Determining when an algebra is an evolution algebra
AArticle
Determining When An Algebra Is An EvolutionAlgebra
Miguel D. Bustamante , Pauline Mellon and M. Victoria Velasco * School of Mathematics and Statistics, University College Dublin, Dublin 4, Ireland; [email protected] School of Mathematics and Statistics, University College Dublin, Dublin 4, Ireland; [email protected] Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain * Correspondence: [email protected]; Tel.: +34-958-241-000† These authors contributed equally to this work.
Abstract:
Evolution algebras are non-associative algebras that describe non-Mendelian hereditaryprocesses and have connections with many other areas. In this paper we obtain necessary and sufficientconditions for a given algebra A to be an evolution algebra. We prove that the problem is equivalentto the so-called SDC problem, that is, the simultaneous diagonalisation via congruence of a given set ofmatrices . More precisely we show that an n -dimensional algebra A is an evolution algebra if, and onlyif, a certain set of n symmetric n × n matrices { M , . . . , M n } describing the product of A are SDC . Weapply this characterisation to show that while certain classical genetic algebras (representing Mendelianand auto-tetraploid inheritance) are not themselves evolution algebras, arbitrarily small perturbations ofthese are evolution algebras. This is intriguing as evolution algebras model asexual reproduction unlikethe classical ones.
Keywords:
Evolution algebra, multiplication structure matrices, simultaneous diagonalisation bycongruence, simultaneous diagonalisation by similarity, linear pencil.
1. Introduction
This is a postprint of our work published in:
Mathematics , , 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 2008by Tian [1] to enlighten the study of non-Mendelian genetics. Since then, a large literature has flourishedon this topic (see for instance [2–17]) motivated by the fact that these algebras have connections withgroup theory, Markov processes, theory of knots, systems and graph theory. For instance, in [2], thetheory of evolution algebras was related to that of pulse processes on weighted digraphs and applicationswere provided by reviewing a report of the National Science Foundation about air pollution achieved bythe Rand Corporation. A pulse process is a structural dynamic model to analyse complex networks bystudying the propagation of changes through the vertices of a weighted digraph, after introducing aninitial pulse in the system at a particular vertex. It is based on a spectral analysis of the correspondingweighted digraph to facilitate large scale decision making processes. Evolution algebras also introduceuseful algebraic techniques into the study of some digraphs because evolution algebras and weighteddigraphs 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 wesay 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 = i , j ∈ Λ with i (cid:54) = j . Such a basis is called natural . Evolution algebras a r X i v : . [ m a t h . R A ] F e b of 15 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 ofa finite-dimensional evolution algebra and a zero-product algebra.In this paper we discuss necessary and sufficient conditions under which a given finite-dimensionalcommutative algebra is an evolution algebra, namely, we determine when such a finite-dimensionalalgebra can be provided with a natural basis. We tackle the problem constructively by assuming anarbitrary basis B with a multiplication table given by equation (2.1) below and then asking whether or notthere is a change of basis from B to a natural basis B ∗ . In Section 2, Theorem 1, we show that this problemis equivalent to the simultaneous diagonalisation via congruence of certain n × n symmetric matrices M , . . . , 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 diagonalisablevia congruence (we will refer to it as the SDC-problem) is one of the 14 open problems posted in 1990by Hiriart-Urruty [18] (see also [19,20]). It has connections with other problems such as blind-sourceseparation in signal processing [21–24]. The SDC-problem was solved recently for complex symmetricmatrices 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 evolutionalgebra if, and only if, A C is an evolution algebra and has a natural basis consisting of elements of A . Thisreduction of the real case to the complex one allows us to apply the results in [25] to both real and complexalgebras.In Theorem 5 we determine if a given algebra A whose annihilator is zero is an evolution algebraand in Theorem 6 we do the same if its annihilator is not zero. A useful characterisation of the property ofbeing an evolution algebra is given in the particular case that one of the multiplication structure matricesis invertible. In this case if M i is invertible then A is an evolution algebra if, and only if, for each k (cid:54) = i the matrix M − i 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 theconditions in the mentioned results are neither redundant nor superfluous.We prove that some classical genetic algebras such as the gametic algebra for simple Mendelianinheritance (Example 2) or the gametic algebra for auto-tetraploid inheritance (Example 5) are not evolutionalgebras. Nevertheless, both of these algebras can be deformed by means of a parameter ε > A ε that is an evolution algebra for every value of the parameter ε , as shown in Example 3 andExample 6 respectively.
