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×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.

Solving the problem of simultaneous diagonalization of complex symmetric matrices via congruence

“…Recently in our paper [3] the problem of determining when a finite dimensional algebra is an evolution algebra was shown to be equivalent to the SDC problem for the structure matrices associated with any given basis. We recall that an evolution algebra is defined as a commutative algebra A for which there exists a basis B * = {e * i : i ∈ Λ} such that e * i e * j = 0, for every i, j ∈ Λ with i = j.…”

“…They were introduced in [18] in the study of non-Mendelian genetics, and the foundations of the theory were provided in 2008 [19]. In [3] we determined when a given algebra A is an evolution algebra. In other words, if B is a basis of A and the multiplication table of A with respect to B is not diagonal, we established the conditions under which there exists a natural basis B * of A, giving A the structure of an evolution algebra.…”

“…. , n. A main result in [3] proves that A is an evolution algebra if, and only if, M 1 (B), ..., M n (B) are simultaneously diagonalisable via congruence.…”

Simultaneous orthogonalization of inner products in infinite-dimensional vector spaces

“…Historically the research about the problem of simultaneous orthogonalization of two inner products over a K-vector space of finite dimension, with K a field satisfying different restrictions, has been largely studied by several authors, see, for example, the works [5], [3], [12], [11], [7] and [1]. The analogous problem for a family with two or more inner products over a K-vector space of finite dimension has been considered in [4] when the ground field is the real or complex numbers. In general, for an arbitrary field K and a finite-dimensional vector space over K, a thorough study has been recently realised in [2].…”

Simultaneous orthogonalization of inner products over arbitrary fields

“…Recently the work [3] deals also with the problem of simultaneously orthogonalization of inner products on finite-dimensional vector spaces over K = R or C. One of the motivations in [3] is that of detecting when a given algebra is an evolution algebra. The issue is equivalent to the simultaneous orthogonalization of a collection of inner products (the projections of the product in the lines generated by each element of a given basis of the algebra).…”