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.


Introduction
This is a postprint of our work published in: Mathematics 2020, 8,1349. https://doi.org/10.3390/ math8081349. It contains a few minor modifications on pages 2 and 4.
Evolution algebras are non-associative algebras with a dynamic nature. They were introduced in 2008 by Tian [1] to enlighten the study of non-Mendelian genetics. Since then, a large literature has flourished on this topic (see for instance [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]) motivated by the fact that these algebras have connections with group theory, Markov processes, theory of knots, systems and graph theory. For instance, in [2], the theory of evolution algebras was related to that of pulse processes on weighted digraphs and applications were provided by reviewing a report of the National Science Foundation about air pollution achieved by the Rand Corporation. A pulse process is a structural dynamic model to analyse complex networks by studying the propagation of changes through the vertices of a weighted digraph, after introducing an initial pulse in the system at a particular vertex. It is based on a spectral analysis of the corresponding weighted digraph to facilitate large scale decision making processes. Evolution algebras also introduce useful algebraic techniques into the study of some digraphs because evolution algebras and weighted digraphs can be canonically identified.
We recall that an algebra is a linear space A provided with a product, that is, a bilinear map from A × A to A via the operation (a, b) → ab. In the particular case that (ab)c = a(bc), for all a, b, c ∈ A we say that A is associative. Meanwhile, if ab = ba, for all a, b ∈ A, then we say that A is commutative.
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. Such a basis is called natural. Evolution algebras are, in general, non-associative. To date most literature on evolution algebras is on finite-dimensional ones. However, in [12] it is shown that every infinite-dimensional Banach evolution algebra is the direct sum of a finite-dimensional evolution algebra and a zero-product algebra.
In this paper we discuss necessary and sufficient conditions under which a given finite-dimensional commutative algebra is an evolution algebra, namely, we determine when such a finite-dimensional algebra can be provided with a natural basis. We tackle the problem constructively by assuming an arbitrary basis B with a multiplication table given by equation (2.1) below and then asking whether or not there is a change of basis from B to a natural basis B * . In Section 2, Theorem 1, we show that this problem is equivalent to the simultaneous diagonalisation via congruence of certain n × n symmetric matrices M 1 , . . . , M n , called the multiplication structure matrices obtained from the given multiplication table.
Finding concrete sufficient conditions for a given set of matrices to be simultaneously diagonalisable via congruence (we will refer to it as the SDC-problem) is one of the 14 open problems posted in 1990 by Hiriart-Urruty [18] (see also [19,20]). It has connections with other problems such as blind-source separation in signal processing [21][22][23][24]. The SDC-problem was solved recently for complex symmetric matrices in [25].
In Theorem 2 we show that if A is a real algebra and B is a basis of A then B also is a basis of A C , the complexification of A (with the same multiplication structure matrices) and that A is an evolution algebra if, and only if, A C is an evolution algebra and has a natural basis consisting of elements of A. This reduction of the real case to the complex one allows us to apply the results in [25] to both real and complex algebras.
In Theorem 5 we determine if a given algebra A whose annihilator is zero is an evolution algebra and in Theorem 6 we do the same if its annihilator is not zero. 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. In this case if M i 0 is invertible then A is an evolution algebra if, and only if, for each k = i 0 the matrix M −1 i 0 M k is diagonalisable by similarity and these matrices pairwise commute. Applications of these results are provided in the final section of this paper. They also show that the conditions in the mentioned results are neither redundant nor superfluous.
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. Nevertheless, both of these algebras can be deformed by means of a parameter ε > 0 to obtain an algebra A ε that is an evolution algebra for every value of the parameter ε, as shown in Example 3 and Example 6 respectively.