2. Characterising evolution algebras by means of simultaneous diagonalisation of matrices bycongruence An n -dimensional algebra A over a field K ( = R or C ) is determined by means of a basis B = { e , . . . , e n } together with a multiplication table e i e j = n ∑ k = m ijk e k , i , j =
1, . . . , n , (2.1)where m ijk ∈ K , for i , j , k =
1, . . . , n . In fact, if a : = n ∑ i = α i e i and b : = n ∑ j = β j e j then, by bilinearity, theproduct ab is obtained from the multiplication table (2.1) as follows ab = (cid:32) n ∑ i = α i e i (cid:33) (cid:32) n ∑ j = β j e j (cid:33) = n ∑ k = (cid:32) n ∑ i , j = α i β j m ijk (cid:33) e k , of 15 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 ( n ∑ i = α i e i ) = α 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 structure constants in n matrices by defining M k ( B ) : = π k ( e e ) π k ( e e n ) ... ... π k ( e n e ) π k ( e n e n ) = m k m nk ... ... m n k m nnk , (2.2)for k =
1, ..., n , then the product of A is given by (cid:32) n ∑ i = α i e i (cid:33) (cid:32) n ∑ j = β j e j (cid:33) = n ∑ k = (cid:16) α T M k ( B ) β (cid:17) e k , (2.3)where α T = ( α , . . . , α n ) , β T = ( β , . . . , β n ) and T indicates the transpose operation. This motivates thefollowing definition. Definition 1. If A is an algebra, B = { e , . . . , e n } is a basis of A and e i e j = n ∑ k = m ijk e k , for i , j =
1, . . . , n ,then the multiplication structure matrices (m-structure matrices for short ) of A with respect to B are the n × n matrices M k ( B ) = (cid:0) π k ( e i e j ) (cid:1) given by (2.2) for k =
1, ..., n . Note that these matrices are symmetricif, and only if, A is commutative.If the basis B is clear from the context then we will write M k : = M k ( B ) for k =
1, ..., n .We recall that an n -dimensional evolution algebra is a commutative algebra A for which there existsa basis B ∗ = { e ∗ , ..., e ∗ n } such that e ∗ i e ∗ j = i , j ∈ { · · · , n } with i (cid:54) = j . Such a basis B ∗ is said tobe a natural basis of A .The next result is a straightforward combination of the concept of evolution algebra with Definition 1. Proposition 1.
An evolution algebra is an algebra A provided with a basis B ∗ = { e ∗ , ..., e ∗ n } such that thecorresponding m-structure matrices M ( B ∗ ) = ( π ( e ∗ i e ∗ j )) , · · · , M n ( B ∗ ) = ( π n ( e ∗ i e ∗ j )) are diagonal. Proof. M k ( B ∗ ) is diagonal for k =
1, ..., n , if, and only if, e ∗ i e ∗ j =
0, for every i (cid:54) = j , or equivalently if B ∗ is anatural basis.In the next theorem we characterise when a given algebra is an evolution algebra. To this end werecall the following property. Definition 2.
Let M , . . . , M m be symmetric n × n matrices. Then these matrices are simultaneouslydiagonalisable via congruence (SDC) if, and only if, there exists a nonsingular n × n matrix P and m diagonal n × n matrices { D j } mj = such that P T M j P = D j , j =
1, . . . , m .It is worth remarking at this point that the general problem of diagonalisation via congruenceconsiders m symmetric matrices of dimension n × n , where m need not be equal to n . This problem hasapplications in statistical signal processing and multivariate statistics [21–24] and was solved for complexsymmetric matrices in [25]. of 15 Theorem 1.
