Determining When an Algebra Is an Evolution Algebra

Mathematics

M.Victoria Velasco¹  Pauline Mellon²  MiguelD. Bustamante² 

[1] Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain;School of Mathematics and Statistics, University College Dublin, Dublin 4, Ireland;

关键词: evolution algebra;    multiplication structure matrices;    simultaneous diagonalisation by congruence;    simultaneous diagonalisation by similarity;    linear pencil;   

DOI  :  10.3390/math8081349

来源: DOAJ

【 摘 要 】

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 $n\times n$ matrices $\{M_1,\dots ,M_n\}$ 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.

【 授权许可】

Unknown