Characterising evolution algebras by means of simultaneous diagonalisation of matrices by congruence
An n-dimensional algebra A over a field K (= R or C) is determined by means of a basis B = {e 1 , . . . , e n } together with a multiplication table where m ijk := π k (e i e j ) and π k : A → K is the projection over the k-th coordinate, that is π k ( These basis-dependent coefficients m ijk are known as structure constants with respect to B (see [26]). For a basis B of A, the structure constants completely determine the algebra A, up to isomorphism.
If we organise the n 3 structure constants in n matrices by defining π k (e n e 1 ) π k (e n e n ) for k = 1, ..., n, then the product of A is given by where α T = (α 1 , . . . , α n ), β T = (β 1 , . . . , β n ) and T indicates the transpose operation. This motivates the following definition. We recall that an n-dimensional evolution algebra is a commutative algebra A for which there exists a basis B * = {e * 1 , ..., e * n } such that e * i e * j = 0 for every i, j ∈ {1, · · · , n} with i = j. Such a basis B * is said to be a natural basis of A.
The next result is a straightforward combination of the concept of evolution algebra with Definition 1.
Proof. M k (B * ) is diagonal for k = 1, ..., n, if, and only if, e * i e * j = 0, for every i = j, or equivalently if B * is a natural basis.
In the next theorem we characterise when a given algebra is an evolution algebra. To this end we recall the following property. It is worth remarking at this point that the general problem of diagonalisation via congruence considers m symmetric matrices of dimension n × n, where m need not be equal to n. This problem has applications in statistical signal processing and multivariate statistics [21][22][23][24] and was solved for complex symmetric matrices in [25]. Theorem 1. Let A be a commutative algebra over K with basis B = {e 1 , . . . , e n }. Let M 1 , . . . , M n be the m-structure matrices of A with respect to B. Then A is an evolution algebra if, and only if, the symmetric matrices M 1 , . . . , M n are simultaneously diagonalisable via congruence.

Proof.
A is an evolution algebra if, and only if, A has a natural basis, say B * = {e * 1 , ..., e * n } (that is a basis such that e * i e * j = 0 if i = j). Let P = (p ij ) be the change of basis matrix from B to B * (that is e * i = ∑ n k=1 p ki e k for i = 1, . . . , n). Then, by (2.3), where α = Pγ i and β = Pγ j with γ i = (0, ..., 0, and hence e * i e * j = 0 if i = j if, and only if, the matrix P T M k P is diagonal for k = 1, ..., n.
Since the problem of simultaneous diagonalisation of matrices via congruence was solved in [25] for complex symmetric matrices, we consider the following.
The complexification of a real algebra A is defined as the complex algebra Note that every basis B of A is trivially a basis of A C so that the real dimension of A and the complex dimension of A C coincide. Theorem 2. Let A be a real algebra. Then A is an evolution algebra if, and only if, A C is an evolution algebra and has a natural basis consisting of elements of A. Moreover, if A is a real evolution algebra then every natural basis of A is a natural basis of A C .
Proof. If A is an evolution algebra and if B is a natural basis of A then obviously B is a natural basis of A C . The converse direction is clear.

Corollary 1.
Let A be a real commutative algebra, B = {e 1 , ..., e n } a basis and M 1 , . . . , M n be the m-structure matrices of A with respect to B. Then A is an evolution algebra if, and only if, the matrices M 1 , . . . , M n (regarded as complex matrices) are simultaneously diagonalisable via congruence by means of a real matrix.
In [25], example 16, we give two real matrices which are diagonalisable via congruence by means of a complex matrix but not by means of any real matrix.

Reviewing the solution of the SDC problem
The aim of this subsection is to review the solution of the SDC problem, that is, determining when m matrices of size n × n are simultaneously diagonalisable via congruence, which was solved in [25] for complex matrices. All matrices considered in this section are complex.
From now on, let M n denote the set of all complex n × n matrices. Moreover, let MS n be the set of all symmetric matrices in M n and GL n be the set of nonsingular matrices in M n .
We recall the following definition of simultaneous diagonalisation of matrices via similarity (SDS), not to be confused with Definition 2 involving simultaneous diagonalisation via congruence (SDC). Nevertheless, the solution of the problem of determining when a set of complex matrices is SDC given in [25] is related to the problem of determining whether a certain set of related matrices is SDS, as we will show below.

