Automorphism groups of Matsuo algebras
Jari Desmet
Abstract
In this paper, we determine the connected component of the automorphism group scheme of Matsuo algebras over fields of characteristic not $3$.
Table of Contents
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Automorphism groups of Matsuo algebras
Authors
Abstract
In this paper, we determine the connected component of the automorphism group scheme of Matsuo algebras over fields of characteristic not .
Paper Structure
This paper contains 17 sections, 28 theorems, 34 equations, 1 figure, 1 table.
Table of Contents
- Introduction
- Outline of the paper
- Notation
- Acknowledgements
- Preliminaries
- Axial algebras
- $3$-transposition groups, Fischer spaces and Matsuo algebras
- Planar relations for derivations
- Derivations in $4$-generated subalgebras
- Subspaces of type $W(D_4)$.
- Subspaces of type $3^3\colon 2$.
- Subspaces of type $3^3:S_4$.
- Subspaces of type $2^6\colon\mathrm{SU}_3(2)^\prime$ and $[3^{10}]\colon 2$.
- Classification of near-solid lines in Matsuo algebras
- Automorphism groups of Matsuo algebras
- ...and 2 more sections
Key Result
Lemma 2.6
Let $(G,D)$ be a $3$-transposition group generated by three transpositions and no less. Then either $G \cong S_4$ or $G \cong \mathrm{SU}_3(2)'/Z$, where $Z$ is a central subgroup of $\mathrm{SU}_3(2)'$.
Figures (1)
- Figure 1: The affine plane of order $3$ and dual affine plane of order $2$.
Theorems & Definitions (62)
- Definition 2.1
- Definition 2.2
- Definition 2.3: JustinSergey
- Definition 2.4
- Example 2.5
- Lemma 2.6
- Proposition 2.7
- proof
- Lemma 2.8
- proof
- ...and 52 more