Some properties of evolution algebras

SOME PROPERTIES OF EVOLUTION ALGEBRAS L. M. CAMACHO, J. R. G´ OMEZ, B. A. OMIROV, R. M. TURDIBAEV Abstract. The paper is devoted to the study of finite dimensional complex evolution algebras. The class of evolution algebras isomorphic to evolution algebras with Jordan form matrices is described. For finite dimensional complex evolution algebras the criteria of nilpotency is established in terms of the properties of corresponding matrices. Moreover, it is proved that for nilpotent n−dimensional complex evolution algebras the possible maximal nilpotency index is 1 + 2n−1.The criteria of planarity for finite graphs is formulated by means of evolution algebras defined by graphs. AMS classifications (2010): 05C25; 17A36; 17D92; 17D99 Keywords: Evolution algebra; commutative algebra; isomorphism; nilpotency; planar graph. 1. Introduction In 20s and 30s of the last century the new object was introduced to mathematics, which was the product of interactions between Mendelian genetics and mathematics. Mendel established the basic laws for inheritance, which are summarized as Mendel’s Law of Segregation and Mendel’s Law of Independent Assortment. This laws were mathematically formulated by Serebrowsky [9], who was also the first to give an algebraic interpretation of the ” ×” sign, which indicated sexual reproduction. Later Glivenkov [5] used the notion of Mendelian algebras in his work. Also Kostitzin [7] independently introduced a ”symbolic multiplication” to express Mendel’s laws. In his several papers Etherington [2]- [4] introduced the formal language of abstract algebra to the study of the genetics. These algebras, in general, are non-associative. However, in the beginning of the XX century in genetics there were discovered several examples of inheritances, where traits do not segregate in accordance with Mendel’s laws. In the present day, non-Mendelian genetics is a basic language of molecular genetics. Non-Mendelian inheritance plays an important role in several disease processes. Naturally, the question arises: what non-Mendelian genetics offers to mathematics? The evolution algebras, introduced in [10] serves as the answer to this question. The concept of evolution algebras lies between algebras and dynamical systems. Algebraically, evolution algebras are non-associative Banach algebra; dynamically, they represent discrete dynamical systems. Evolution algebras have many connections with various branches of mathematics, such as graph theory, group theory, stochastic processes, mathematical physics etc. Since evolution algebras are not defined by identities, they can not belong to any well-known classes of non-associative algebras, as Lie, alternative and Jordan algebras. The foundation of evolution algebra theory and applications in non-Mendelian genetics and Markov chains are developed, with pointers to some further research topics was given in book [11]. In this paper, we study some properties of finite dimensional complex evolution algebras. Since any evolution algebra in a natural basis is defined by a quadratic matrix, we 1 2 L. M. CAMACHO, J. R. G ´ OMEZ, B. A. OMIROV, R. M. TURDIBAEV study the connection between the algebraic structure of evolution algebras and matrices. More precise results are obtained for evolution algebras with non-singular matrices. For