Remark 1.
Concerning the statement of the above theorem in [27] we point out that the fact that the symmetric matrices N 1 , ..., N m commute guarantees that N 1 , ..., N m are simultaneously diagonalisable by similarity only when each of N 1 , ..., N m are diagonalisable matrices (and obviously not otherwise).
In [25], to solve the SDC problem, Theorem 3 and Theorem 4 below were proved. To state them, we recall the next definition.
Consequently, this supremum must be achieved so that there exists λ 0 ∈ C m with λ 0 = 1 such that and we say that r 0 is the maximum pencil rank of M 1 , ..., M m .
The next theorem corresponds to Theorem 7 in [25] and deals with the case when the maximum pencil rank of the matrices is n. Given 1 ≤ r < n, and matrices M r ∈ M r and N n−r ∈ M n−r , denote by M r ⊕ N n−r the n × n matrix given by When the pencil rank of M 1 , ..., M m ∈ MS n is strictly less than n then the SDC problem can be reduced to a similar one in a reduced dimension as the following result (Theorem 9 in [25]) shows.
Moreover, if either of the above conditions is satisfied, then the pencil D associated with the r × r matrices D 1 , ..., D m is non-singular. Indeed, if λ 0 ∈ C m with λ 0 = 1 is such that r = rank M(λ 0 ) then D(λ 0 ) ∈ GL r .

Checking when an algebra is an evolution algebra
We apply the above results to the m-structure matrices M 1 , . . . , M n of an algebra A with respect to a basis B = {e 1 , ..., e n } as in (2.2). For a real algebra A we consider the complexification A C provided with the same basis B.
We recall that the annihilator of an algebra A is the set Ann(A) = {b ∈ A : ab = ba = 0, for every a ∈ A}.
This set is an ideal of A.  Proof. Assume A is as stated. Then there exists λ 0 = 1 such that the pencil M(λ 0 ) is invertible if, and only if, the maximum pencil rank of M k ( B) is r. If this happens then dim(∩ n j=1 ker M j ( B)) = n − r, as dim Ann(A) = dim(∩ n j=1 ker M j ( B)) by Lemma 1. Therefore if M(λ 0 ) is invertible then, by Corollary 2, we have that M 1 , ..., M n are SDC if, and only if, each of the matrices M(λ 0 ) −1 M 1 , ..., M(λ 0 ) −1 M n is diagonalisable by similarity and they pairwise commute. Since the matrices M 1 , ..., M n are SDC (by P r ∈ GL r ) if, and only if, the matrices M 1 ( B), ..., M n ( B) are SDC (by P n := P r ⊕ I n−r ), the result follows from Theorem 1.

Remark 2.
The above result shows that the condition that A/Ann(A) be an evolution algebra is a necessary condition for A to be an evolution algebra. This is known because it was proved in [3] that the quotient of an evolution algebra by an ideal is an evolution algebra. However, Theorem 6 proves that this condition is not sufficient (which is new). In fact, if dim Ann(A) := r < n, and we consider a basis B, as in Theorem We conclude this section by providing a procedure, obtained from Theorems 1, 5, 3 and 6 above, to determine in a finite number of steps whether or not a given commutative algebra A with fixed basis B = {e 1 , ..., e n } is an evolution algebra. Let M 1 , ..., M n be the m-structure matrices of A with respect to B.
While one can try to check directly, see Example 1 below, if the matrices M 1 , ..., M n are SDC this is generally not easy to do. Alternatively, to determine if A is an evolution algebra we can proceed as follows.
Check if any one of the matrices M 1 , ..., M n is invertible. M n are all diagonalisable (by similarity) and they pairwise commute then we can conclude that A is an evolution algebra, and otherwise we conclude that A is not an evolution algebra.
(b) If none of the matrices M 1 , ..., M n is invertible then we determine Ann(A), that is, by means of (2.3), we describe those elements a ∈ A such that ae i = 0 for every i = 1, ..., n.
invertible as an r × r matrix. In particular, this is the case whenever M i 0 is invertible for some 1 ≤ i 0 ≤ n (in which case we can choose M(λ 0 ) = M i 0 ). If such a λ 0 does not exist then we conclude that A is not an evolution algebra. Otherwise, we have that A is an evolution algebra if, and only if, the matrices M(λ 0 ) −1 M 1 , ..., M(λ 0 ) −1 M n are all diagonalisable (by similarity) and they pairwise commute.

