Some properties of evolution algebras

TL;DR

This paper explores properties of finite dimensional complex evolution algebras, including their classification, nilpotency criteria, maximal nilpotency index, and a connection to graph planarity.

Contribution

It provides new criteria for nilpotency, classifies evolution algebras via Jordan form matrices, and links algebra properties to graph planarity.

Findings

Classification of evolution algebras with Jordan form matrices

Nilpotency criteria based on matrix properties

Maximal nilpotency index for nilpotent evolution algebras is 1+2^{n-1}

Abstract

The paper is devoted to the study of finite dimensional complex evolution algebras. The class of evolution algebras isomorphic to evolution algebras with Jordan form matrices is described. For finite dimensional complex evolution algebras the criteria of nilpotency is established in terms of the properties of corresponding matrices. Moreover, it is proved that for nilpotent $n-$dimensional complex evolution algebras the possible maximal nilpotency index is $1+2^{n-1}.$ The criteria of planarity for finite graphs is formulated by means of evolution algebras defined by graphs.