Let A be a commutative algebra over K with basis B = { e , . . . , e n } . Let M , . . . , M n be them-structure matrices of A with respect to B . Then A is an evolution algebra if, and only if, the symmetric matricesM , . . . , 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 ∗ , ..., e ∗ n } (that is a basissuch that e ∗ i e ∗ j = i (cid:54) = j ). Let P = ( p ij ) be the change of basis matrix from B to B ∗ (that is e ∗ i = ∑ nk = p ki e k for i =
1, . . . , n ). Then, by (2.3), e ∗ i e ∗ j = (cid:32) n ∑ k = p ki e k (cid:33) (cid:32) n ∑ k = p kj e k (cid:33) = n ∑ k = (cid:16) α T M k β (cid:17) e k , (2.4)where α = P γ i and β = P γ j with γ i = (
0, ..., 0, ( i − th ) ) T ∈ M n × ( K ) . Thus e ∗ i e ∗ j = n ∑ k = (cid:16) γ Ti P T M k P γ j (cid:17) e k =
0, for i (cid:54) = j , (2.5)and hence e ∗ i e ∗ j = i (cid:54) = 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] forcomplex symmetric matrices, we consider the following.The complexification of a real algebra A is defined as the complex algebra A C : = A ⊕ iA = { a + ib : a , b ∈ A } , where for a , b , c , d ∈ A and r , s ∈ R , ( a + ib ) + ( c + id ) = ( a + b ) + i ( b + d ) , ( r + is )( a + ib ) = ra − sb + i ( rb + sa ) , ( a + ib )( c + id ) = ( ac − bd ) + i ( ad + bc ) .Note that every basis B of A is trivially a basis of A C so that the real dimension of A and the complexdimension 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 andhas a natural basis consisting of elements of A. Moreover, if A is a real evolution algebra then every natural basis ofA 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 , ..., e n } a basis and M , . . . , M n be the m-structurematrices of A with respect to B . Then A is an evolution algebra if, and only if, the matrices M , . . . , M n (regarded ascomplex 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 acomplex matrix but not by means of any real matrix. of 15
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] forcomplex 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 ofall 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 willshow below. Definition 3.
Let N , ..., N m ∈ M n . These matrices are said to be simultaneously diagonalisable bysimilarity (SDS) if, and only if, there exists P ∈ GL n such that P − N k P is diagonal for every k =
1, ..., m .The following result is well known [27, Theorem 1.3.12 and Theorem 1.3.21]. Proposition 2.
Let N , ..., N m ∈ M n . These matrices are simultaneously diagonalisable by similarity (SDS) if,and only if, they are each diagonalisable by similarity and they pairwise commute.
Remark 1.
Concerning the statement of the above theorem in [27] we point out that the fact that thesymmetric matrices N , ..., N m commute guarantees that N , ..., N m are simultaneously diagonalisable bysimilarity only when each of N , ..., 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, werecall the next definition. Definition 4.
Given M , ..., M m ∈ M n , define the associated linear pencil to be the map M : C m → M n given by M ( λ ) : = m ∑ j = λ j M j , for every λ = ( λ , ..., λ m ) in C m . Since, for λ (cid:54) = M ( λ ) = rank M (cid:18) λ (cid:107) λ (cid:107) (cid:19) , it follows thatsup { rank M ( λ ) : λ ∈ C m } = sup { rank M ( λ ) : λ ∈ C m with (cid:107) λ (cid:107) = } ∈ {
0, 1, ..., n } .Consequently, this supremum must be achieved so that there exists λ ∈ C m with (cid:107) λ (cid:107) = r : = rank M ( λ ) = max { rank M ( λ ) : λ ∈ C m } ,and we say that r is the maximum pencil rank of M , ..., M m .The next theorem corresponds to Theorem 7 in [25] and deals with the case when the maximumpencil rank of the matrices is n . of 15 Theorem 3.
Let M , ..., M m ∈ MS n have maximum pencil rank n . Let λ ∈ C m be such that r : = rankM ( λ ) = n . Then M , ..., M m are SDC if, and only if, M ( λ ) − M , ..., M ( λ ) − M m are SDS. Proposition 2 gives the following result.
Corollary 2.
Let M , ..., M m ∈ MS n , and λ ∈ C m be such thatr : = rank M ( λ ) = n . Then M , ..., M m are SDC if, and only if, M ( λ ) − M , ..., M ( λ ) − M m are all diagonalisable by similarity andpairwise commute .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 matrixgiven by (cid:32) M r r × ( n − r ) ( n − r ) × r N n − r (cid:33) .When the pencil rank of M , ..., M m ∈ MS n is strictly less than n then the SDC problem can bereduced to a similar one in a reduced dimension as the following result (Theorem 9 in [25]) shows. Theorem 4.
