Generalization of n-ary Nambu algebras and beyond

Abstract

Abstract. The aim of this paper is to introduce n-ary Hom-algebra structures generalizing the n-ary algebras of Lie type enclosing n-ary Nambu algebras, n-ary Nambu-Lie algebras, n-ary Lie algebras, and n-ary algebras of associative type enclosing n-ary totally associative and n-ary partially associative algebras. Also, we provide a way to construct examples starting from an n-ary algebra and an n-ary algebras endomorphism. Several examples could be derived using this process.

Hom-Lie superalgebras and Hom-Lie admissible superalgebras

“…Applying Theorem 2.7 we obtain a family of Hom-Lie superalgebras osp(1, 2) λ = (osp (1,2), [·,·] α λ , α λ ) where the Hom-Lie superalgebra bracket [·,·] α λ on the basis elements is given, for λ = 0, by…”

“…We show that starting with an ordinary Lie superalgebra and a superalgebra endomorphism, we may construct a Hom-Lie superalgebra. Moreover, we construct a one parameter family of Hom-Lie superalgebras deforming the orthosymplectic superalgebra osp (1,2). In Section 3, we introduce Hom-Lie admissible superalgebras and more general G-Hom-associative superalgebras, where G is a subgroup of the permutation group S 3 .…”

On Deformations of n-Lie Algebras

“…See [9] and [4] for triple commutator leading to 3-Lie algebras and ternary Hom-Nambu-Lie algebras [8]. More general construction of (n + 1)-Lie algebras induced by n-Lie algebras was studied in [5] …”

Ternary and n-ary f-distributive structures

“…These type of algebraic structures, called sometimes Hom-algebras, appeared rst in quantum deformations of algebras of vector elds, motivated by physical aspects. A systematic study and mathematical aspects were provided for Lie type algebras by Hartwig-Larsson and Silvestrov in [7] whereas associative and other nonassociative algebras were discussed by the fourth author and Silvestrov in [8] and n-ary Hom-type algebras in [9]. The main feature of all these generalization is that the usual identities are twisted by a homomorphism.…”

“…Recall that this construction was introduced rst by Yau to deform a Lie algebra to a Hom-Lie algebra along a Lie algebra morphism. It was generalized to di erent situation, in particular to n-ary algebras in [9].…”