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

Abstract

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...

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

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Structure and cohomology of 3-Lie-Rinehart superalgebras

“…Interesting constructions of ternary Lie superalgebras in connection to superspace extension of Nambu-Hamilton equation is considered in [5]. In [17], a method was demonstrated of how to construct n-ary multiplications from the binary multiplication of a Hom-Lie algebra and a ðn À 2Þ-linear function satisfying certain compatibility conditions. Solvability and Nilpotency for n-Hom-Lie algebras and (n þ 1)-Hom-Lie algebras induced by n-Hom-Lie algebras have been considered in [43].…”

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