Let M , ..., M m ∈ MS n have maximum pencil rank r . Then the following assertions are equivalent: ( i ) M , ..., M m are SDC; ( ii ) dim ( ∩ mj = ker M j ) = n − r and there exists P ∈ GL n satisfying P T M j P = (cid:101) D j ⊕ n − r where (cid:101) D j ∈ MS r is diagonal for ≤ j ≤ m . Moreover, if either of the above conditions is satisfied, then the pencil (cid:101)
D associated with the r × r matrices (cid:101) D , ..., (cid:101) D m is non-singular. Indeed, if λ ∈ C m with (cid:107) λ (cid:107) = is such that r = rank M ( λ ) then (cid:101) D ( λ ) ∈ GL r . We apply the above results to the m-structure matrices M , . . . , M n of an algebra A with respect to abasis B = { e , ..., e n } as in (2.2). For a real algebra A we consider the complexification A C provided withthe same basis B .We recall that the annihilator of an algebra A is the setAnn ( A ) = { b ∈ A : ab = ba =
0, for every a ∈ A } .This set is an ideal of A . Lemma 1.
Let A be a commutative algebra and B = { e , ..., e n } be a basis of A . Let M , . . . , M n be the m-structurematrices of A with respect to B. Then Ann ( A ) = { n ∑ i = β i e i : ( β , ..., β n ) T ∈ ∩ nj = ker M j } . Proof.
Since (cid:18) n ∑ i = α i e i (cid:19) (cid:32) n ∑ j = β j e j (cid:33) = n ∑ k = (cid:0) α T M k β (cid:1) e k , as shown in (2.3) we have that if ( β , ..., β n ) T ∈∩ nj = ker M j then b : = n ∑ j = β j e j ∈ Ann ( A ) as ab = ba = a ∈ A (because M k β = ). of 15 Conversely, if b : = n ∑ j = β j e j ∈ Ann ( A ) then e i b = i =
1, ..., n . It follows that, (
0, ..., 0, ( i − th ) ) M k ( β , ..., β n ) T = i , k ∈ {
1, ..., n } . Fixing k and running i we deduce that, for each k =
1, ..., n , ( β , ..., β n ) T ∈ ker M k ,Consequently, ( β , ..., β n ) T ∈ ∩ nj = ker M j , as desired. Theorem 5.
Let A be a complex commutative algebra with
Ann ( A ) = { } . Let B = { e , . . . , e n } be a basis of Aand let M , . . . , M n be the m-structure matrices of A with respect to B. ( i ) If M , ..., M n have maximum pencil rank n and λ ∈ C n with (cid:107) λ (cid:107) = is such that rank M ( λ ) = nthen A is an evolution algebra if, and only if, each of the matrices M ( λ ) − M , ..., M ( λ ) − M n is diagonalisableby similarity and they pairwise commute. ( ii ) If M , . . . , M n have maximum pencil rank r < n then A is not an evolution algebra. Proof. (i) If λ ∈ C n with (cid:107) λ (cid:107) = M ( λ ) = n then, by Corollary 2, we conclude that A is an evolution algebra if, and only if, the matrices M ( λ ) − M , ..., M ( λ ) − M n are diagonalisable andthey pairwise commute. (ii) Otherwise the maximum pencil rank of { M , . . . , M n } is r < n and, by theabove lemma, dim Ann ( A ) = ∩ nj = ker M j = (cid:54) = n − r . Consequently, by Theorem 4, we conclude that A is not an evolution algebra. Corollary 3.
Let A be a complex commutative algebra and let B = { e , . . . , e n } be a basis of A. Let M , . . . , M n bethe m-structure matrices of A with respect to B. If M i is invertible for some ≤ i ≤ n then Ann ( A ) = { } , andA is an evolution algebra if, and only if, each of the matrices M − i M , ..., M − i M n is diagonalisable (by similarity)for j =
1, ... n and they pairwise commute.
Proof.
