On the characterization of the space of derivations in evolution algebras

TL;DR

This paper investigates the structure of derivations in finite-dimensional evolution algebras, revealing how the associated graph's structure influences derivation spaces and providing classifications for specific cases.

Contribution

It offers new insights into how the twin partition of the associated graph determines the derivation space, including conditions for simplification and classification in 3-dimensional cases.

Findings

Derivation space is zero for twin-free associated graphs.

Sufficient conditions are provided for simplifying derivation analysis.

Classification of derivations for 3-dimensional evolution algebras is achieved.

Abstract

We study the space of derivations for some finite-dimensional evolution algebras, depending on the twin partition of an associated directed graph. For evolution algebras with a twin-free associated graph we prove that the space of derivations is zero. For the remaining families of evolution algebras we obtain sufficient conditions under which the study of such a space can be simplified. We accomplish this task by identifying the null entries of the respective derivation matrix. Our results suggest how strongly the associated graph's structure impacts in the characterization of derivations for a given evolution algebra. Therefore our approach constitutes an alternative to the recent developments in the research of this subject. As an illustration of the applicability of our results we provide some examples and we exhibit the classification of the derivations for non-degenerate irreducible $3$-dimensional evolution algebras.