aa r X i v : . [ m a t h . A C ] J u l EVOLUTION ALGEBRA OF A “CHICKEN” POPULATION
M. LADRA, U. A. ROZIKOV
Abstract.
We consider an evolution algebra which corresponds to a bisexual popula-tion with a set of females partitioned into finitely many different types and the maleshaving only one type. We study basic properties of the algebra. This algebra is com-mutative (and hence flexible), not associative and not necessarily power associative, ingeneral. Moreover it is not unital. A condition is found on the structural constantsof the algebra under which the algebra is associative, alternative, power associative,nilpotent, satisfies Jacobi and Jordan identities. In a general case, we describe the fullset of idempotent elements and the full set of absolute nilpotent elements. The set ofall operators of left (right) multiplications is described. Under some conditions on thestructural constants it is proved that the corresponding algebra is centroidal. Moreoverthe classification of 2-dimensional and some 3-dimensional algebras are obtained.
Key words.
Evolution algebra; bisexual population; associative algebra; centroidal;idempotent; nilpotent; unital.
Mathematics Subject Classifications (2010).
Introduction
Description of a sex linked inheritance with algebras involves overcoming the obstacleof asymmetry in the genetic inheritance rules. Inheritance which is not sex linked issymmetrical with respect to the sexes of the organisms [4], while sex linked inheritanceis not (see [3], [5]). The main problem for a given algebra of a sex linked population is tocarefully examine how the basic algebraic model must be altered in order to compensatefor this lack of symmetry in the genetic inheritance system. In [1] Etherington beganthe study of this kind of algebras with the simplest possible case.Now the methods of mathematical genetics have become probability theory, stochasticprocesses, nonlinear differential and difference equations and non-associative algebras.The book [4] describes some mathematical apparats of studying algebras of genetics.This book mainly considers a free population , which means random mating in the popu-lation. Evolution of a free population can be given by a dynamical system generated bya quadratic stochastic operator (QSO) and by an evolution algebra of a free population.In [4] an evolution algebra associated to the free population is introduced and usingthis non-associative algebra, many results are obtained in explicit form, e.g., the explicitdescription of stationary quadratic operators, and the explicit solutions of a nonlinearevolutionary equation in the absence of selection, as well as general theorems on conver-gence to equilibrium in the presence of selection. In [2] some recently obtained results and also several open problems related to the theory of QSOs are discussed. See also [4]for more detailed theory of QSOs.Recently in [3] an evolution algebra B is introduced identifying the coefficients ofinheritance of a bisexual population as the structure constants of the algebra. The basicproperties of the algebra are studied. Moreover a detailed analysis of a special case ofthe evolution algebra (of bisexual population in which type “1” of females and maleshave preference) is given. Since the structural constants of the algebra B are given bytwo cubic matrices, the study of this algebra is difficult. To avoid such difficulties onehas to consider an algebra of bisexual population with a simplified form of matricesof structural constants. In this paper we consider a such simplified model of bisexualpopulation and study corresponding evolution algebra.The paper is organized as follows. In Section 2 we define our algebra as an evolutionalgebra which corresponds to a bisexual population with a set of females partitioned intofinitely many different types and the males having only one type. Then we study basicproperties (associativity, non-associativity, commutativity, power associativity, nilpo-tency, unitality, etc.) of the algebra. Section 3 is devoted to subalgebras, absolutenilpotent elements and idempotent elements of the algebra. In Section 4 the set of alloperators of left (right) multiplications is described. In Section 5, under some conditionson the structural constants, it is proved that the corresponding algebra is centroidal.The last section gives a classification of 2-dimensional and some 3-dimensional algebras.2. Definition and basic properties of the EACP
We consider a set { h i , i = 1 , . . . , n } (the set of “hen”s) and r (a “rooster”). Definition 1.
