On a ternary generalization of Jordan algebras

TL;DR

This paper introduces n-ary Jordan algebras as a generalization of classical Jordan algebras, explores a specific ternary example, and constructs ternary derivation algebras using Cayley-Dickson algebras.

Contribution

It defines n-ary Jordan algebras based on generalized derivation properties and provides explicit ternary examples and derivation algebra constructions.

Findings

Defined n-ary Jordan algebras via generalized derivation conditions

Constructed a ternary example of these algebras

Presented a family of ternary derivation algebras using Cayley-Dickson algebras

Abstract

Based on the relation between the notions of Lie triple systems and Jordan algebras, we introduce the $n$-ary Jordan algebras,an $n$-ary generalization of Jordan algebras obtained via the generalization of the following property $\left[ R_{x},R_{y}\right] \in Der\left( \mathcal{A}\right)$, where $\mathcal{A}$ is an $n$-ary algebra. Next, we study a ternary example of these algebras. Finally, based on the construction of a family of ternary algebras defined by means of the Cayley-Dickson algebras, we present an example of a ternary $D_{x,y}$-derivation algebra ($n$-ary $D_{x,y}$-derivation algebras are the non-commutative version of $n$-ary Jordan algebras).