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

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.