Since Ann ( A ) ⊆ ker M i by Lemma 1, we obtain that if M i is invertible then Ann ( A ) = { } .Moreover, for λ = (
0, ..., 0, ( i − th ) ) we haverank ( M ( λ )) = rank ( M i ) = n and the result follows from Theorem 5.If A is an algebra with Ann ( A ) (cid:54) = { } (suppose that dim Ann ( A ) = r >
0) then we can fix a basis ofAnn ( A ) which can be extended to a basis of A . Therefore we obtain a basis (cid:101) B = { e , ..., e r , e r + , ..., e n } of A such that { e r + , ..., e n } is a basis of Ann ( A ) and the m-structure matrices M ( (cid:101) B ) , . . . , M n ( (cid:101) B ) of A withrespect to (cid:101) B satisfy M k ( (cid:101) B ) = (cid:101) M k ⊕ n − r , for certain r × r matrices (cid:101) M k ∈ MS r . Theorem 6.
Let A be a commutative complex algebra with
Ann ( A ) (cid:54) = { } . Let (cid:101) B = { e , ..., e r , e r + , ..., e n } bea basis of A such that { e r + , ..., e n } is a basis of Ann ( A ) . Let M ( (cid:101) B ) , ..., M n ( (cid:101) B ) be the m-structure matrices of Awith respect to (cid:101) B with M k ( (cid:101) B ) = (cid:101) M k ⊕ n − r , where (cid:101) M k ∈ MS r . Then A is an evolution algebra if, and only if,there exists (cid:107) λ (cid:107) = such that the pencil (cid:101) M ( λ ) is invertible, each of the matrices (cid:101) M ( λ ) − (cid:101) M ..., (cid:101) M ( λ ) − (cid:101) M n , is diagonalisable by similarity and they pairwise commute. of 15 Proof.
Assume A is as stated. Then there exists (cid:107) λ (cid:107) = (cid:101) M ( λ ) is invertible if, andonly if, the maximum pencil rank of M k ( (cid:101) B ) is r . If this happens then dim ( ∩ nj = ker M j ( (cid:101) B )) = n − r , asdim Ann ( A ) = dim ( ∩ nj = ker M j ( (cid:101) B )) by Lemma 1. Therefore if (cid:101) M ( λ ) is invertible then, by Corollary2, we have that (cid:101) M , ..., (cid:101) M n are SDC if, and only if, each of the matrices (cid:101) M ( λ ) − (cid:101) M , ..., (cid:101) M ( λ ) − (cid:101) M n isdiagonalisable by similarity and they pairwise commute. Since the matrices (cid:101) M , ..., (cid:101) M n are SDC (by P r ∈ GL r ) if, and only if, the matrices M ( (cid:101) B ) , ..., M n ( (cid:101) B ) are SDC (by P n : = P r ⊕ I n − r ), the result followsfrom Theorem 1. Remark 2.
The above result shows that the condition that A /Ann ( A ) be an evolution algebra is a necessarycondition for A to be an evolution algebra. This is known because it was proved in [3] that the quotient ofan evolution algebra by an ideal is an evolution algebra. However, Theorem 6 proves that this condition isnot sufficient (which is new). In fact, if dim Ann ( A ) : = r < n , and we consider a basis (cid:101) B , as in Theorem6 above, with m-structure matrices given by M k ( (cid:101) B ) = (cid:101) M k ⊕ n − r for k =
1, ..., n , then A is an evolutionalgebra if, and only if, (cid:101) M ..., (cid:101) M n are SDC. Suppose now that (cid:101) M ..., (cid:101) M r are SDC but that (cid:101) M ..., (cid:101) M n are notSDC. It turns out that A /Ann ( A ) is an evolution algebra but A is not (because the m-structure matrices of A /Ann ( A ) with respect to the basis (cid:101) B A /Ann ( A ) = { e + Ann ( A ) , ..., e r + Ann ( A ) } are precisely (cid:101) M ..., (cid:101) M r ).It is easy to come up with particular examples of this situation (see Remark 3 below).