Two dimensional perfect evolution algebras over domains

TL;DR

This paper classifies two-dimensional perfect evolution algebras over domains, introducing new concepts like quasiperfect algebras and a colored graph approach to distinguish isomorphism classes.

Contribution

It extends the theory of evolution algebras over fields to domains, introduces quasiperfect algebras, and develops a colored graph classification method.

Findings

Classification of 2D perfect evolution algebras over domains

Introduction of quasiperfect algebra concept

Development of a colored graph classification method

Abstract

We will study evolution algebras $A$ which are free modules of dimension $2$ over domains. Furthermore, we will assume that these algebras are perfect, that is $A^2=A$. We start by making some general considerations about algebras over domains: they are sandwiched between a certain essential $D$-submodule and its scalar extension over the field of fractions of the domain. We introduce the notion of quasiperfect algebras and modify slightly the procedure to associate a graph to an evolution algebra over a field given in \cite{ElduqueGraphs}. Essentially, we introduce color in the connecting arrows, depending on a suitable criterion related to the squares of the natural basis elements. Then we classify the algebras under scope parametrizing the isomorphic classes by convenient moduli.