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.
Highlights
Evolution algebras are non-associative algebras with a dynamic nature
We prove that some classical genetic algebras such as the gametic algebra for simple Mendelian inheritance (Example 2) or the gametic algebra for auto-tetraploid inheritance (Example 5) are not evolution algebras
In this paper we determine completely whether a given algebra A is an evolution algebra, by translating the question to a recently solved problem, namely the problem of simultaneous diagonalisation via congruence of the m-structure matrices of A. This is relevant because evolution algebras have strong connections with areas such as group theory, Markov processes, theory of knots, and graph theory, among others
Summary
Evolution algebras are non-associative algebras with a dynamic nature. They were introduced in by Tian [1] to enlighten the study of non-Mendelian genetics. A useful characterisation of the property of being an evolution algebra is given in the particular case that one of the multiplication structure matrices is invertible. We recall that an n-dimensional evolution algebra is a commutative algebra A for which there exists a basis B∗ = {e1∗ , ..., en∗ } such that ei∗ e∗j = 0 for every i, j ∈ {1, · · · , n} with i 6= j. An evolution algebra is an algebra A provided with a basis B∗ = {e1∗ , ..., en∗ } such that the corresponding m-structure matrices M1 ( B∗ ) = (π1 (ei∗ e∗j )), · · · , Mn ( B∗ ) = (πn (ei∗ e∗j )) are diagonal. M1 , . . . , Mn (regarded as complex matrices) are simultaneously diagonalisable via congruence
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