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 , ..., e n } is an evolution algebra. Let M , ..., 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 , ..., M n are SDC this isgenerally 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 , ..., M n is invertible.(a) Suppose that M i is invertible, for some 1 ≤ i ≤ n . If M − i M , ..., M − i M n are all diagonalisable(by similarity) and they pairwise commute then we can conclude that A is an evolution algebra, andotherwise we conclude that A is not an evolution algebra.(b) If none of the matrices M , ..., 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 = i =
1, ..., n .(b.1) If Ann ( A ) = { } then we check if there exists some λ = ( λ , ..., λ n ) ∈ C n with (cid:107) λ (cid:107) = M ( λ ) : = n ∑ i = λ i M i is invertible. If such a λ does not exist then we conclude that A is notan evolution algebra. Otherwise we have that A is an evolution algebra if, and only if, the matrices M ( λ ) − M , ..., M ( λ ) − M n are all diagonalisable (by similarity) and they pairwise commute.(b.2) If Ann ( A ) (cid:54) = { } then we construct a basis (cid:101) B = { (cid:101) e , ..., (cid:101) e r , (cid:101) e r + , ..., (cid:101) e n } , such that { (cid:101) e r + , ..., (cid:101) e n } is abasis of Ann ( A ) (cid:54) = { } . We then have M k ( (cid:101) B ) = (cid:101) M k ⊕ n − r for k =
1, ..., n and r × r matrices (cid:101) M , ..., (cid:101) M n .Next, we check if there exists λ = ( λ , ..., λ n ) ∈ C n with (cid:107) λ (cid:107) = (cid:101) M ( λ ) : = n ∑ i = λ i (cid:101) M i isinvertible as an r × r matrix. In particular, this is the case whenever (cid:101) M i is invertible for some 1 ≤ i ≤ n (in which case we can choose (cid:101) M ( λ ) = (cid:101) M i ). If such a λ does not exist then we conclude that A is notan evolution algebra. Otherwise, we have that A is an evolution algebra if, and only if, the matrices (cid:101) M ( λ ) − (cid:101) M , ..., (cid:101) M ( λ ) − (cid:101) M n are all diagonalisable (by similarity) and they pairwise commute.
3. Some examples and applications
We discuss some examples where our approach is useful to determine whether or not certain classicalgenetic algebras are evolution algebras. Mostly these algebras are defined in the literature as real algebras of 15 but, in our case, they can be regarded as complex algebras (with the same basis, and hence with the samem-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 furtherbackground is not necessary to decide if these algebras are evolution algebras or not, we nevertheless referthe reader to [29] and [30] for a review of these algebras.
Example 1.
Let A be the algebra with basis B = { e , e } and e = e , e e = e = e e , e = e . Define ξ : A → K by ξ ( α e + β e ) = α + β . Obviously ξ is linear and if a = α e + β e and if b = γ e + δ e then ab = ( αγ + βδ ) e + ( αδ + βγ ) e ,so that ξ ( ab ) = ( αγ + βδ ) + ( αδ + βγ ) = ( α + β )( γ + δ ) = ξ ( a ) ξ ( b ) , and hence ξ is a non-zero algebrahomomorphism. Consequently A is a baric algebra [28].The corresponding m-structure matrices with respect to B are M = (cid:32) (cid:33) and M = (cid:32) (cid:33) .Since for P = (cid:32) − (cid:33) we have that P T M P = (cid:32) (cid:33) and P T M P = (cid:32) − (cid:33) , by Theorem1 , we obtain that A is an evolution algebra. In fact, (cid:101) B = { (cid:101) e , (cid:101) e } , with (cid:101) e = e − e and (cid:101) e = e + e , is anatural basis of A , as (cid:101) e (cid:101) e = Remark 3.
