On nilpotent index and dibaricity of evolution algebras

TL;DR

This paper characterizes nilpotent evolution algebras, providing criteria for maximal nilpotent index, classifying finite-dimensional cases, and exploring their dibaricity properties.

Contribution

It establishes a criterion for maximal nilpotent index, classifies complex evolution algebras with this property, and analyzes dibaricity of real evolution algebras.

Findings

Maximal nilpotent index is $2^{n-1}+1$ for certain nilpotent evolution algebras.

Classified finite-dimensional complex evolution algebras with maximal nilpotent index.

Proved nilpotent evolution algebras are not dibaric and provided criteria for dibaricity in 2D cases.

Abstract

An evolution algebra corresponds to a quadratic matrix $A$ of structural constants. It is known the equivalence between nil, right nilpotent evolution algebras and evolution algebras which are defined by upper triangular matrices $A$. We establish a criterion for an $n$-dimensional nilpotent evolution algebra to be with maximal nilpotent index $2^{n-1}+1$. We give the classification of finite-dimensional complex evolution algebras with maximal nilpotent index. Moreover, for any $s=1,...,n-1$ we construct a wide class of $n$-dimensional evolution algebras with nilpotent index $2^{n-s}+1$. We show that nilpotent evolution algebras are not dibaric and establish a criterion for two-dimensional real evolution algebras to be dibaric.