Evolution algebras with one-dimensional square

TL;DR

This paper classifies evolution algebras with one-dimensional squares by leveraging inner product space theory, resulting in three main classes based on the properties of a generator of the square.

Contribution

It introduces a classification framework for evolution algebras with one-dimensional squares using inner product space concepts, distinguishing three broad algebra classes.

Findings

Three classes of evolution algebras identified

Classification based on annihilator and isotropy properties

Provides a structural understanding of these algebras

Abstract

Evolution algebras with one dimensional square are classified using the theory of inner product spaces. More precisely, for $A$ an evolution algebra with $\dim(A^2) = 1$ and $a$ a generator of $A^2$, the product of $A$ is given by $xy = \langle x,y\rangle a$. Three broad classes of algebras are obtained: (1) $a\in\hbox{Ann}(A)$; (2) $a\notin\text{Ann}(A)$ and $a$ is isotropic relative to $\langle\cdot, \cdot\rangle$; (3) $a\notin\text{Ann}(A)$ and $a$ is anisotropic relative to $\langle\cdot, \cdot\rangle$.