example, the only automorphisms for such algebras are the composition of basis permutation and the multiplication of basic vectors to scalars. Since in the matrix theory the Jordan form of the matrix is essential topic, we investigate a class of evolution algebras isomorphic to evolution algebras with Jordan form matrices. Thus we can distinguish the class of evolution algebras with a matrix in which the eigenvalues are known. Therefore, corresponding algebras can be investigated by the eigenvalues in algebraical point of view. Namely, the problem of reconstruction of Markov chains on trees [8] which depends on the second eigenvalue can be studied by above evolution algebras. In [4] it was pointed out for general genetic algebras that the nilpotent property is essential to these algebras and the definition as train algebras and baric algebras were formulated. By this means, we define nil, solvable, right-nilpotent and nilpotent evolution algebras as in [1] and study some properties of n−dimensional nilpotent evolution algebras. The notions as right nilpotency and nility for finite dimensional evolution algebras are equivalent [1]. In this work, we prove that any n−dimensional rightnilpotent evolution algebra is nilpotent. Moreover, for evolution algebras of dimension nwe describe some possible values for indexes of nilpotency and prove that 1 + 2n−1is a maximal nilpotency index. In [11] the relation between graph theory and evolution algebras was given. The last section of this work is dedicated to the study of some evolution algebras defined by graphs, namely we find some algebraic properties of evolution algebras defined by complete and complete bipartite graphs and reformulate the graph planarity criteria in terms of evolution algebras. 2. Preliminaries Now we define the main object of the paper. Definition 2.1. [11] Let Ebe a vector space over a field Kwith defined multiplication ·and a basis {e1, e2,...}such that ei·ej= 0, i 6=j, ei·ei=X k aikek, i ≥1, then Eis called evolution algebra and basis {e1, e2,...}is said to be natural basis. From the above definition it follows that evolution algebras are commutative (therefore, flexible). Let Ebe a finite dimensional evolution algebra with natural basis {e1,...,en},then ei·ei= n X j=1 aijej,1≤i≤n, where remaining products are equal to zero. The matrix A= (aij )n i,j=1 is called matrix of the algebra Ein natural basis {e1,...,en}. In [11] conditions for basis transformations that preserve naturalness of the basis are given. Also, the relation between the matrices in a new and old natural basis is established in terms of new defined operation on matrices. Since this approach is not SOME PROPERTIES OF EVOLUTION ALGEBRAS 3 practical for our further purposes, below we give the following brief version in terms of its matrix elements. Now let us consider non-singular linear transformation of the given natural basis {e1,...,en}by matrix T= (tij)n i,j=1 : fi= n X j=1 tijej,1≤i≤n. This transformation is isomorphism if and only if fifj= 0 for all i6=j. Thus, fi·fj= n X p=1 tiptjp(ep·ep) = n X k=1 n X p=1 tiptjpapk!ek= 0. Hence, if Tis an isomorphism, then for i6=jand 1 ≤k≤nwe have n X p=1 tiptjpapk = 0.