Classifying Evolution Algebras of Dimensions Two and Three

M Eugenia Celorrio,M Victoria Velasco

doi:10.3390/math7121236

Abstract

We classified evolution algebras of dimensions two and three. Evolution algebras of dimensions three were classified recently obtaining 116 non-isomorphic types of algebras. Herein, with a new approach, we classify these algebras into 14 non-isomorphic types of algebra, so that this new classification is easier to deal with.

Highlights

Mathematics and biology are intimately tied, and genetic algebras are an example of this link, as these are algebras with biological meanings
In this paper we are going to work with evolution algebras, kinds of genetic algebra introduced by Tian in [1] in 2008 that are used to model non-Mendelian genetics laws, this is not their only application
In [2], the authors studied the relationship between evolution algebras and the spaces of functions defined by the Gibbs measure of a graph, which led into direct applications in biology, physics and mathematics itself

Summary

Introduction

Mathematics and biology are intimately tied, and genetic algebras are an example of this link, as these are algebras with biological meanings. In this paper we are going to work with evolution algebras, kinds of genetic algebra introduced by Tian in [1] in 2008 that are used to model non-Mendelian genetics laws, this is not their only application They are strongly connected with group theory, Markov processes, the theory of knots, dynamic systems and graph theory. In [2], the authors studied the relationship between evolution algebras and the spaces of functions defined by the Gibbs measure of a graph, which led into direct applications in biology, physics and mathematics itself. Mathematics 2019, 7, 1236 dimension three were classified into 116 types of non-isomorphic evolution algebras. We classify three-dimensional evolution algebras into 14 non-isomorphic types (Theorem 11). Since we reduce the study of evolution algebras of dimension three to 14 non-isomorphic types, this classification is much more practical than the classification provided in [17]. As a matter of fact, Markov processes are evolution algebras whose structure matrix is stochastic

Basic Background

Classification of Two-Dimensional Evolution Algebras

Classification of Three-Dimensional Evolution Algebras

The Non-Degenerate Case

The Degenerate Case

The Main Result