Let M and M be as above and consider a matrix M that does not commute with M , sayfor instance M = (cid:32) − (cid:33) . Then we have that M − M and M − M do not commute so that, by theproof of Theorem 6 (or alternatively using [25, Section 3.3]), the 3 × M ⊕ × , M ⊕ × and M ⊕ × are not SDC, while M and M are SDC. Therefore the algebra (cid:101) A with basis (cid:101) B = { e , e , e } andproduct e = e + e , e = e − e , e = e e = e = e e , e e = e e = e e = e e = ( (cid:101) A ) = K e . By Theorem 6 (see also Remark 2) we have that (cid:101) A is therefore not an evolutionalgebra whereas (cid:101) A /Ann ( (cid:101) A ) is an evolution algebra isomorphic to the evolution algebra A in Example 1. Example 2 (Gametic algebra for simple Mendelian inheritance) . Let A denote a commutative2-dimensional algebra over R , corresponding to the gametic algebra describing simple Mendelianinheritance (see [30]). In terms of the basis B = { e , e } the multiplication table is e = e , e e = e e = ( e + e ) , e = e .The associated m-structure matrices M , M can be read off easily: M = (cid:32) (cid:33) , M = (cid:32) (cid:33) . It is easy to check that A is a baric algebra, with weight function defined by ξ ( e ) = ξ ( e ) = . Note that M − = (cid:32) − (cid:33) while M − M = (cid:32) − (cid:33) (cid:32) (cid:33) = (cid:32) − − (cid:33) is not diagonalisable by similarity, as λ = − A is not an evolution algebra. (This last assertioncan also be deduced from Theorem 1, with more tedious calculations, by directly checking that M and M 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 thealgebra A of the previous example. We denote these deformed algebras by A ε , which depend on the freeparameter ε ∈ R . In terms of the basis B = { e , e } , the multiplication table for A ε is given by e = ( − ε ) e + ε e , e e = e e = ( e + e ) , e = e .The associated m-structure matrices M , M are now: M = (cid:32) − ε (cid:33) , M = (cid:32) ε (cid:33) .For genetic applications we restrict 0 < ε ≤ A ε is baric with weight function defined by ξ ( e ) = ξ ( e ) =
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 = . Let us consider whether A ε is an evolution algebra by using Theorem 5. First of all, the maximalrank of the linear pencil M ( λ ) = λ M + λ M is r = M is nonsingular for all ε , so we cantake λ = (
1, 0 ) . Thus M ( λ ) = M . To see that A ε is an evolution algebra we prove that M − M isdiagonalisable by similarity. It is easy to check that M − M = (cid:32) ε − ε − (cid:33) and that if P = (cid:32) − ε − (cid:33) then P − M − M P == (cid:32) ε ( ε − ) − ε ε ε (cid:33) (cid:32) ε − ε − (cid:33) (cid:32) − ε − (cid:33) = (cid:32) − ε − (cid:33) .Since P T M ( λ ) P = P T M P = (cid:32) −
11 2 ε − (cid:33) (cid:32) − ε (cid:33) (cid:32) − ε − (cid:33) = (cid:32) − ε ε (cid:33) ,and det P = ε , we conclude by Theorem 5 that the algebra A ε is an evolution algebra if, and only if, ε (cid:54) =
0. For completeness we show the diagonalisation of the original matrices: P T M P = (cid:32) − ε ε (cid:33) , P T M P = (cid:32) ε ε ( ε − ) (cid:33) ,which shows by Theorem 1, that A ε is an evolution algebra for every ε >
0, having B = { e − e , e + ( ε − ) e } as a natural basis. Example 4.
The annihilator of every algebra A ε in the above example is zero as one of its m-structurematrices is invertible. To get a similar example with algebras having non-zero annihilator, consider forinstance the algebra A ε with natural basis (cid:98) B = { e , e , e } and product given by e = ( − ε ) e + ε e − ε e , e = e − e ; e = e e = e e = ( e + e − e ) , e e = e e = e e = e e = M k ( (cid:98) B ) = M k ⊕ i =
1, 2, 3 ) , where 0 denotes the 1 × M and M are given in the above example and M = − M . Hence if P = − ε − we obtain, from the calculations in the above example, that P T M k ( (cid:98) B ) P is diagonal for every k =
1, 2, 3 andhence A ε is an evolution algebra. Nevertheless, for ε = A then the quotient algebra A /Ann ( A ) is exactly the algebra A in Example 2which 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 denote a 3-dimensional commutativealgebra over R , considered the simplest case of special train algebras in polyploidy [28, Chapter 15] (seealso [29] and [30]). In terms of the basis { e , e , e } the multiplication table is given by e = e , e = e e = ( e + e + e ) , e = e , e e = ( e + e ) , e e = ( e + e ) .The corresponding m-structure matrices M , M , M are M =