Some examples and applications
We discuss some examples where our approach is useful to determine whether or not certain classical genetic algebras are evolution algebras. Mostly these algebras are defined in the literature as real algebras but, in our case, they can be regarded as complex algebras (with the same basis, and hence with the same m-structure matrices) as shown in Theorem 2 and Corollary 1.
We will consider the class of gametic algebras discussed by Etherington [28]. Gametic algebras, widely used in genetics, are simply baric algebras: they are endowed with a weight function. While further background is not necessary to decide if these algebras are evolution algebras or not, we nevertheless refer the reader to [29] and [30] for a review of these algebras.
Since for P = 1 1 1 −1 we have thatP T M 1 P = 2 0 0 2 and P T M 2 P = 2 0 0 −2 , by Theorem 1, we obtain that A is an evolution algebra. In fact, B = { e 1 , e 2 }, with e 1 = e 1 − e 2 and e 2 = e 1 + e 2 , is a natural basis of A, as e 1 e 2 = 0. Therefore the algebra A with basis B = {e 1 , e 2 , e 3 } and product e 2 1 = e 1 + e 3 , e 2 2 = e 1 − e 3 , e 2 3 = 0, e 1 e 2 = e 2 = e 2 e 1 , e 1 e 3 = e 3 e 1 = e 2 e 3 = e 3 e 2 = 0 is an algebra such that Ann( A) = Ke 3 . By Theorem 6 (see also Remark 2) we have that A is therefore not an evolution algebra whereas A/Ann( A) is an evolution algebra isomorphic to the evolution algebra A in Example 1.
Example 2 (Gametic algebra for simple Mendelian inheritance). Let A 0 denote a commutative 2-dimensional algebra over R, corresponding to the gametic algebra describing simple Mendelian inheritance (see [30]). In terms of the basis B = {e 1 , e 2 } the multiplication table is e 2 1 = e 1 , e 1 e 2 = e 2 e 1 = The associated m-structure matrices M 1 , M 2 can be read off easily: It is easy to check that A 0 is a baric algebra, with weight function defined by ξ(e 1 ) = ξ(e 2 ) = 1. Note that is not diagonalisable by similarity, as λ = −1 is the unique eigenvalue and the associated eigenspace has dimension 1. Therefore, by Corollary 3, we obtain that A 0 is not an evolution algebra. (This last assertion can also be deduced from Theorem 1, with more tedious calculations, by directly checking that M 1 and M 2 are not SDC).
We will now deform this algebra in order to construct an evolution algebra.

Example 3 (Evolution algebra for deformed Mendelian inheritance)
. Consider a deformation of the algebra A 0 of the previous example. We denote these deformed algebras by A ε , which depend on the free parameter ε ∈ R. In terms of the basis B = {e 1 , e 2 }, the multiplication table for A ε is given by The associated m-structure matrices M 1 , M 2 are now: For genetic applications we restrict 0 < ε ≤ 1 so that all coefficients in these matrices are non-negative. Moreover, A ε is baric with weight function defined by ξ(e 1 ) = ξ(e 2 ) = 1, for any ε. In fact ξ(e i e j ) = ξ(e i )ξ(e j ) = 1, for i, j = 1, 2. Obviously, the undeformed case corresponds to ε = 0. Let us consider whether A ε is an evolution algebra by using Theorem 5. First of all, the maximal rank of the linear pencil M(λ) = λ 1 M 1 + λ 2 M 2 is r = 2 because M 1 is nonsingular for all ε, so we can take λ 0 = (1, 0). Thus M(λ 0 ) = M 1 . To see that A ε is an evolution algebra we prove that M −1 1 M 2 is diagonalisable by similarity. It is easy to check that Since and det P = 2ε, we conclude by Theorem 5 that the algebra A ε is an evolution algebra if, and only if, ε = 0. For completeness we show the diagonalisation of the original matrices: which shows by Theorem 1, that A ε is an evolution algebra for every ε > 0, having B = {e 1 − e 2 , e 1 + (2ε − 1)e 2 } as a natural basis.

