CHAINS OF THREE-DIMENSIONAL EVOLUTION ALGEBRAS: A DESCRIPTION

Abstract

In this paper we describe locally all the chains of three-dimensional evolution algebras (3-dimensional CEAs). These are families of evolution algebras with the property that their structure matrices respect to a certain natural basis satisfy the Chapman-Kolmogorov equation. We do it by describing all 3-dimensional CEAs whose structure matrices have a range equal to 3,2 and 1 respectively. Later we show that arbitrary CEAs are locally CEAs with a fix rank. Since every evolution algebra can be regarded as a weighted digraph, this allows us to understand and visualize time-dependent weighed digraphs with 3 nodes.