## Abstract

In the first half of this paper, we define axial algebras: nonassociative commutative algebras generated by axes, that is, semisimple idempotents-the prototypical example of which is Griess' algebra [2] for the Monster group. When multiplication of eigenspaces of axes is controlled by fusion rules, the structure of the axial algebra is determined to a large degree. We give a construction of the universal Frobenius axial algebra on n generators with specified fusion rules, of which all n-generated Frobenius axial algebras with the same fusion rules are quotients. In the second half, we realise this construction in the Majorana/Ising/Vir(4, 3)-case on 2 generators, and deduce a result generalising Sakuma's theorem in VOAs [13].

Original language | English |
---|---|

Pages (from-to) | 394-424 |

Number of pages | 31 |

Journal | Journal of Algebra |

Volume | 421 |

DOIs | |

Publication status | Published - 1 Jan 2015 |

## Keywords

- Fusion rules
- Idempotents
- Monster group
- Nonassociative algebras