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 [R_x; R_y] \in Der (A); where 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).