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Determining When an Algebra Is an Evolution Algebra

Author

Listed:
  • Miguel D. Bustamante

    (School of Mathematics and Statistics, University College Dublin, Dublin 4, Ireland
    These authors contributed equally to this work.)

  • Pauline Mellon

    (School of Mathematics and Statistics, University College Dublin, Dublin 4, Ireland
    These authors contributed equally to this work.)

  • M. Victoria Velasco

    (Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
    These authors contributed equally to this work.)

Abstract

Evolution algebras are non-associative algebras that describe non-Mendelian hereditary processes and have connections with many other areas. In this paper, we obtain necessary and sufficient conditions for a given algebra A to be an evolution algebra. We prove that the problem is equivalent to the so-called SDC problem, that is, the simultaneous diagonalisation via congruence of a given set of matrices. More precisely we show that an n -dimensional algebra A is an evolution algebra if and only if a certain set of n symmetric n × n matrices { M 1 , … , M n } describing the product of A are SDC. We apply this characterisation to show that while certain classical genetic algebras (representing Mendelian and auto-tetraploid inheritance) are not themselves evolution algebras, arbitrarily small perturbations of these are evolution algebras. This is intringuing, as evolution algebras model asexual reproduction, unlike the classical ones.

Suggested Citation

  • Miguel D. Bustamante & Pauline Mellon & M. Victoria Velasco, 2020. "Determining When an Algebra Is an Evolution Algebra," Mathematics, MDPI, vol. 8(8), pages 1-14, August.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:8:p:1349-:d:398069
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    Cited by:

    1. Farrukh Mukhamedov & Izzat Qaralleh, 2023. "On Extendibility of Evolution Subalgebras Generated by Idempotents," Mathematics, MDPI, vol. 11(12), pages 1-18, June.

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