Let ( C , · ) be an algebra over a field K (with characteristic = 2 ). If itadmits a basis { h , . . . , h n , r } , such that h i r = rh i = (cid:16)P nj =1 a ij h j + b i r (cid:17) ,h i h j = 0 , i, j = 1 , . . . , n ; rr = 0 , (2.1) then this algebra is called an evolution algebra of a “chicken” population (EACP). Wecall the basis { h , . . . , h n , r } a natural basis. Remark 1. If n X j =1 a ij = 1; b i = 1; for all i = 1 , , . . . , n (2.2) then the corresponding C is a particular case of an evolution algebra of a bisexual popu-lation, B , introduced in [3]. The study of the algebra B is difficult, since it is determinedby two cubic matrices. While the algebra C is more simpler, since it is defined by a VOLUTION ALGEBRA OF A “CHICKEN” POPULATION 3 rectangular n × ( n + 1) -matrix M = a a . . . a n b a a . . . a n b ... ... ... ... ... a n a n . . . a nn b n , which is called the matrix of structural constants of the algebra C . This simplicity allowsto obtain deeper results on C than on B . Moreover, in this paper we do not require thecondition (2.2). The general formula for the multiplication is the extension of (2.1) by bilinearity, i.e.for x, y ∈ C , x = n X i =1 x i h i + ur, y = n X i =1 y i h i + vr using (2.1), we obtain xy = 12 n X j =1 n X i =1 ( vx i + uy i ) a ij ! h j + 12 n X i =1 ( vx i + uy i ) b i ! r (2.3)and x = xx = n X j =1 n X i =1 ( ux i ) a ij ! h j + n X i =1 ( ux i ) b i ! r. (2.4)We recall the following definitions: If x, y and z denote arbitrary elements of an algebrathenAssociative: ( xy ) z = x ( yz ).Commutative: xy = yx .Anticommutative: xy = − yx .Jacobi identity: ( xy ) z + ( yz ) x + ( zx ) y = 0.Jordan identity: ( xy ) x = x ( yx ).Power associative: For all x , any three nonnegative powers of x associate. That is if a, b and c are nonnegative powers of x , then a ( bc ) = ( ab ) c . This is equivalent to sayingthat x m x n = x n + m for all nonnegative integers m and n .Alternative: ( xx ) y = x ( xy ) and ( yx ) x = y ( xx ).Flexible: x ( yx ) = ( xy ) x .It is known that these properties are related by • associative implies alternative implies power associative ; • associative implies Jordan identity implies power associative ; • Each of the properties associative, commutative, anticommutative, Jordan iden-tity, and Jacobi identity individually imply flexible.
M. LADRA, U. A. ROZIKOV
For a field with characteristic not two, being both commutative and anticommutativeimplies the algebra is just { } .By [3, Theorem 4.1.] we have(1) Algebra C is not associative, in general.(2) Algebra C is commutative, flexible.(3) C is not power-associative, in general.Now we shall give conditions on the matrix M under which C will be associative. Theorem 1.
The algebra C is associative iff the elements of the corresponding matrix M satisfy the following n X j =1 a ij a jk = 0 , b i = 0 , for any i, k = 1 , , . . . , n. (2.5) Proof.
Necessity.
Assume the algebra C is associative. First we consider the equality( xy ) z = x ( yz ) for basis elements. If x, y, z ∈ { h , . . . , h n } or x = y = z = r then theequality is obvious.From ( h i r ) h j = h i ( rh j ) we get a jk b i = a ik b j , for all i, j, k = 1 , . . . , n. The equality ( h i h j ) r = h i ( h j r ) gives a ik b j = 0 , b i b j = 0 for all i, j, k = 1 , . . . , n. From ( rh i ) r = r ( h i r ) = ( h i r ) r = h i ( rr ) = 0 we obtain n X j =1 a ij a jk = 0 , for all i, k = 1 , . . . , n. These conditions imply the condition (2.5).
Sufficiency.
Assume the condition (2.5) is satisfied, then the following lemma showsthat the algebra C is associative. (cid:3) Lemma 1.