12 1612 16 , M =
12 2312 23 1223 12 , M =
16 1216 12 .The algebra T is baric, with weight function defined by ξ ( e j ) = j =
1, 2, 3 . To see that this algebra is not an evolution algebra note that M − = − −
18 18 ,and that M − M = − −
18 18
12 2312 23 1223 12 = − − − is not diagonalisable by similarity because it has a single eigenvalue ( λ = −
2) and the dimension of theassociated eigenspace is 1 (indeed, ( −
2, 1 ) T generates it). Consequently, A is not an evolution algebra byCorollary 3.On the other hand, M − M = − −
18 18
16 1216 12 = − − −
153 6 10 so that M − M M − M = M − M M − M = − − − − − − .This proves that, in Theorem 5, the condition that the matrices M ( λ ) − M , ..., M ( λ ) − M n pairwisecommute is not sufficient to ensure that the given algebra is an evolution algebra (see also Proposition 2). Example 6 (Evolution algebra for deformed auto-tetraploid inheritance) . Consider now a deformation ofthe algebra T of the previous example. We denote this deformed algebra by T ε , which depends on the freeparameter ε ∈ R . In terms of the basis { e , e , e } the multiplication table for T ε is: e = e + ε ( e + e ) , e = ( e + e + e ) − ε ( e − e ) , e = e + ε e , e e = ( e + e + e ) + ε e , e e = ( e + e ) + ε e , e e = ( e + e ) + ε e .The corresponding m-strucuture matrices M , M , M are M = + ε
12 1612 16 , M = ε
12 2312 23 − ε , M = ε + ε ε + ε + ε + ε + ε + ε .For genetic applications, we restrict 0 < ε ≤ T ε is baric, with weight function defined by ξ ( e j ) = + 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 M + λ M + λ M is r = M is nonsingular for all ε , so we can take λ = (
1, 0, 0 ) .Thus M ( λ ) = M . By Theorem 5, a necessary condition is that the matrices M − M and M − M aresimultaneously diagonalisable by similarity: in particular, they must commute. Let us write these matricesexplicitly: M − M = − − − ε
33 18 ε − , M − M = + ε + ε + ε − ( + ε ) − ( + ε ) − ( + ε ) ( − ε − ε ) ( − ε − ε ) ( − ε − ε ) .It is straightforward to show that these matrices commute for all ε (even for ε = M − M and M − M we find that if ε > P such that P − M − M P is diagonal: P − M − M P = − − − ε − S ε
00 0 2 − ε + S ε , S ε = (cid:113) ε ( ε + ) .Explicitly, in terms of the radical S ε , P = − − − ε − S ε − − ε + S ε − ε + ε + S ε + ε − S ε .We find det P = − ε S ε which shows there is a problem at ε =
0. It is easy to show that at ε = M − M is not diagonal. For ε > M − M is diagonal and so is the Jordanform of M − M : P − M − M P = − ε − ε + ε + S ε
00 0 1 + ε − S ε . For completeness we show the diagonalisation of the original matrices: P T M P = ε − + ε + S ε
00 0 4 + ε − S ε , P T M P = − ε − + ε + ε + ( ε + ) S ε
00 0 4 + ε + ε − ( ε + ) S ε , P T M P = ε α + ε + ε +( ε + ) S ε
00 0 4 + ε + ε − ( ε + ) S ε ,where α = − + ε + ε .
4. Conclusions and Discussion
In this paper we determine completely whether a given algebra A is an evolution algebra, bytranslating the question to a recently solved problem, namely, the problem of simultaneous diagonalisationvia congruence of the m-structure matrices of A . This is relevant because evolution algebras have strongconnections with areas such as group theory, Markov processes, theory of knots, and graph theory, amongstothers. In fact, every evolution algebra can be canonically regarded as a weighted digraph when a naturalbasis is fixed, and because of this evolution algebras may introduce useful algebraic techniques into thestudy of some digraphs.We also consider applications of our results to classical genetic algebras. Strikingly, the classical casesof Mendelian and auto-tetraploid inheritance are not evolution algebras, while slight deformations ofthem produce evolution algebras. This is interesting because evolution algebras are supposed to describeasexual reproduction, unlike these classical cases. In future work we will study more closely the relationbetween 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 andStatistics, University College Dublin to the third author
Conflicts of Interest:
The authors declare no conflict of interest.
Mathematics Subject Classification [2010]: Primary 17D92 and 15A60.
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