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).

Keywords:

• Jordan algebras
• non-commutative Jordan algebras
• derivations
• n-ary algebras
• Lie triple systems
• generalized Lie algebras
• Cayley–Dickson construction
• TKK construction

AMS Subject Classifications:

• 17A42
• 17C50

Acknowledgements

The authors are thankful to the referee for the valuable remarks.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

FEUC, Universidade de Coimbra, Coimbra, Portugal. CeBER, Universidade de Coimbra, Coimbra, Portugal.

Additional information

Funding

The work was supported by RFBR [grant number 17-01-00258].