If the condition (2.5) is satisfied then xyz = 0 , for all x, y, z ∈ C . (2.6) Proof.
Under condition (2.5) from (2.3) we get xy = 12 n X j =1 n X i =1 ( vx i + uy i ) a ij ! h j . (2.7)Consequently, for z = P nm =1 z m h m + wr , using condition (2.5) we get( xy ) z = 12 n X j =1 w n X i =1 ( vx i + uy i ) a ij ! ( h j r ) = VOLUTION ALGEBRA OF A “CHICKEN” POPULATION 5 n X m =1 w n X j =1 n X i =1 ( vx i + uy i ) a ij ! a jm h m =14 n X m =1 n X i =1 w ( vx i + uy i ) n X j =1 a ij a jm h m = 0 . (cid:3) By this lemma and above mentioned properties we get
Corollary 1.
If the condition (2.5) is satisfied then algebra C is alternative, powerassociative, satisfies Jacobi and Jordan identities. We note that the conditions (2.2) and (2.5) cannot be satisfied simultaneously, so thecorresponding algebra B of a bisexual population is not associative. Example 1.
The following matrix M (for n = 2 ) satisfies the condition (2.5): M = (cid:18) a b c − a (cid:19) , for any a, b, c with a = − bc . Definition 2.
An element x of an algebra A is called nil if there exists n ( a ) ∈ N suchthat ( · · · (( x · x ) · x ) · · · x | {z } n ( a ) ) = 0 . The algebra A is called nil if every element of the algebrais nil. For k ≥
1, we introduce the following sequences: A (1) = A , A ( k +1) = A ( k ) A ( k ) . A < > = A , A
An algebra A is called (i) solvable if there exists n ∈ N such that A ( n ) = 0 and the minimal such numberis called index of solvability; (ii) right nilpotent if there exists n ∈ N such that A
M. LADRA, U. A. ROZIKOV
Corollary 2.
If the condition (2.5) is satisfied then algebra C is nilpotent with nilpotencyindex equal 3. Recall that an algebra is unital or unitary if it has an element e with ex = x = xe forall x in the algebra. Proposition 1.
The algebra C is not unital.Proof. Assume e = P ni =1 a i h i + br be a unity element. We then have eh i = h i whichgives ba jj = 1; ba jm = 0 , m = j ; bb j = 0 , for any j = 1 , . . . , n. (2.8)From er = r we get n X i =1 a i a ij = 0 , for any j = 1 , . . . , n ; n X i =1 a i b i = 1 . (2.9)From system (2.8) we get b = 0 and b i = 0 for all i . But for this b i the second equationof the system (2.9) is not satisfied. This completes the proof. (cid:3) An algebra A is a division algebra if for every a, b ∈ A with a = 0 the equations ax = b and xa = b are solvable in A . Proposition 2.
