## Automorphism group of a Matsuo algebra

dc.contributor.advisor | Shpectorov, Sergey | |

dc.contributor.author | Alharbi, Faarie | |

dc.date.accessioned | 2024-06-10T11:11:14Z | |

dc.date.available | 2024-06-10T11:11:14Z | |

dc.date.issued | 2024-06 | |

dc.description.abstract | Axial algebras are commutative, not necessarily associative algebras generated by axes, where axes are idempotents whose adjoint actions are semisimple and such that the product of eigenvectors is controlled by a specific fusion law. Matsuo algebras, defined using Fischer spaces and $3$-transposition groups, are examples of axial algebras of Jordan type $\eta$. In this thesis, we show that Matsuo algebras with $\eta \neq \frac{1}{2}$ almost never contain additional axes, and consequently, their automorphism groups almost always coincide with the automorphism group of the underlying groups of $3$-transpositions. Furthermore, we identify all exceptions to this general rule in the irreducible case. Namely, we find three infinite series and an additional sporadic example of Matsuo algebras having extra axes and a larger group of automorphisms. In particular, for specific $\eta$, we show that $M=M_\eta(S_n)$ is isomorphic to a quotient of $M'=M_\eta(S_{n+1})$, and similarly $M=M_\eta(O^{\epsilon}_{2m}(2))$ is isomorphic to a quotient of $M'=M_\eta(Sp_{2m}(2))$. These are examples of what we call aligned pairs of groups of $3$-transpositions. We use the information about the spectrum of the diagram of irreducible $3$-transposition groups, provided in \cite{JS}, to find all aligned pairs of irreducible $3$-transposition groups. We conclude that, apart from the examples above, there is only one additional sporadic aligned pair $(Fi_{23},\Omega_8(3):S_3)$. | |

dc.format.extent | 136 | |

dc.identifier.uri | https://hdl.handle.net/20.500.14154/72291 | |

dc.language.iso | en_US | |

dc.publisher | University of Birmingham | |

dc.subject | Axial algebras | |

dc.subject | Matsuo algebras | |

dc.subject | Aligned pairs | |

dc.title | Automorphism group of a Matsuo algebra | |

dc.type | Thesis | |

sdl.degree.department | Mathematics | |

sdl.degree.discipline | Algebra | |

sdl.degree.grantor | University of Birmingham | |

sdl.degree.name | Doctor of Philosophy |