(2.1) Observe that fi·fi= n X p=1 t2 ip(ep·ep) = n X p=1 t2 ip n X k=1 apkek= n X k=1 n X p=1 t2 ipapk!ek. Now let Tij be the elements of matrix T−1.Then ek= n X s=1 Tksfsand fi·fi= n X k=1 n X p=1 t2 ipapk!n X s=1 Tksfs= n X s=1 n X k=1 n X p=1 t2 ipapkTks!fs. Hence, for the elements of the matrix B= (bis)i,s=1,n of evolution algebra Ein natural basis {f1,...,fn}we have bis = n X k=1 n X p=1 t2 ipapkTks.(2.2) Definition 2.2. An element aof evolution algebra Eis called nil if there exists n(a)∈N such that (...((a·a)·a)·...)·a) |{z } n(a)times = 0.Evolution algebra Eis called nil if any element of the algebra is nil. We introduce the following sequences: E(1) =E, E(k+1) =E(k)E(k), k ≥1 E<1>=E, E<k+1>=E<k>E, k ≥1 E1=E, Ek= k−1 X i=1 EiEk−i, k ≥1 Note that is not difficult to prove the following inclusions for k≥1 : E<k> ⊆Ek, E(k+1) ⊆E2k. 4 L. M. CAMACHO, J. R. G ´ OMEZ, B. A. OMIROV, R. M. TURDIBAEV Also, note that since Eis commutative algebra we obtain Ek=X 1≤i≤k−i EiEk−i. Definition 2.3. An evolution algebra Eis called (i) solvable if there exists n∈Nsuch that E(n)= 0 and the minimal such number is called index of solvability; (ii) right nilpotent if there exists n∈Nsuch that E<n> = 0 and the minimal such number is called index of right nilpotency; (iii) nilpotent if there exists n∈Nsuch that En= 0 and the minimal such number is called index of nilpotency. Observe that if evolution algebra is nilpotent, then it is right nilpotent and solvable. The following example shows that solvable evolution algebra is not necessarily a right nilpotent algebra. Example 2.4. Let Ebe an evolution algebra with natural basis {e1,...en}and the following multiplication: eiei=e1+···+en,1≤i≤n−1 enen= (1 −n)(e1+···+en). Then E(3) = 0,but Ek=he1+···+enifor k≥2. The example described above in fact is a particular case of the following Proposition 2.5. pro-sol Let Ebe an n−dimensional complex evolution algebra such that dim E(2) = 1.Then E(3) = 0 if and only if Eis isomorphic to an evolution algebra with natural basis {e1,...,en}with the following multiplication: eiei=λi(e1+···+ek),1≤i≤n, where λi∈C, k X j=1 λj= 0, n X j=1 |λj|26= 0 and 1≤k≤n. Proof. Since E(2) = 1 and E(2) is spanned by eiei,1≤i≤nwe obtain that they are collinear to a non-zero vector x=a1e1+···+anen. With the suitable natural basis change, one can assume that x=e1+···+ekfor some 1 ≤k≤n. Let eiei=λix, 1≤i≤nand n X j=1 |λj|26= 0.Then E(3) is spanned by xx = (e1+···+ek)2= k X j=1 λjx. Hence, E(3) = 0 if and only if k X j=1 λj= 0. Remark 2.6. Actually, the multiplication obtained in Proposition ?? can be divided into two disjoint classes. First one, when λi= 0 for all 1≤i≤k, then this evolution algebra is nilpotent. The second one is when λi6= 0 for some 1≤i≤k. Then by natural basis transformation one can assume that e1e1=e1+···+ekand hence, this evolution algebra is not nilpotent. In [1] the equivalence of right nilpotency and nility for finite dimensional complex evolution algebras is proved. SOME PROPERTIES OF EVOLUTION ALGEBRAS 5 Theorem 2.7. The following statements are equivalent: a) The matrix of an evolution algebra Ecan be transformed by natural basis permutation to A=      0a12 a13 . . . a1n 0 0 a23 . . . a2n 0 0 0 . . . a3n . . .. . .. . ..... . . 0 0 0 ... 0      ; (2.3) b) Evolution algebra Eis right nilpotent algebra; c) Evolution algebra Eis nil algebra. 