Half-axes in power associative algebras

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3 Scopus citations

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 languageEnglish
Pages (from-to)1-23
Number of pages23
JournalJournal of Algebra
Volume510
DOIs
StatePublished - 15 Sep 2018

Keywords

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

ASJC Scopus subject areas

  • Algebra and Number Theory

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