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 ade (multiplication by e) is one dimensional. In this paper we consider the identities (⁎) x2x2=x4 and x3x2=xx4. We show that if identities (⁎) hold strictly in A, then one gets (very) interesting identities between elements in the eigenspaces of ade (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