On primitive axial decomposition algebras of Majorana type with degenerate eigenvalues

TL;DR

This paper introduces and classifies a specific class of axial decomposition algebras of Majorana type with three eigenvalues, focusing on cases where the eigenvalue ta is outside the set ,1,, providing new insights into their structure.

Contribution

It classifies a new class of axial decomposition algebras of Majorana type with specific eigenvalues, expanding understanding of their algebraic structure and automorphisms.

Findings

Classified axial decomposition algebras with ta,1,.

Included specializations of 2-generated Majorana type algebras.

Identified structural properties related to Miyamoto automorphisms.

Abstract

A class of axial decomposition algebras with Miyamoto group generated by two Miyamoto automorphisms and three eigenvalues $0,1$ and $\eta$ is introduced and classified in the case with $\eta\notin\{0,1,\frac{1}{2}\}$. This class includes specializations of 2-generated axial algebras of Majorana type $(\xi,\eta)$ to the case with $\xi=\eta$.