On real chains of evolution algebras

TL;DR

This paper investigates chains of evolution algebras linked to permutations, characterizing their triviality, nilpotency, and properties like baric and idempotent elements over time.

Contribution

It introduces a new framework for chains of evolution algebras associated with permutations and provides classifications and constructions for various dimensions.

Findings

A chain of evolution algebras is trivial iff the permutation has no fixed points.

A chain is nilpotent at all times iff it is trivial.

Behavior of baric property and idempotent elements varies over time.

Abstract

In this paper we define a chain of $n$-dimensional evolution algebras corresponding to a permutation of $n$ numbers. We show that a chain of evolution algebras (CEA) corresponding to a permutation is trivial (consisting only algebras with zero-multiplication) iff the permutation has not a fixed point. We show that a CEA is a chain of nilpotent algebras (independently on time) iff it is trivial. We construct a wide class of chains of 3-dimensional EAs and a class of symmetric $n$-dimensional CEAs. A construction of arbitrary dimensional CEAs is given. Moreover, for a chain of 3-dimensional EAs we study the behavior of the baric property, the behavior of the set of absolute nilpotent elements and dynamics of the set of idempotent elements depending on the time.