Quadratic $n$-ary Hom-Nambu algebras

TL;DR

This paper introduces quadratic $n$-ary Hom-Nambu algebras, exploring their construction, properties, and connections with representation theory, and shows how to derive higher arity algebras and reduce to lower arity cases.

Contribution

It provides the first systematic study of quadratic $n$-ary Hom-Nambu algebras, including construction methods and their structural relationships.

Findings

Constructed quadratic $n$-ary Hom-Nambu algebras using twisting, tensor products, and T*-extensions.

Established connections between these algebras and representation theory.

Demonstrated derivation of higher arity algebras and reduction to lower arity cases.

Abstract

The purpose of this paper is to introduce and study quadratic $n$-ary Hom-Nambu algebras, which are $n$-ary Hom-Nambu algebras with an invariant, nondegenerate and symmetric bilinear forms that are also $\alpha$-symmetric and $\beta$-invariant where $\alpha$ and $\beta$ are twisting maps. We provide constructions of these $n$-ary algebras by using twisting principles, tensor product and T*-extension. Also is discussed their connections with representation theory and centroids. Moreover we show that one may derive from quadratic $n$-ary Hom-Nambu algebra ones of increasingly higher arities and that under suitable assumptions it reduces to a quadratic $(n-1)$-ary Hom-Nambu algebra.