Primitive axial algebras of Jordan type

By Jonathan I. Hall, Yoav Segev, Sergey V. Shpectorov. 2014

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Abstract An axial algebra over the field F is a commutative algebra generated by idempotents whose adjoint action has multiplicity-free minimal polynomial. For semisimple associative algebras this leads to sums of copies of F . Here we consider the first nonassociative case, where adjoint minimal polynomials divide ( x − 1 ) x ( x − η ) for fixed 0 ≠ η ≠ 1 . Jordan algebras arise when η = 1 2 , but our motivating examples are certain Griess algebras of vertex operator algebras and the related Majorana algebras. We study a class of algebras, including these, for which axial automorphisms like those defined by Miyamoto exist, and there classify the 2-generated examples. Always for η ≠ 1 2 and in identifiable cases for η = 1 2 this implies that the Miyamoto involutions are 3-transpositions, leading to a classification.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/40bb10e79b6e3feec2171fcefca3baa22bc244e5