Half-axes in power associative algebras

2017/07/19 by Segev, Yoav
#17A05 (Primary) #17B69 (Secondary) #17C99 #FOS: Mathematics #Group Theory (math.GR) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1707.05906

Abstract

Let A be a commutative, non-associative algebra over a field \mathbbF 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 \rm ad_e (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 \rm ad_e (note that if |\mathbbF|>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.

Related

• Axes in non-associative algebras
• Quasi-definite axial algebras of Jordan type half
• Solid subalgebras in algebras of Jordan type half
• Axes of Jordan type in non-commutative algebras
• Miyamoto involutions in axial algebras of Jordan type half