# D\'erivations dans les alg\`ebres d'\'evolution \`a puissances associatives

## TL;DR

This paper studies derivations in power-associative evolution algebras, focusing on nilalgebras, and provides explicit calculations for indecomposable cases up to dimension six, including their derivation algebras and inner derivations.

## Contribution

It offers a detailed analysis and explicit descriptions of derivation algebras for indecomposable evolution nilalgebras, extending previous work to higher dimensions.

## Key findings

- Derived the derivation algebra for n-dimensional indecomposable associative evolution nilalgebras.
- Provided explicit descriptions of derivations and inner derivations for nilalgebras up to dimension 6.
- Showed that derivations in decomposable algebras can be reduced to those in indecomposable components.

## Abstract

In this paper, we investigate the derivations in evolution algebras that are power-associative. This problem is reduced to that of power-associative evolution nilalgebras. We show how to calculate derivations in decomposable algebras. This caculation shows that it is enough to describe derivations in indecomposable evolution algebras. We first determine the derivation algebra of $n$-dimensional indecomposable associative evolution nilalgebras with one-dimensional annihilator. We describe the derivation algebra of indecomposable nilalgebras, up to dimension $6$, that are associative or not. In each cases, we give the commutator of two derivations. We also describe the ideal of inner derivations.