On the characterization of the space of derivations in evolution algebras

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.

Characterization theorems for the spaces of derivations of evolution algebras associated to graphs

“…Then in [11] Tian studies the connection between this type of algebra and other structures such as graphs, Markov chains, groups, dynamical systems, among others. For a review of some advances in this type of algebra we refer the reader to [1][2][3][4][5][6][7][8][9][10]13,14] and the references therein.…”

A Characterization of Associative Evolution Algebras

“…In this section we provide a result on the space of derivations of evolution algebras, in the case of these being associative, that can be added to those already obtained regarding this concept in the case of evolution algebras in general (see [11][12][13][14][15] in this respect, for instance).…”

“…Examples of usual topics in the literature, which have proven to be a very convenient approach, are the study of derivation spaces [11][12][13][14][15] and the classification of some family of evolution algebras sharing interesting properties. For example, nilpotent evolution algebras are characterized in [16][17][18] and power-associative evolution algebras are classified in [19], up to dimension six.…”

Hilbert evolution algebras and its connection with discrete-time Markov chains

“…Here we mention some of the recent works, and we refer the reader to the references therein for a deeper study of the theory. In [3][4][5][6][7] the reader may find a survey of properties and results of general evolution algebras; the works in [1,2,8,14] are devoted to the connections between evolution algebras and graphs together with some related properties; and in [10,12] one may see a good review of results with relevance in genetics and other applications.…”