3. Isomorphisms In case of evolution algebras with non-singular evolution matrices the problem of finding isomorphic algebras to the given one can be solved more precisely. Let Ebe an evolution algebra with matrix Asuch that det A6= 0. Proposition 3.1. Aut(E) = {Tπ|π∈Sn},where Tπ= (tij )1≤i,j≤nsuch that tij 6= 0 if and only if j=π(i).Moreover, if Tπis an automorphism of evolution algebra Eand B= (bij)1≤i,j≤nis the matrix of Ein basis Tπ(e1),...,Tπ(en)then bij =t2 i,π(i) tj,π(j) ·aπ(i)π(j).(3.1) Proof. Consider (2.1) as a linear homogeneous system of equations in terms of unknowns ti1tj1,...,tintjn.If Ais a non-singular matrix then from (2.1) we obtain        ti1tj1= 0 ti2tj2= 0 ... tintjn = 0 where i6=j. Since matrix Tis non-singular, in every row there is at least one non-zero element. But for any non-zero element tip (in the i−th row) we have tiptjp = 0 for all j6=i. Therefore, tjp = 0 for j6=i. Now if for some m6=pwe have tim 6= 0,then similarly, we obtain tjm for all j6=m. But this contradicts to non-singularity of matrix T. Therefore, in every row and every column we have exactly one non-zero element, i.e., the matrix Thas the form described in the statement of the proposition. Note that det T= (−1)σ(π)t1π(1)t2π(2) ···tnπ(n),where σ(π) is a signature of π. We obtain that the group of automorphisms of Eis {Tπ|π∈Sn}and Tπ◦Tτ=Tτ◦π. Let us fix one π∈Sn.Then T(ei) = ti,π(i)eπ(i)for all 1 ≤i≤n. Now T(ei)·T(ei) = t2 i,π(i)(eπ(i)·eπ(i)) = t2 i,π(i) n X k=1 aπ(i)kek= t2 i,π(i) n X k=1 aπ(i)π(k)eπ(k)= n X k=1 t2 i,π(i) tk,π(k) aπ(i)π(k)T(ek). Hence, the elements of evolution matrix B= (bij)i,j=1,n of isomorphic algebra to E satisfy (3.1). 6 L. M. CAMACHO, J. R. G ´ OMEZ, B. A. OMIROV, R. M. TURDIBAEV For a π∈Sndenote by sπ:{1,2,...,n} \ {π−1(n)} → {1,2,...,n}a one-to-one mapping defined by sπ(i) = π−1(1 + π(i)). Proposition 3.2. Let A= (aij)1≤i,j≤nbe a matrix of an evolution algebra isomorphic to an evolution algebra with Jordan cell matrix with non-zero eigenvalue λ. Then the only non-zero elements of Aare the diagonal elements and ai,sπ(i)for all i6=π−1(n) and λ=a2 ii ai,sπ(i)asπ(i)sπ(i) for all i6=π−1(n). Proof. First consider the isomorphism of evolution algebra with Jordan cell matrix with non-zero eigenvalue λ. Since the matrix is non-singular, by the proof of Proposition 3.1 we obtain that it is in the form Tπ. For fixed π∈Snwe put Tπ(ei) = fiand derive fi·fi=ti,π(i)λfi+t2 i,π(i) tsπ(i),π(sπ(i)) fsπ(i). Hence the matrix of the new evolution algebra is a sum of non-singular diagonal matrix and a matrix that has exactly one non-zero element on each row except the π−1(n)−th, which is a zero row and at most one non-zero element in each column. Now let us fix a permutation π∈Snand consider matrix A= (aij)n i,j=1 with zero elements except the diagonal elements and ai,sπ(i)for all i6=π−1(n) and sπ(i) = π−1(1+ π(i)).If this evolution algebra is isomorphic to an evolution algebra with Jordan cell matrix with eigenvalue λthen aii =λti,π(i)for all 1 ≤i≤nand ai,sπ(i)=t2 i,π(i) tsπ(i),π(sπ(i)) . Since ti,π(i)=1 λaii and tsπ(i),π(sπ(i)) =1 λasπ(i)sπ(i)we obtain ai,sπ(i)=a2 ii λ2·λ asπ(i)sπ(i) =1 λ·a2 ii asπ(i)sπ(i) and hence λ=a2 ii ai,sπ(i)asπ(i)sπ(i) . Hence, if matrix Asatisfies λ=a2 ii ai,sπ(i)asπ(i)sπ(i)for all i6=π−1(n),then evolution algebra with matrix Ais isomorphic to evolution algebra with Jordan cell matrix with eigenvalue λ. This isomorphism has the matrix which is the inverse to T= (tij)n i,j=1, where tiπ(i)=1 λaii and zero otherwise.  