Primitive axial algebras of Jordan type

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 - eta) for fixed 0 not equal eta not equal 1. Jordan algebras arise when eta = 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 eta = 1/2 and in identifiable cases for eta not equal 1/2 this implies that the Miyamoto involutions are 3-transpositions, leading to a classification. (C) 2015 Elsevier Inc. All rights reserved.