On simple evolution algebras of dimension two and three. Constructing simple and semisimple evolution algebras

TL;DR

This paper classifies three-dimensional simple evolution algebras over arbitrary fields using graph and algebraic tools, providing explicit models and methods to construct higher-dimensional simple evolution algebras.

Contribution

It introduces explicit models for three-dimensional simple evolution algebras and offers a method to construct higher-dimensional simple evolution algebras from smaller ones.

Findings

Complete classification of 3D simple evolution algebras over any field.

Explicit construction models for these algebras.

Method to build higher-order simple evolution algebras from smaller ones.

Abstract

This work classifies three-dimensional simple evolution algebras over arbitrary fields. For this purpose, we use tools such as the associated directed graph, the moduli set, inductive limit group, Zariski topology and the dimension of the diagonal subspace. Explicitly, in the three-dimensional case, we construct some models $_i\mathbf{III}_{\lambda_1,\ldots,\lambda_n}^{p,q}$ of such algebras with $1\le i\le 4$, $\lambda_i\in\mathbb{K}^\times$, $p,q\in\mathbb{N}$, such that any algebra is isomorphic to one (and only one) of the given in the models and we further investigate the isomorphic question within each one. Moreover, we show how to construct simple evolution algebras of higher-order from known simple evolution algebras of smaller size.