Determining when an algebra is an evolution algebra

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 $\{M_1,\dots ,M_n\}$ 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 intriguing as evolution algebras model asexual reproduction unlike the classical ones.

References in corpus (5)

• Simultaneous Diagonalization of Matrices and its Applications in Quadratically Constrained Quadratic Programming
• Some properties of evolution algebras
• The derivations of some evolution algebras
• Nilpotent evolution algebras over arbitrary fields
• Solving the problem of simultaneous diagonalization of complex symmetric matrices via congruence

Cited by in corpus (3)

• A historical perspective of Tian's evolution algebras
• Solving the problem of simultaneous diagonalization of complex symmetric matrices via congruence
• Non-degenerate evolution algebras