Determining When an Algebra Is an Evolution Algebra

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 s...

Full description

Saved in:
Bibliographic Details
Published in:Mathematics (Basel) 2020-08, Vol.8 (8), p.1349
Main Authors: Bustamante, Miguel, Mellon, Pauline, Velasco, M.
Format: Article
Language:English
Subjects:
Algebra
Evolution
evolution algebra
Inheritances
linear pencil
Mathematics
multiplication structure matrices
Signal processing
simultaneous diagonalisation by congruence
simultaneous diagonalisation by similarity
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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 {M1,…,Mn} 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.
ISSN:2227-7390
2227-7390
DOI:10.3390/math8081349