## Abstract

Let A be a commutative, non-associative algebra over a field F of characteristic ≠2. A half-axis in A is an idempotent e∈A such that e satisfies the Peirce multiplication rules in a Jordan algebra, and, in addition, the 1-eigenspace of ad_{e} (multiplication by e) is one dimensional. In this paper we consider the identities (⁎) x^{2}x^{2}=x^{4} and x^{3}x^{2}=xx^{4}. We show that if identities (⁎) hold strictly in A, then one gets (very) interesting identities between elements in the eigenspaces of ad_{e} (note that if |F|>3 and the identities (⁎) hold in A, then they hold strictly in A). Furthermore we prove that if A is a primitive axial algebra of Jordan type half (i.e., A is generated by half-axes), and the identities (⁎) hold strictly in A, then A is a Jordan algebra.

Original language | English |
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Pages (from-to) | 1-23 |

Number of pages | 23 |

Journal | Journal of Algebra |

Volume | 510 |

DOIs | |

State | Published - 15 Sep 2018 |

## Keywords

- Axial algebra
- Half-axis
- Jordan algebra
- Power associative algebra

## ASJC Scopus subject areas

- Algebra and Number Theory