On a ternary generalization of Jordan algebras

• Main Author: Kaygorodov, Ivan
• Publication Date: 2019
• Other Authors: Pozhidaev, Alexander, Saraiva, Paulo
• Format: Article
• Language: eng
• Source: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Download full: http://hdl.handle.net/10316/89485; https://doi.org/10.1080/03081087.2018.1443426
• topic: Jordan algebras; non-commutative Jordan algebras; derivations; n-ary algebras; Lie triple systems; generalized Lie algebras; Cayley–Dickson construction; TKK construction

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).