Determining when an algebra is an evolution algebra

TL;DR

This paper provides necessary and sufficient conditions for an algebra to be an evolution algebra, linking the problem to simultaneous diagonalisation via congruence of matrices, and explores implications for genetic algebras.

Contribution

It establishes a new characterization of evolution algebras through matrix simultaneous diagonalisation, connecting algebraic properties with matrix theory.

Findings

Characterization of evolution algebras via SDC problem.

Classical genetic algebras are not evolution algebras, but small perturbations can be.

Evolution algebras model asexual reproduction, unlike classical genetic algebras.

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 $n\times n$ matrices $\{M_{1}, \ldots, M_{n}\}$ 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 intriguing as evolution algebras model asexual reproduction unlike the classical ones.