hom Lie algebras based on twisted discretizations of vector fields [24] and Hom-Lie admissible algebras have been extended to Hom-Lie superalgebras, a subclass of graded quasi-Lie algebras [35,37].…”

Section: Introductionmentioning

confidence: 99%

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Extensions of hom-Lie color algebras

Armakan

Silvestrov

Farhangdoost

2019

Georgian Mathematical Journal

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In this paper we study (non-Abelian) extensions of a given hom-Lie color algebra and provide a geometrical interpretation of extensions. In particular, we characterize an extension of a hom-Lie algebra g by another hom-Lie algebra h and we discuss the case where h has no center. We also deal with the setting of covariant exterior derivatives, Chevalley derivative, curvature and the Bianchi identity for the possible extensions in differential geometry. Moreover, we find a cohomological obstruction to the existence of extensions of hom-Lie color algebras, i. e. we show that in order to have an extendible hom-Lie color algebra, there should exist a trivial member of the third cohomology.M.S.C. 2010: 17B56, 17B75, 17B40.

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Section: Extensions Of Hom-lie Color Algebrasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Extensions of hom-Lie color algebras

Armakan

Silvestrov

Farhangdoost

2019

Georgian Mathematical Journal

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“…Now, we recall the definition of Hom-Jordan algebra as [17]. Definition 3.1: Let (L,µ,α) be a Hom-algebra.…”

Section: That Means D ∉C(a)mentioning

confidence: 99%

Generalized Derivations of BiHom-Lie Algebras

Bh¹

2017

J Generalized Lie Theory Appl

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BiHom-Lie algebra is a generalized Hom-Lie algebra endowed with two commuting multiplicative linear maps. This paper is devoted to investigate the generalized derivation of BiHom-Lie algebra. We generalize the main results of Leger and Luks to the case of BiHom-Lie algebra. Firstly we review some concepts associated with BiHom-Lie algebra L. Furthermore, we give the definitions of the generalized derivationand quasicentroid QC(L). Later one, we give some useful proprieties and connections between these derivations. In particular, we prove that GDer(L)=QDer(L)+QC(L). We also prove that QDer(L) can be embedded as derivations in larger BiHom-Lie algebra.

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“…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 of graded quasi-Lie algebras [37,39].…”

mentioning

confidence: 99%

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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Constructions and Cohomology of Hom–Lie Color Algebras

Cited by 43 publications

References 35 publications

Extensions of hom-Lie color algebras

Extensions of hom-Lie color algebras

Generalized Derivations of BiHom-Lie Algebras

Enveloping algebras of color hom-Lie algebras

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