Solid lines in axial algebras of Jordan type \tfrac12 and Jordan algebras

2024/03/27 by Jari Desmet, Desmet, Jari
Mathematics · Materials Science · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Synthesis and properties of polymers

paper · pdf · doi:10.48550/arxiv.2403.18808

Abstract

We show that a primitive axial algebra of Jordan type η= \tfrac12 is a Jordan algebra if and only if every 2-generated subalgebra is solid, a notion introduced recently by Ilya Gorshkov, Sergey Shpectorov and Alexei Staroletov. As a byproduct, we show that a subalgebra generated by axes a,b is solid if and only if the associator [L_a,L_b] is a derivation. Moreover, we show that 2-generated subalgebras that are not solid contain precisely 3 axes.

Related

• Solid subalgebras in algebras of Jordan type half
• Primitive axial algebras are of Jordan type
• Axial identities
• Central extensions of axial algebras
• Axial algebras of Jordan and Monster type