Example 4.
The annihilator of every algebra A ε in the above example is zero as one of its m-structure matrices is invertible. To get a similar example with algebras having non-zero annihilator, consider for instance the algebra A ε with natural basis B = {e 1 , e 2 , e 3 } and product given by we obtain, from the calculations in the above example, that P T M k ( B)P is diagonal for every k = 1, 2, 3 and hence A ε is an evolution algebra. Nevertheless, for ε = 0 we do not obtain an evolution algebra. Indeed, if we denote this algebra by A then the quotient algebra A/Ann(A) is exactly the algebra A 0 in Example 2 which is not an evolution algebra and, consequently, A is not an evolution algebra (see Remark 2).
Example 5 (Gametic algebra for auto-tetraploid inheritance). Let T 0 denote a 3-dimensional commutative algebra over R, considered the simplest case of special train algebras in polyploidy [28,Chapter 15] (see also [29] and [30]). In terms of the basis {e 1 , e 2 , e 3 } the multiplication table is given by The algebra T 0 is baric, with weight function defined by ξ(e j ) = 1, j = 1, 2, 3.
The corresponding m-strucuture matrices M 1 , M 2 , M 3 are For genetic applications, we restrict 0 < ε ≤ 2/9, so all coefficients in the above matrices are non-negative. The algebra T ε is baric, with weight function defined by ξ(e j ) = 1 + 10 ε, j = 1, 2, 3. Let us consider whether T ε is an evolution algebra. First of all, the maximal rank of the linear pencil M(λ) = λ 1 M 1 + λ 2 M 2 + λ 3 M 3 is r = 3 because M 1 is nonsingular for all ε, so we can take λ 0 = (1, 0, 0). Thus M(λ 0 ) = M 1 . By Theorem 5, a necessary condition is that the matrices M −1 1 M 2 and M −1 1 M 3 are simultaneously diagonalisable by similarity: in particular, they must commute. Let us write these matrices explicitly: It is straightforward to show that these matrices commute for all ε (even for ε = 0). Regarding the Jordan decomposition for M −1 1 M 2 and M −1 1 M 3 we find that if ε > 0 then these matrices are simultaneously diagonalisable: in fact, there is a nonsingular matrix P such that P −1 M −1 1 M 2 P is diagonal: Explicitly, in terms of the radical S ε , We find det P = −24εS ε which shows there is a problem at ε = 0. It is easy to show that at ε = 0 the Jordan form of M −1 1 M 2 is not diagonal. For ε > 0 the Jordan form of M −1 1 M 2 is diagonal and so is the Jordan form of M −1 1 M 3 : 14 of 15 For completeness we show the diagonalisation of the original matrices: where α = −2 + 144ε + 1440ε 2 .

Conclusions and Discussion
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, amongst others. In fact, every evolution algebra can be canonically regarded as a weighted digraph when a natural basis is fixed, and because of this evolution algebras may introduce useful algebraic techniques into the study of some digraphs.
We also consider applications of our results to classical genetic algebras. Strikingly, the classical cases of Mendelian and auto-tetraploid inheritance are not evolution algebras, while slight deformations of them produce evolution algebras. This is interesting because evolution algebras are supposed to describe asexual reproduction, unlike these classical cases. In future work we will study more closely the relation between baric algebras and evolution algebras, in order to better understand this phenomenon.
Author Contributions: All authors contributed equally to this manuscript Funding: This work was partially supported by Project MTM216-76327-C3-2-P . This work was also supported by the award of the Distinguished Visitor Grant of the School of Mathematics and Statistics, University College Dublin to the third author