Generalization of n-ary Nambu algebras and beyond

TL;DR

This paper introduces generalized n-ary Hom-algebra structures that encompass various n-ary algebras of Lie and associative types, providing a framework for constructing new examples from existing algebras and endomorphisms.

Contribution

It extends the theory of n-ary algebras by defining Hom-algebra structures that unify and generalize multiple existing algebraic systems, along with a method to generate examples.

Findings

Defined n-ary Hom-algebra structures generalizing existing n-ary algebras.

Provided a construction method for new examples from known n-ary algebras and endomorphisms.

Demonstrated the applicability of the framework through multiple examples.

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.