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 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.