Primitive axial algebras of Jordan type
Abstract
An axial algebra over the field $\mathbb F$ is a commutative algebra generated by idempotents whose adjoint action has multiplicityfree minimal polynomial. For semisimple associative algebras this leads to sums of copies of $\mathbb F$. Here we consider the first nonassociative case, where adjoint minimal polynomials divide $(x1)x(x\eta)$ for fixed $0\neq\eta\neq 1$. Jordan algebras arise when $\eta=\frac{1}{2}$, 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. For $\eta \neq \frac{1}{2}$ this implies that the Miyamoto involutions are $3$transpositions, leading to a classification.
 Publication:

arXiv eprints
 Pub Date:
 March 2014
 arXiv:
 arXiv:1403.1898
 Bibcode:
 2014arXiv1403.1898H
 Keywords:

 Mathematics  Rings and Algebras;
 Mathematics  Group Theory
 EPrint:
 41 pages