The algebra C is not a division algebra.Proof. Since C is a commutative algebra we shall check only ax = b . For coordinates ofany a = P ni =1 α i h i + αr , b = P ni =1 β i h i + βr , x = P ni =1 x i h i + ur the equation ax = b has the following form( P ni =1 a ij α i ) u + α P ni =1 a ij x i = 2 β j , j = 1 , . . . , n, ( P ni =1 b i α i ) u + α P ni =1 b i x i = 2 β. (2.10)So this is a linear system with n + 1 unknowns x , . . . , x n , u . This system can be writtenas M y = B where y T = ( x , . . . , x n , u ), B = 2( β , . . . , β n , β ) and M = α n · a a . . . a n P ni =1 a i α i a a . . . a n P ni =1 a i α i ... ... ... ... ... a n a n . . . a nn P ni =1 a in α i b b . . . b n P ni =1 b i α i . By the very known Kronecker-Capelli theorem the system of linear equations M y = B has a solution if and only if the rank of matrix M is equal to the rank of its augmentedmatrix ( M | B ). Since the last column of the matrix M is a linear combination of theother columns of the matrix, we have det( M ) = 0. Consequently rank M ≤ n . Moreoversince dimension of the algebra C is n + 1 one can choose b , i.e. the vector B such thatrank( M | B ) = 1 + rank M . Then for such b the equation ax = b is not solvable. Thiscompletes the proof. (cid:3) VOLUTION ALGEBRA OF A “CHICKEN” POPULATION 7 Evolution subalgebras and operator corresponding to C By analogues of [6, Definition 4, p. 23] we give the following
Definition 4. Let C be an EACP, and C be a subspace of C . If C has a nat-ural basis, { h ′ , h ′ , . . . , h ′ m , r ′ } , with multiplication table like (2.1), we call C anevolution subalgebra of a CP. Let I ⊂ C be an evolution subalgebra of a CP. If C I ⊆ I , we call I an evolutionideal of a CP. Let C and D be EACPs, we say a linear homomorphism f from C to D isan evolution homomorphism, if f is an algebraic map and for a natural basis { h , . . . , h n , r } of C , { f ( r ) , f ( h i ) , i = 1 , . . . , n } spans an evolution subalgebra of aCP in D . Furthermore, if an evolution homomorphism is one to one and onto,it is an evolution isomorphism. An EACP, C is simple if it has no proper evolution ideals. C is irreducible if it has no proper subalgebras. The following proposition gives some evolution subalgebras of a CP.
Proposition 3.
Let C be an EACP with the natural basis { h , . . . , h n , r } and matrix M = a . . . b a a . . . b ... ... ... ... ... ... a n a n a n . . . a nn b n . Then for each m , ≤ m ≤ n , the algebra C m = h h , . . . , h m , r i ⊂ C is an evolutionsubalgebra of a CP.Proof. For given M it is easy to see that C m is closed under multiplication. The chosensubset of the natural basis of C satisfies (2.1). (cid:3) The following is an example of a subalgebra of C , which is not an evolution subalgebraof a CP. Example 2.
Let C be EACP with basis { h , h , h , r } and multiplication defined by h i r = h i + r , i = 1 , , . Take u = h + r , u = h + r . Then ( au + bu )( cu + du ) = acu +( ad + bc ) u u + bdu = (2 ac + ad + bc ) u +(2 bd + ad + bc ) u . Hence, F = Ku + Ku is a subalgebra of C , but it is not an evolution subalgebra of aCP. Indeed, assume v , v be a basis of F . Then v = au + bu and v = cu + du forsome a, b, c, d ∈ K such that D = ad − bc = 0 . We have v = (2 a +2 ab ) u +(2 b +2 ab ) u and v = (2 c + 2 cd ) u + (2 d + 2 cd ) u . We must have v = v = 0 , i.e. a + ab = 0 , b + ab = 0 , c + cd = 0 , d + cd = 0 . M. LADRA, U. A. ROZIKOV
From this we get a = − b and c = − d . Then D = 0 , a contradiction. If a = 0 then b = 0 (resp. c = 0 then d = 0 ), we reach the same contradiction. Hence v = 0 and v = 0 , and consequently F is not an evolution subalgebra of a CP. Let C be an EACP on the field K = R , with a basis set { h , . . . , h n , r } and x = P ni =1 x i h i + ur ∈ C . Formula (2.4) can be written as x = V ( x ) = n X j =1 x ′ j h j + u ′ r, (3.1)where the evolution operator V : x ∈ C → x ′ = V ( x ) ∈ C is defined as the following V : ( x ′ j = u P ni =1 a ij x i , j = 1 , . . . , n,u ′ = u P ni =1 b i x i . (3.2)If we write x [ k ] for the power ( . . . ( x ) . . . ) ( k times) with x [0] = x then the trajectorywith initial x is given by k times iteration of the operator V , i.e. V k ( x ) = x [ k ] . Thisalgebraic interpretation of the trajectory is useful to connect powers of an element ofthe algebra and with the dynamical system generated by the evolution operator V . Forexample, zeros of V , i.e. V ( x ) = 0 correspond to absolute nilpotent elements of C andfixed points of V , i.e. V ( x ) = x correspond to idempotent elements of C .For x = P ni =1 x i h i + ur define a functional b as b ( x ) = n X i =1 b i x i . The following proposition fully describes the set N of absolute nilpotent elements of C with ground field K = R . Proposition 4.
