Weil Algebra, 3-Lie Algebra and B.R.S. Algebra

Abramov, Viktor

doi:10.1007/978-3-030-41850-2_1

Cited by 9 publications

(3 citation statements)

References 19 publications

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“…It is called the (n + 1)-hom-Lie algebra induced by n-hom-Lie algebra. In this context, Abramov gave a new approach of this construction (see [1]). Now, we generalize this approach in the Hom case.…”

Section: N-hom-lie Algebras Induced By Hom-lie Algebrasmentioning

confidence: 99%

Some Constructions of Multiplicative $$\varvec{n}$$-ary hom–Nambu Algebras

Hassine

Mabrouk

Ncib

2019

Adv. Appl. Clifford Algebras

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We show that given a hom-Lie algebra one can construct the n-ary hom-Lie bracket by means of an (n − 2)-cochain of the given hom-Lie algebra and find the conditions under which this n-ary bracket satisfies the Filippov-Jacobi identity, thereby inducing the structure of n-hom-Lie algebra. We introduce the notion of a hom-Lie n-tuple system which is the generalization of a hom-Lie triple system. We construct hom-Lie n-tuple system using a hom-Lie algebra.Mathematics Subject Classification. 17A30, 17A36, 17A40, 17A42.In [6], generalizations of n-ary algebras of Lie type and associative type by twisting the identities using linear maps were introduced. The notions of representations, derivations, cohomology and deformations were studied in [3,12,15,21,24]. These generalizations include n-ary Hom-algebra structures generalizing the n-ary algebras of Lie type including n-ary Nambu algebras, n-Lie algebras (called also n-ary Nambu-Lie algebras) and n-ary Lie algebras, and n-ary algebras of associative type including n-ary totally associative and n-ary partially associative algebras. In [4], a method was demonstrated how to construct ternary multiplications from the binary multiplication of a hom-Lie algebra, a linear twisting map, and a trace function satisfying certain compatibility conditions; and it was shown that this method can be used to construct ternary hom-Nambu-Lie algebras from hom-Lie algebras. This construction was generalized to n-Lie algebras and n-hom-Nambu-Lie algebras in [5].It is well known that algebras of derivations and generalized derivations are very important in the study of Lie algebras and its generalizations. The notion of δ-derivation appeared in the paper of Filippov [14]. The results for δ-derivations and generalized derivations were studied by many authors. For example, Zhang and Zhang [26] generalized the above results to the case of Lie superalgebras; Chen, Ma, Ni and Zhou considered the generalized derivations of color Lie algebras, hom-Lie superalgebras and Lie triple systems [10,11]. Derivations and generalized derivations of n-ary algebras were considered in [17,18] and other papers. In [9], the authors generalize these results in the color n-ary hom-Nambu case.This paper is organized as follows. In Sect. 1, we review some basic concepts of hom-Lie algebras, n-ary hom-Nambu algebras and n-hom-Lie algebras. We also recall the definitions of derivations, α k -derivations, α kquasiderivations and α k -centroid. In Sect. 2, we provide a construction procedure of n-hom-Lie algebras starting from a binary bracket of a hom-Lie algebra and multilinear form satisfying certain conditions. To this end, we give the relation between α k -derivations, (resp. α k -quasiderivations and α kcentroid) of hom-Lie algebras and α k -derivations (resp. α k -quasiderivations and α k -centroid) of n-hom-Lie algebras. In Sect. 3, we introduce the notion of a hom-Lie n-tuple system which is the generalization of a Lie n-tuple system which is introduced in [13]. We construct a hom-Lie n-tuple system using a ho...

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Section: N-hom-lie Algebras Induced By Hom-lie Algebrasmentioning

confidence: 99%

Some Constructions of Multiplicative $$\varvec{n}$$-ary hom–Nambu Algebras

Hassine

Mabrouk

Ncib

2019

Adv. Appl. Clifford Algebras

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“…The structure of 3-Lie algebras induced by Lie algebras, classification of 3-Lie algebras, and application to constructions of B.R.S. algebras were considered in [51][52][53]. Interesting constructions of ternary Lie superalgebras in connection to superspace extension of the Nambu-Hamilton equation is considered in [54].…”

Section: Introductionmentioning

confidence: 99%

On Properties and Classification of a Class of 4-Dimensional 3-Hom-Lie Algebras with a Nilpotent Twisting Map

Kitouni,

Silvestrov

2024

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The aim of this work is to investigate the properties and classification of an interesting class of 4-dimensional 3-Hom-Lie algebras with a nilpotent twisting map α and eight structure constants as parameters. Derived series and central descending series are studied for all algebras in this class and are used to divide it into five non-isomorphic subclasses. The levels of solvability and nilpotency of the 3-Hom-Lie algebras in these five classes are obtained. Building upon that, all algebras of this class are classified up to Hom-algebra isomorphism. Necessary and sufficient conditions for multiplicativity of general (n+1)-dimensional n-Hom-Lie algebras, as well as for algebras in the considered class, are obtained in terms of the structure constants and the twisting map. Furthermore, for some algebras in this class, it is determined whether the terms of the derived and central descending series are weak subalgebras, Hom-subalgebras, weak ideals, or Hom-ideals.

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“…The structure of 3-Lie algebras induced by Lie algebras, classification of 3-Lie algebras and application to constructions of B.R.S. algebras have been considered in [1,3,4]. Interesting constructions of ternary Lie superalgebras in connection to superspace extension of Nambu-Hamilton equation is considered in [5].…”

Section: Introductionmentioning

confidence: 99%