Determining When an Algebra Is an Evolution Algebra

Miguel D. Bustamante, Pauline Mellon and M. Victoria Velasco
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Miguel D. Bustamante: School of Mathematics and Statistics, University College Dublin, Dublin 4, Ireland
Pauline Mellon: School of Mathematics and Statistics, University College Dublin, Dublin 4, Ireland
M. Victoria Velasco: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain

Mathematics, 2020, vol. 8, issue 8, 1-14

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.

Keywords: evolution algebra; multiplication structure matrices; simultaneous diagonalisation by congruence; simultaneous diagonalisation by similarity; linear pencil (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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