We have N = { ( x, u ) ∈ C : u = 0 }∪ ( { (0 , . . . , , u ) ∈ C : u = 0 } , if det( A ) = 0 { ( x, u ) ∈ C : u = 0 , A x = 0 , b ( x ) = 0 } if det( A ) = 0 , where x = ( x , . . . , x n ) , A = ( a ij ) .Proof. An absolute nilpotent element ( x , . . . , x n , u ) satisfies ( u P ni =1 a ij x i = 0 , j = 1 , . . . , n,u P ni =1 b i x i = 0 . (3.3)The proof follows from a simple analysis of this system. (cid:3) Now we shall describe idempotent elements of C , these are solutions to x = x . Suchan element x = ( x , . . . , x n , u ) satisfies the following ( u P ni =1 a ij x i = x i , j = 1 , . . . , n,u P ni =1 b i x i = u. (3.4) VOLUTION ALGEBRA OF A “CHICKEN” POPULATION 9
Case u = 0. If u = 0 then from (3.4) we get x i = 0 for all i = 1 , . . . , n . Hence x = 0is a unique idempotent element. Case u = 0. Consider a matrix T u = ( t ij ) i,j =1 ,...,n such that t ij = ( ua ji if i = j,ua ii − i = j. Then first n equations of the system (3.4) can be written as T u x = 0 , (3.5)where x = ( x , . . . , x n ).Consider now u ∈ R \ { } as a parameter, then equation (3.5) has a unique solution x = 0 if det( T u ) = 0, which gives u = 0, i.e. this is a contradiction to the assumptionthat u = 0.If det( T u ) = 0 then we fix a solution u = u ∗ = 0 of the equation det( T u ) = 0. In thiscase there are infinitely many solutions x ∗ = ( x ∗ , . . . , x ∗ n ) of T u ∗ x = 0. Substituting asolution x ∗ in the last equation of the system (3.4), we get b ( x ∗ ) = n X i =1 b i x ∗ i = 1 . (3.6)Denote by I d ( C ) the set of idempotent elements of C . Hence we have proved the following Proposition 5.
We have I d ( C ) = { } ∪ { ( x ∗ , . . . , x ∗ n , u ∗ ) : u ∗ = 0 , T u ∗ x ∗ = 0 , b ( x ∗ ) = 1 , det( T u ∗ ) = 0 } . The enveloping algebra of an EACP
For a given algebra A with ground field K , we recall that multiplication by elementsof A on the left or on the right give rise to left and right K -linear transformations of A given by L a ( x ) = ax and R a ( x ) = xa . The enveloping algebra , denoted by E ( A ), of anon-associative algebra A is the subalgebra of the full algebra of K -endomorphisms of A which is generated by the left and right multiplication maps of A . This envelopingalgebra is necessarily associative, even though A may be non-associative. In a sense thismakes the enveloping algebra “the smallest associative algebra containing A ”.Since an EACP, C , is a commutative algebra the right and left operators coincide, sowe use only L a . Theorem 2.
Let C be an EACP with a natural basis { h , . . . , h n , r } and matrix ofstructural constants M = A ⊕ b . If det( A ) = 0 then { L , . . . , L n , L r } (where L i = L h i )spans a linear space, denoted by span ( L, C ) , which is the set of all operators of leftmultiplication. The vector space span ( L, C ) and C have the same dimension. Proof.
