## 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} )∈ 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).

Original language | English |
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Pages (from-to) | 1074-1102 |

Number of pages | 29 |

Journal | Linear and Multilinear Algebra |

Volume | 67 |

Issue number | 6 |

DOIs | |

Publication status | Published - 3 Jun 2019 |

## Keywords

- Cayley–Dickson construction
- derivations
- generalized Lie algebras
- Jordan algebras
- Lie triple systems
- n-ary algebras
- non-commutative Jordan algebras
- TKK construction
- 17C50
- 17A42
- IDENTITIES
- LIE
- Cayley-Dickson construction