The above result can be generalized to the case of Jordan form matrices. Let J= J1⊕J2⊕ · · · ⊕ Jr,where Jiare Jordan cells of dimension niwith non-zero eigenvalue λi. Now let us denote µk=λifor n1+···+ni−1+ 1 ≤k≤n1+···+ni,1≤i≤r. Take π∈Snand denote s′ π:{1,...,n}\{π−1(n1),...,π−1(nr)} → {1,...,n}a one-to-one mapping defined by s′ π(i) = π−1(1 + π(i)). Corollary 3.3. Let A= (aij )1≤i,j≤nbe a matrix of an evolution algebra isomorphic to an evolution algebra with Jordan form matrix J. Then the only non-zero elements of A are the diagonal elements and ai,s′ π(i)such that a2 ii ai,s′ π(i)as′ π(i)s′ π(i) =µ2 i µs′ π(i) for i6∈ {π−1(n1),...,π−1(nr)}. 4. Nilpotency of evolution algebras Let us now consider an evolution algebra Ewith Jordan cell with eigenvalue λ. Proposition 4.1. If λ6= 0 then Eis neither solvable nor right nilpotent and therefore is not nilpotent. SOME PROPERTIES OF EVOLUTION ALGEBRAS 7 Proof. Since λ6= 0 then evolution matrix is non-degenerated. Therefore, E2=E(2) = E<2>=E. By simple induction we obtain Ek=E(k)=E<k> =Eand the statement of the proposition is verified.  Proposition 4.2. For an evolution algebra with Jordan cell matrix and eigenvalue λ= 0 the following statements hold: (i) Eis one generated; (ii) Eis solvable with index of solvability n+ 1; (iii) Eis right nilpotent with index of right nilpotency n+ 1; (iv) Eis nilpotent with index of nilpotency 2n−1+ 1. Proof. (i) From λ= 0 it follows that for basis elements eiof Ewe have [ei, ei] = ei+1 for all 1 ≤i≤n−1 and [en, en] = 0. Therefore, Eis one-generated: E=idhe1i. (ii) First observe that E(2) =he2,...,eni. If for some kwe have E(k)=hek, ek+1,...,eni, then for k+ 1 we obtain E(k+1) =E(k)E(k)=hek+1,...,eni. Therefore, E(n)=heniand E(n+1) = 0 and (ii) is verified. (iii) is similar to (ii). (iv) We claim that E2k+1 =E2k+2 =···=E2k+1 =hek+2,...,eni for all 0 ≤k≤n−2. Indeed, for k= 0 we have E2=EE =he2,...,eni. For k= 1 we have E2+1 =E3=EE2=he3,...,eni, E22=E4=EE3+E2E2=he3,...,eni. Assume that E2k−1+1 =E2k−1+2 =···=E2k=hek+1,...,eni. Using this assumption we obtain E2k+1 =EE2k+E2E2k−1+···+E2k−1E2k−1+1 = EE2k+E2E2k+···+E2k−1E2k= (E+E2+E3+···+E2k−1)E2k=EE2k=hek+2,...,eni. Also E2k+1 =EE2k+1−1+E2E2k+1−2+···+E2kE2k⊇ E2kE2k=hek+2,...,eni. So we obtain hek+2,...,eni=E2k+1 ⊇E2k+2 ⊇ · · · ⊇ E2k+1 ⊇ hek+2,...,eni. Hence, E2k+1 =E2k+2 =···=E2k+1 =hek+2,...,eni. 8 L. M. CAMACHO, J. R. G ´ OMEZ, B. A. OMIROV, R. M. TURDIBAEV Therefore, E2n−1=heniand E2n−1+1 = 0. Hence, Eis nilpotent with nilpotency index equal to 1 + 2n−1. Remark 4.3. We should note that the statements (ii)−(iv)of Proposition 4.2 are equivalent, since one can show that each of them is equivalent to λ= 0.However, statement (i)is not equivalent to λ= 0 since for λ= 1 one can prove that Eis generated by the element e1+e2. Observe that any evolution subalgebra of an evolution algebra is an ideal. Therefore if we consider an evolution algebra EJwith matrix Jin Jordan form J=J1⊕J2⊕· · ·⊕Jr where Jiare Jordan cells of dimension niwith eigenvalues λi,then EJ=E1⊕E2⊕ · · · ⊕ Er, where Ei=heni−1+1,...,enii. Now we have Ek J=Ek 1⊕Ek 2⊕ · · · ⊕ Ek rand therefore EJis nilpotent if and only if every Eiis nilpotent. Since we have obtained the criteria of nilpotency of Jordan blocks, we obtain Corollary 4.4. EJis nilpotent (with index of nilpotency equal to max1≤i≤r{1 +2ni−1}) if and only if