On a ternary generalization of Jordan 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).

17 pages

References in corpus (1)

• Jordan structures in mathematics and physics

Cited by in corpus (9)

• The algebraic and geometric classification of nilpotent binary Lie algebras
• Non-associative algebraic structures: classification and structure
• The algebraic classification of nilpotent $\mathfrak{CD}$-algebras
• Split Regular $Hom$-Leibniz Color $3$-Algebras
• The geometric classification of nilpotent $\mathfrak{CD}$-algebras
• Decompositions of linear spaces induced by $n$-linear maps
• The algebraic classification of nilpotent commutative $\mathfrak{CD}$-algebras
• The geometric classification of nilpotent commutative $\mathfrak{CD}$-algebras
• Generalized derivations, quasiderivations and centroids of ternary Jordan algebras