On $\exp(Der(E))$ of nilpotent evolution algebras

TL;DR

This paper studies the exponential of the derivation algebra of nilpotent evolution algebras, showing it forms a normal subgroup of automorphisms and calculating its index, with implications for algebraic structure analysis.

Contribution

It introduces a Banach algebra norm on evolution algebras, describes derivations and automorphisms, and analyzes the structure of the exponential of derivations in nilpotent cases.

Findings

$ ext{exp}(Der(E))$ is a normal subgroup of $Aut(E)$

The index of $ ext{exp}(Der(E))$ in $Aut(E)$ is computed

Provides structural insights into nilpotent evolution algebras

Abstract

In the present paper, every evolution algebra is endowed with Banach algebra norm. This together with the description of derivations and automorphisms of nilpotent evolution algebras, allows to investigated the set $\exp(Der(E))$. Moreover, it is proved that $\exp(Der(E))$ is a normal subgroup of $Aut(E)$, and its corresponding index is calculated.