Primitive axial algebras of Jordan type

An axial algebra   over the field FF 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 FF. Here we consider the first nonassociative case, where adjoint minimal polynomials divide (x−1)x(x−η)(x−1)x(x−η) for fixed 0≠η≠10≠η≠1. Jordan algebras arise when η=12, 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≠12 and in identifiable cases for η=12 this implies that the Miyamoto involutions are 3-transpositions, leading to a classification.