For x = P ni =1 x i h i + ur ∈ C by linearity of multiplication in C we can write L x as the following L x = n X i =1 x i L i + uL r . If L x = L y , for y = P ni =1 y i h i + vr ∈ C , then n X i =1 x i h i + ur ! h j = n X i =1 y i h i + vr ! h j implies urh j = vrh j , i.e. u = v . From n X i =1 x i h i + ur ! r = n X i =1 y i h i + vr ! r we get n X j =1 n X i =1 ( x i − y i ) a ij ! h j + n X i =1 ( x i − y i ) b i ! r = 0 . Hence n X i =1 ( x i − y i ) a ij = 0 , n X i =1 ( x i − y i ) b i = 0 . By assumption det( A ) = 0 from the last system we get x i = y i for all i = 1 , . . . , n . Thus x = y . This means that L x is an injection. So the linear space that is spanned by alloperators of left multiplication can be spanned by the set { L i , i = 1 , . . . , n, r } . This setis a basis for span( L, C ). (cid:3) Proposition 6.
For any x ∈ C and any i, i , i , . . . , i m ∈ { , , . . . , n } the following hold L i m ◦ L i m − ◦ · · · ◦ L i ( x ) = m − m − Y j =1 b i j L i m ( x ) , (4.1) L r ◦ L i ( x ) = 12 n X j =1 a ij L j ( x ) , (4.2) L i ◦ L r ( x ) = b ( x )2 L i ( r ) . (4.3) Proof.
For x = P ni =1 x i h i + ur we note that L j ( x ) = uh j r , j = 1 , . . . , n .1) To prove (4.1) we use mathematical induction over m . For m = 2 we have( L i ◦ L i )( x ) = h i ( h i x ) = h i ( uh i r )= h i u n X i =1 a i j h j + b i r ! = ( b i / uh i r ) = ( b i / L i ( x ) . VOLUTION ALGEBRA OF A “CHICKEN” POPULATION 11
Assume now that the formula (4.1) is true for m , we shall prove it for m + 1: L i m +1 ◦ L i m ◦ · · · ◦ L i ( x ) = L i m +1 ◦ m − m − Y j =1 b i j L i m ( x ) = m − m − Y j =1 b i j L i m +1 ◦ L i m ( x ) = m m Y j =1 b i j L i m +1 ( x ) .
2) Proof of (4.2): L r ◦ L i ( x ) = r ( h i x ) = r ( uh i r ) = 12 n X j =1 a ij ( urh j ) = 12 n X j =1 a ij L j ( x ) .
3) Proof of (4.3): L i ◦ L r ( x ) = h i ( rx ) = h i n X j =1 x j ( rh j ) = h i n X m =1 n X j =1 a im x j h m + n X j =1 x j b j r = b ( x )2 L i ( r ) . (cid:3) The centroid of an EACP
We recall (see [6]) that the centroid Γ( A ) of an algebra A is the set of all lineartransformations T ∈ Hom( A , A ) that commute with all left and right multiplicationoperators T L x = L x T, T R y = R y T, for all x, y ∈ A . An algebra A over a field K is centroidal if Γ( A ) ∼ = K . Theorem 3.
