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Skew Axial Algebras of Monster Type

Michael Turner

Abstract

Skew axets were first defined by McInroy and Shpectorov where they used the term of axets to classify shapes of an algebra. When they first submitted their paper, it was not known if skew axial algebras exist and now we will present such examples with axet $X'(1+2)$. Looking at $2$-generated primitive axial algebras of Monster type, we will be able to state and prove the classification of such algebras with axet $X'(1+2)$. We will conclude by looking at larger skew axets and give a suggestion on how one could extend the classification.

Skew Axial Algebras of Monster Type

Abstract

Skew axets were first defined by McInroy and Shpectorov where they used the term of axets to classify shapes of an algebra. When they first submitted their paper, it was not known if skew axial algebras exist and now we will present such examples with axet . Looking at -generated primitive axial algebras of Monster type, we will be able to state and prove the classification of such algebras with axet . We will conclude by looking at larger skew axets and give a suggestion on how one could extend the classification.
Paper Structure (19 sections, 23 theorems, 81 equations, 2 figures, 6 tables)

This paper contains 19 sections, 23 theorems, 81 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Prerequisites
  3. Axial Algebras
  4. Axets
  5. The Known Examples
  6. Three Dimensions
  7. A Variation
  8. Four Dimensions
  9. In characteristic not equal to 5
  10. In characteristic 5
  11. The Foundations
  12. Skew Relations
  13. Proof of Theorem \ref{['Theorem']}
  14. Orthogonal axes
  15. Non-Orthogonal Case
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\mathbb{F}$ be a field of characteristic not equal to $2$. Suppose $(A,X)$ is a $2$-generated primitive $\mathcal{M}(\alpha,\beta)$-axial algebra over $\mathbb{F}$ and $\langle X\rangle$ is isomorphic to $X'(1+2)$. Then either

Figures (2)

  • Figure 1: $X'(1+2)$
  • Figure 2: $X'(2+4)$

Theorems & Definitions (42)

  • Theorem 1.1
  • Remark
  • Corollary 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.6
  • Definition 2.7
  • Example 2.8
  • ...and 32 more