## Abstract

We introduce decomposition algebras as a natural generalization of axial algebras, Majorana algebras and the Griess algebra. They remedy three limitations of axial algebras:

(1) They separate fusion laws from specific values in a field, thereby allowing repetition of eigenvalues; (2) They allow for decompositions that do not arise from multiplication by idempotents; (3) They admit a natural notion of homomorphisms, making them into a nice category.

We exploit these facts to strengthen the connection between axial algebras and groups. In particular, we provide a definition of a universal Miyamoto group which makes this connection functorial under some mild assumptions.

We illustrate our theory by explaining how representation theory and association schemes can help to build a decomposition algebra for a given (permutation) group. This construction leads to a large number of examples.

We also take the opportunity to fix some terminology in this rapidly expanding subject.

(1) They separate fusion laws from specific values in a field, thereby allowing repetition of eigenvalues; (2) They allow for decompositions that do not arise from multiplication by idempotents; (3) They admit a natural notion of homomorphisms, making them into a nice category.

We exploit these facts to strengthen the connection between axial algebras and groups. In particular, we provide a definition of a universal Miyamoto group which makes this connection functorial under some mild assumptions.

We illustrate our theory by explaining how representation theory and association schemes can help to build a decomposition algebra for a given (permutation) group. This construction leads to a large number of examples.

We also take the opportunity to fix some terminology in this rapidly expanding subject.

Original language | English |
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Pages (from-to) | 287-314 |

Number of pages | 28 |

Journal | Journal of Algebra |

Volume | 556 |

Early online date | 31 Mar 2020 |

DOIs | |

Publication status | Published - 15 Aug 2020 |

## Keywords

- 17A99, 20F29
- Association schemes
- Axial algebras
- Decomposition algebras
- Fusion laws
- Griess algebra
- Majorana algebras
- Norton algebras
- Representation theory
- math.GR
- math.RA