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

TL;DR

This paper characterizes pseudo-composition and train algebras of rank 3 as axial algebras with specific fusion laws, and describes their small subalgebras to aid classification.

Contribution

It introduces a new perspective by linking these algebras to axial algebras with fusion laws from Peirce decompositions, advancing their structural understanding.

Findings

Pseudo-composition and train algebras of rank 3 are characterized as $ ext{PC}( exteta)$-axial algebras.

Descriptions of 2- and 3-generated subalgebras are provided.

The work lays groundwork for classifying these algebras.

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 $\mathcal{PC}(\eta)$-axial algebras, where $\eta$ is an element of the ground field. As a first step towards their classification, we describe $2-$ and $3$-generated subalgebras of such algebras.