Jhas only zero eigenvalues. The same assertion holds for right nilpotency and solvability with corresponding indexes equal to 1 + max1≤i≤r{ni}. Note that from the Corollary 4.4 it follows that for every 1 ≤k≤nwe obtain an example of nilpotent evolution algebra with index of nilpotency equal to 1 + 2k−1. The following theorem represents the criteria of nilpotency of finite dimensional evolution algebra. Theorem 4.5. Let Ebe an n−dimensional evolution algebra. Then Eis nilpotent if and only if the matrix of evolution algebra Acan be transformed by the natural basis permutation to form (2.3). Moreover, the index of nilpotency of evolution algebra Eis not greater then 2n−1+ 1. Proof. Let Ebe a nilpotent. Then it is right nilpotent and therefore, by Theorem 2.7 the matrix of this evolution algebra can be transformed by the natural basis permutation to from (2.3). Now let the matrix Aof Ecan be transformed by the natural basis permutation to form (2.3). Assume that a12a23 . . . an−1n6= 0.Similar to the proof of (iv) in Proposition 4.2 one can verify E2k+1 =E2k+2 =···=E2k+1 =hek+2,...,eni for all 0 ≤k≤n−2. Therefore, E2n−1=heniand E2n−1+1 = 0. Hence, Eis nilpotent with nilpotency index equal to 1 + 2n−1. Now assume that a12a23 ...an−1n= 0.In this case we claim that hek+2,...,eni ⊇ E2k+1 for all 0 ≤k≤n−2. Indeed, for k= 0 we have E2=EE ⊆ he2,...,eni. For k= 1 we have E2+1 =E3=EE2⊆ he3,...,eni. SOME PROPERTIES OF EVOLUTION ALGEBRAS 9 Assume that hek+1,...,eni ⊇ E2k−1+1. Using this assumption we obtain E2k+1 =EE2k+E2E2k−1+···+E2k−1E2k−1+1 ⊆ EE2k−1+1 +E2E2k−1+1 +···+E2k−1E2k−1+1 = (E+E2+E3+···+E2k−1)E2k−1+1 ⊆EE2k−1+1 ⊆ hek+2,...,eni. So we obtain hek+2,...,eni ⊇ E2k+1. Therefore, heni ⊇ E2n−2+1. Hence, E2n−1+1 =EE2n−1+E2E2n−1−1···+E2n−2E2n−2+1 ⊆ (E+E2+. . . E2n−2)heni ⊆ Eheni= 0. Thus, Eis nilpotent with nilpotency index not greater then 1 + 2n−1. Corollary 4.6. For finite dimensional complex evolution algebra notions as nil, nilpotent and right nilpotent algebras are equivalent. However, the indexes of nility, right nilpotency and nilpotency do not coincide in general. The following proposition excludes significantly many possible values that a nilpotency indexes of n−dimensional evolution algebras can take. Proposition 4.7. Let Ebe a nilpotent evolution algebra with index of nilpotency not equal to 2n−1+ 1.Then it is not greater then 2n−2+ 1. Proof. Since Eis nilpotent, we assume that the matrix Aof Ein the natural basis {e1,...,en}is in the form (2.3). From the proof of Proposition 4.5 it follows that a12a23 . . . an−1n= 0 and hek+2,...,eni ⊇ E2k+1 for all 0 ≤k≤n−2. Assume that Eis nilpotent with index of nilpotency greater then 2n−2+ 1 and not equal to 2n−1+ 1. Then heni ⊇ E2n−2+1 and since E2n−2+1 6= 0 we obtain E2n−2+1 =heni. Therefore, hen−1, eni ⊇ E2n−3+1 ⊇E2n−3+2 ⊇ · · · ⊇ E2n−2⊇ heni. Now if E2n−3+1 =E2n−3+2 =···=E2n−2=henithen E2n−2+1 =EE2n−2+E2E2n−2−1···+E2n−3E2n−3+1 ⊆ (E+E2+···+E2n−3)heni=Eheni= 0 which is a contradiction. Hence, hen−1, eni=E2n−3+1. Now assume that hen−k,...,eni=E2n−k−2+1. Then hen−k−1, en−k,...,eni ⊇ E2n−k−3+1 ⊇E2n−k−3+2 ⊇ · · · ⊇ E2n−k−2⊇ hen−k,...,eni. If E2n−k−3+1 6=hen−k−1, en−k,...,enithen E2n−k−3+1 =E2n−k−3+2 =··· =E2n−k−2= hen−k,...,eniand E2n−k−2+1 =EE2n−k−2+···+E2n−k−3E2n−k−3+1 =Ehen−k, en−k,...,eni ⊆ hen−k+1, en−k,...,eni which contradicts to hen−k,...,eni=E2n−k−2+1.