Let C be an EACP with a natural basis { h , . . . , h n , r } and matrix ofstructural constants M = A ⊕ b . If det( A ) = 0 then C is centroidal.Proof. Let T ∈ Γ( C ). Assume T ( h i ) = n X j =1 t ij h j + t i r, T ( r ) = n X k =1 τ k h k + τ r. We have
T L j ( h i ) = T ( h j h i ) = 0 = L j T ( h i ) = h j n X k =1 t ik h k + t i r ! = t i h j r. This gives t i = 0 , for all i = 1 , . . . , n. (5.1) Now consider
T L j ( r ) = T ( h j r ) = 12 T n X m =1 a jm h m + b j r ! = 12 n X m =1 a jm T ( h m ) + b j T ( r )= 12 n X k =1 n X m =1 a jm t mk + b j τ k ! h k + 12 n X m =1 a jm t m + b j τ ! r. In another way we have L j T ( r ) = h j n X k =1 τ k h k + τ r ! = τ h j r = τ n X k =1 a jk h k + b j r ! . According to (5.1) we should have n X m =1 a jm t mk + b j τ k = τ a jk . (5.2)Furthermore, T L r ( h j ) = 12 n X k =1 " n X m =1 a jm t mk + b j τ k h k + 12 " n X m =1 a jm t m + b j τ r and L r T ( h j ) = 12 n X k =1 " n X m =1 a mk t jm h k + 12 " n X m =1 t jm b m r. These equalities imply P nm =1 a jm t mk + b j τ k = P nm =1 a mk t jm P nm =1 a jm t m + b j τ = P nm =1 t jm b m . (5.3)Finally, T L r ( r ) = T ( rr ) = 0 = L r T ( r ) = 12 n X j =1 n X k =1 a kj τ k ! h j + 12 n X k =1 τ k b k ! r. Consequently, n X k =1 a kj τ k = 0 , j = 1 , . . . , n ; n X k =1 b k τ k = 0 . (5.4)Since det( A ) = 0 from (5.4) we get τ i = 0 , for all i = 1 , . . . , n. (5.5)Using (5.5), from (5.2) we obtain n X m =1 a jm t mk = τ a jk , j, k = 1 , . . . , n. (5.6) VOLUTION ALGEBRA OF A “CHICKEN” POPULATION 13
Again using det( A ) = 0, by the Cramer’s rule, from (5.6) we get the following solution t mk = ( , if m = kτ, if m = k. (5.7)Note that solutions (5.1), (5.5) and (5.7) satisfy system (5.3). Hence we obtain T ( h i ) = τ h i , T ( r ) = τ r, where τ is a scalar in the ground field K . That is T is a scalar multiplication. Conse-quently, Γ( C ) ∼ = K and C is centroidal. (cid:3) Classification of 2 and 3-dimensional EACP
Let C be a 2-dimensional EACP and { h, r } be a basis of this algebra.It is evident that if dim C = 0 then C is an abelian algebra, i.e. an algebra with allproducts equal to zero. Proposition 7.
Any 2-dimensional, non-trivial EACP C is isomorphic to one of thefollowing pairwise non isomorphic algebras: C : rh = hr = h , h = r = 0 , C : rh = hr = ( h + r ) , h = r = 0 .Proof. For an EACP C we have hr = 12 ( ah + br ) , h = r = 0 . Case: a = 0, b = 0. By change of basis h ′ = h and r ′ = a r we get the algebra C . Case: a = 0, b = 0. Take h ′ = r and r ′ = b h then we get the algebra C . Case: a = 0, b = 0. The change h ′ = a r , and r ′ = b h implies the algebra C .Since C C = 0 and C C = 0, the algebras C and C are not isomorphic. (cid:3) We note that the algebra C is known as the sex differentiation algebra [5].Let now C be a 3-dimensional EACP and { h , h , r } be a basis of this algebra. Theorem 4.
