On evolution algebras

TL;DR

This paper studies the structure and classification of evolution algebras, establishing equivalences, classifying 2D cases, and providing an algorithm to determine isomorphism between finite-dimensional examples.

Contribution

It introduces a classification of 2D complex evolution algebras and develops an algorithm to decide isomorphism, linking algebraic properties to matrix forms.

Findings

Equivalence between nil, right nilpotent, and upper triangular matrix-based evolution algebras.

Complete classification of 2-dimensional complex evolution algebras.

An algorithm in Mathematica to determine isomorphism between finite-dimensional evolution algebras.

Abstract

The structural constants of an evolution algebra is given by a quadratic matrix $A$. In this work we establish equivalence between nil, right nilpotent evolution algebras and evolution algebras, which are defined by upper triangular matrix $A$. The classification of 2-dimensional complex evolution algebras is obtained. For an evolution algebra with a special form of the matrix $A$ we describe all its isomorphisms and their compositions. We construct an algorithm running under Mathematica which decides if two finite dimensional evolution algebras are isomorphic.