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arXiv · 2102.04493

Determining when an algebra is an evolution algebra

Miguel D. Bustamante·Pauline Mellon·M. Victoria Velasco
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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.

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Miguel D. Bustamante, Pauline Mellon, M. Victoria Velasco. 2021-02-08. Determining when an algebra is an evolution algebra. https://doi.org/10.3390/math8081349

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