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 three-dimensional evolution algebras.

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Acknowledgements

Part of this work was carried out during a visit of Y.C., M.L.R and P.M.R. at the Universidade Federal do ABC—UFABC in Santo André, SP, Brazil. They are grateful for their hospitality and support. The authors also thank the Fundação Getulio Vargas—FGV, Rio de Janeiro, RJ, Brazil, host organization of the ILAS 2019 conference, for their hospitality. Special thanks are also due to the anonymous reviewer for his/her helpful comments and suggestions.

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Correspondence to Pablo M. Rodriguez.

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This work was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo—FAPESP (Grants 2016/11648-0, 2017/10555-0, 2018/06925-0), and Conselho Nacional de Desenvolvimento Científico e Tecnológico—CNPq (Grant 304676/2016-0).

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Cabrera Casado, Y., Cadavid, P., Rodiño Montoya, M. et al. On the characterization of the space of derivations in evolution algebras. Annali di Matematica (2020). https://doi.org/10.1007/s10231-020-01012-2

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Keywords

  • Genetic algebra
  • Evolution algebra
  • Derivation
  • Graph
  • Twin partition

Mathematics Subject Classification

  • 17A36
  • 05C25
  • 17D92