Any 3-dimensional EACP C with dim ( C ) = 1 is isomorphic to one of thefollowing pairwise non isomorphic algebras: C : h r = r ; C : h r = h ; C : h r = h + r .In each algebra we take rh i = h i r , i = 1 , and all omitted products are zero.Proof. For a 3-dimensional EACP C we have h r = rh = 12 ( ah + bh + Ar ) , h r = rh = 12 ( ch + dh + Br ) , h = h = h h = r = 0 . First we note that non-zero coefficients of h r can be taken 1. Indeed, if abA = 0then the change of basis h ′ = A h , h ′ = baA h , r ′ = a r makes all coefficients of h r equal 1. In case some a, b, A is equal 0 then one can choose a suitable change of basisto make non-zero coefficients equal to 1. Therefore we have three parametric families: h r = rh = ( ch + dh + Br ) with one of the following conditions(i) h r = rh = r, (ii) h r = rh = h , (iii) h r = rh = h + r, (iv) h r = rh = h + r, (v) h r = rh = h + h + r, (vi) h r = rh = h , (vii) h r = rh = h + h . If dim( C ) = 1 then h r is proportional to h r . From above-mentioned (i)-(vii) itfollows the following cases for h r and h r . Case (i): In this case h r = r and h r = cr for some c ∈ K . If c = 0 we get thealgebra C . If c = 0 then by the change h ′ = h , h ′ = − h + 1 c h , r ′ = r we again obtain the algebra C . Case (ii): In this case h r = h and h r = ch for some c ∈ K . If c = 0 we get thealgebra C . If c = 0 then by the change h ′ = 1 c r, h ′ = ch − h , r ′ = h we get the algebra C . Case (iii): In this case we have h r = h + r and h r = c ( h + r ) for some c ∈ K . If c = 0 we get the algebra C . If c = 0 then by the change h ′ = h , h ′ = 1 c h − h , r ′ = r we get the algebra C . Case (iv): We have h r = h + r and h r = c ( h + r ) for some c ∈ K . If c = 0 thenby change h ′ = h , h ′ = h , r ′ = h + r we get the algebra C . If c = 0 then by the change h ′ = 1 c h , h ′ = 1 c h − h , r ′ = 1 c r we get the algebra C . Case (v): We have h r = h + h + r and h r = c ( h + h + r ) for some c ∈ K . If c = − h ′ = 11 + c ( h + h ) , h ′ = 11 + c ( − ch + h ) , r ′ = 11 + c r we get the algebra C . If c = − h ′ = h , h ′ = h + h , r ′ = h + h + r we get the algebra C . VOLUTION ALGEBRA OF A “CHICKEN” POPULATION 15
Case (vi): We have h r = h and h r = ch for some c ∈ K . In this case by thechange h ′ = r, h ′ = ch − h , r ′ = h we get the algebra C . Case (vii): In this case h r = h + h and h r = c ( h + h ) for some c ∈ K . Takingthe change h ′ = r c , h ′ = h − ch , r ′ = h + h we get the algebra C .The obtained algebras are pairwise non-isomorphic this may be checked by comparisonof the algebraic properties listed in the following table. C i C i = 0 Nilpotent C Yes No C Yes Yes C No No (cid:3)
Acknowledgements
The first author was supported by Ministerio de Ciencia e Innovaci´on (EuropeanFEDER support included), grant MTM2009-14464-C02-01. The second author thanksthe Department of Algebra, University of Santiago de Compostela, Spain, for providingfinancial support of his many visits to the Department. He was also supported by theGrant No.0251/GF3 of Education and Science Ministry of Republic of Kazakhstan. Wethank B.A. Omirov for his helpful discussions.
References [1] I.M.H. Etherington, Non-associative algebra and the symbolism of genetics, Proc. Roy. Soc. Edin-burgh 61 (1941) 24–42.[2] R.N. Ganikhodzhaev, F.M. Mukhamedov, U.A. Rozikov, Quadratic stochastic operators and pro-cesses: results and open problems. Inf. Dim. Anal. Quant. Prob. Rel. Fields., 14(2) (2011), 279–335.[3] M. Ladra, U.A. Rozikov, Evolution algebra of a bisexual population, J. Algebra. 378 (2013) 153–172.[4] Y.I. Lyubich, Mathematical structures in population genetics, Springer-Verlag, Berlin, 1992.[5] M.L. Reed, Algebraic structure of genetic inheritance, Bull. Amer. Math. Soc. (N.S.) 34 (2) (1997)107–130.[6] J.P. Tian, Evolution algebras and their applications, Lecture Notes in Mathematics, 1921, Springer-Verlag, Berlin, 2008.
M. Ladra, Departamento de ´Algebra, Universidad de Santiago de Compostela, 15782Santiago de Compostela, Spain
E-mail address : [email protected] U. A. Rozikov, Institute of mathematics, 29, Do’rmon Yo’li str., 100125, Tashkent,Uzbekistan.
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