Few remarks on evolution algebras

TL;DR

This paper investigates algebraic properties of evolution algebras, focusing on special types like idempotents and nilpotent elements, and explores their structure and associative enveloping algebras.

Contribution

It introduces reductions to specific algebra types and analyzes three-dimensional cases with infinite-period basis elements, advancing understanding of evolution algebra structures.

Findings

Reduction of permutation evolution algebras to special types

Characterization of three-dimensional evolution algebras with infinite-period basis elements

Description of associative enveloping algebras for certain evolution algebras

Abstract

In the present paper we study some algebraic properties of evolution algebras. Moreover, we reduce the study of evolution algebras of permutations to two special types of evolution algebras, idempotents and absolute nilpotent elements of the algebra. We study three-dimensional evolution algebras whose each element of evolution basis has infinite period. In addition, for an evolution algebra with some properties we describe its associative enveloping algebra.