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Axial view on pseudo-composition algebras and train algebras of rank 3

Ilya Gorshkov, Andrey Mamontov, Alexey Staroletov

Abstract

We show that pseudo-composition algebras and train algebras of rank 3 generated by idempotents are characterized as axial algebras with fusion laws derived from the Peirce decompositions of idempotents in these classes of algebras. The corresponding axial algebras are called $\mathcal{PC}(η)$-axial algebras, where $η$ is an element of the ground field. As a first step towards their classification, we describe $2-$ and $3$-generated subalgebras of such algebras.

Axial view on pseudo-composition algebras and train algebras of rank 3

Abstract

We show that pseudo-composition algebras and train algebras of rank 3 generated by idempotents are characterized as axial algebras with fusion laws derived from the Peirce decompositions of idempotents in these classes of algebras. The corresponding axial algebras are called -axial algebras, where is an element of the ground field. As a first step towards their classification, we describe and -generated subalgebras of such algebras.
Paper Structure (4 sections, 22 theorems, 55 equations, 4 tables)

This paper contains 4 sections, 22 theorems, 55 equations, 4 tables.

Table of Contents

  1. Introduction
  2. $\mathcal{PC}(\eta)$-axial algebras generated by two primitive axes
  3. Proof of Theorem \ref{['th:1']}
  4. $\mathcal{PC}(\eta)$-axial algebras generated by three primitive axes

Key Result

Theorem 1.1

Suppose that $\mathbb{F}$ is a field of characteristic not $2$ or $3$. Suppose that $A$ is a primitive $\mathcal{PC}(\eta)$-axial algebra over $\mathbb{F}$, where $\eta\in\mathbb{F}$ and $\eta\not\in\{1, \frac{1}{2}\}$. Then the following statements hold:

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1
  • proof
  • Corollary 1
  • proof
  • Remark 1
  • Proposition 2
  • proof
  • Proposition 3
  • ...and 31 more