A class of nilpotent evolution algebras

TL;DR

This paper extends a recent technique for classifying nilpotent evolution algebras to higher dimensions, constructing new examples and analyzing the completeness of the classification.

Contribution

It develops the classification technique for high-dimensional nilpotent evolution algebras and constructs new examples beyond previous cases.

Findings

Constructed nilpotent evolution algebras of arbitrary type

Identified limitations of the existing classification technique

Extended the classification framework to higher dimensions

Abstract

Recently, by A. Elduque and A. Labra a new technique and a type of an evolution algebra are introduced. Several nilpotent evolution algebras defined in terms of bilinear forms and symmetric endomorphisms are constructed. The technique then used for the classification of the nilpotent evolution algebras up to dimension five. In this paper we develop this technique for high dimensional evolution algebras. We construct nilpotent evolution algebras of any type. Moreover, we show that, except the cases considered by Elduque and Labra, this construction of nilpotent evolution algebras does not give all possible nilpotent evolution algebras.