Two dimensional perfect evolution algebras over domains

2022/04/18 by Yolanda Cabrera Casado, Casado, Yolanda Cabrera, Dolores Martı́n Barquero +3 · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2204.08410

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 A²=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 \citeElduqueGraphs. 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.

Cited by

• On formal deformations and degenerations of evolution algebras

Related

• Classification of 2-dimensional evolution algebras, their groups of automorphisms and derivation algebras
• On simple evolution algebras of dimension two and three. Constructing simple and semisimple evolution algebras
• On evolution algebras
• Evolution algebras of arbitrary dimension and their decompositions
• On some structural properties of evolution algebras