Dealing with the Resolubility of Evolution Algebras

Math.Comput.Sci. (2022) 16:4 https://doi.org/10.1007/s11786-022-00524-9 Mathematics in Computer Science Dealing with the Resolubility of Evolution Algebras D. Fernández-Ternero ·V. M. Gómez-Sousa · J. Núñez-Valdés Received: 21 June 2021 / Revised: 10 November 2021 / Accepted: 15 February 2022 / Published online: 22 March 2022 © The Author(s) 2022 Abstract Although since their introduction by Tian in 2004, evolution algebras have been the subject of a very deep study in the last years due to their numerous applications to other disciplines, this study is not easy since these algebras lack an identity that characterizes them, such as the identity of Jacobi, for Lie algebras or those of Leibniz and Malcev for those corresponding algebras. In this paper we deal with the concepts of solvability and nilpotency of these evolution algebras. Some novel results on them obtained from using the evolution operator of these algebras are given and some examples illustrating these results are also shown. The main result obtained states that an evolution algebra is solvable if and only if its structure matrix is nilpotent, which implies, in turn, that the solvability and the nilpotency indices of that algebra coincide provided the corresponding evolution operator is an endomorphism of the algebra. Keywords Evolution algebras ·Solvability ·Nilpotency Mathematics Subject Classification 17B30 ·17B40 ·17B51 1 Introduction Since Tian introduced them making them the subject of his Ph.D. Thesis in 2004 [12], evolution algebras have proven to be a very useful tool to address different aspects of several disciplines, not only scientific, as could be Biology, Mathematics, Physics or Engineering, for instance, but also economic ones or logistics. Let us recall that an evolution algebra is an algebra endowed with a basis (which is called natural basis), for which the product of two different generators is always zero, that is, an algebra E≡(E,+,·)over a field K,with a basis B={ei,i∈}of E, where is an index set, such that ei·ej=0, if i= j. Note that this definition involves that ej·ej=i∈aijei, with D. Fernández-Ternero ·V. M. Gómez-Sousa ·J. Núñez-Valdés (B) Dpto. de Geometría y Topología, Facultad de Matemáticas, Universidad de Sevilla, Seville, Spain e-mail: jnv[email protected] D. Fernández-Ternero e-mail: [email protected] V. M. Gómez-Sousa e-mail: jnv[email protected] 4Page 2 of 8 D. Fernández-Ternero et al. aij ∈K, where only a finite quantity of aij (which are called structure constants) are non-zero for each j∈fixed. In this way, the product on Eis determined by the structure matrix A =(aij). Note, therefore, that the definition of evolution algebra implies that the generators of these algebras behave in exactly the opposite way in relation to those of a Lie algebra, since in these last algebras, the product (bracket) of every generator by itself is always zero. Taking into consideration this definition, if in addition the product of all generators by themselves (which is called the square of each generator) is different from zero, the evolution algebra is called non-degenerate. Otherwise, the algebra is called degenerate. A particular case of non-degenerate evolution algebras is the so-called non-zero trivial evolution algebra, in which ei·ei=kiei, where kiis a nonzero scalar. On the other hand, as an example of degenerate algebra, we have the zero trivial evolution algebra, which is the one in which the product of any pair of generators is zero. Starting from the publication by Tian and Petr Vojtechovsky in 2006 [14], and later by Tian himself in 2008 [13], lots of researchers have dealt with these algebras, both for their own importance and for their useful applications to other disciplines. As examples of these works can be mentioned, among others, those which relate evolution algebras to graphs and combinatorial structures [5,9,10], those which link them with Markov chains [2,8] and the ones that use these algebras to study distinct biological populations, [4,6,7,11], for instance. However, there are still quite a few topics to discuss about them, such as, for example, the problem of their classification, the determination of new properties or the study of their hierarchy, subalgebras and ideals. With respect to it, one of these aspects on which there is very scarce literature to date is the issue of the solvability and nilpotency of these algebras, an aspect that is precisely the goal of this paper. Note that these two algebraic concepts might be interpreted biologically as the fact that some of the original gametes (that is, generators, in the biological interpretation of these algebras) become extinct after a certain number of generations (see [3,7], for instance). 2 Preliminaries In this section we briefly recall the main definitions and results on evolution algebras which are related with the topic of the paper. As we have already indicated in the Introduction, an algebra E≡(E,+,·)over a field Kis said to be an evolution algebra if it admits a basis B={ei,i∈},where is an index set, such that ei·ej=0, if i= jand ej·ej=i∈aijei, with aij ∈K.The scalars aij are called structure constants. In this way, the product on Eis determined by the so-called structure matrix A =(aij). Tian defined in [13]theevolution operator associated with Bas the endomorphism L:E→Ewhich maps each generator into its square, that is, L(ej)=e2 j=i∈aijei, for all j∈. Note that the matrix representation of the evolution operator with respect to the basis Bis also the structure matrix A=(aij)and note also that the evolution operator of an evolution algebra may or may not be an endomorphism of the algebra. A vector subspace E⊆Eof an evolution algebra Eis a subalgebra of Eif it is closed with respect to the second inner law of Eand it is an evolution subalgebra of Eif it is both subalgebra and of evolution, that is, if there exists a natural basis B={ei,i∈}of Esuch that ei·ej=0, if i= j. In this case, we say that Everifies the extension property if a natural basis Bof Ecan be extended to a natural basis Bof E. The subset Eis an ideal of Eif E·E⊆E. It is easy to see that every ideal of an evolution algebra is a subalgebra of the algebra. Previous concepts allows us to define the ones of evolution ideal and evolution ideal with the extension property. In fact, every subalgebra with the extension property is an ideal, therefore if Ehas the extension property, then the concepts of evolution subalgebra and evolution ideal are equivalent. An element aof an evolution algebra Eis called nil if there exists n(a)∈N,such that (···((a·a)·a)···a)   n(a) =0. An evolution algebra Eis called nil if every element of the algebra is nil. Dealing with the Resolubility of Evolution Algebras Page 3 of 8 4 Let us consider the following sequences E[0]=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 i=1EiEk−i,k≥2. In general, any algebra A,whether it is evolution algebra or not, is called 1. Solvable if there exists n∈Nsuch that E[n]=0.The minimal such number nis called solvability index. 2. Right nilpotent if there exists n∈Nsuch that En=0.The minimal such number nis called right nilpotency index. 3. Nilpotent if there exists n∈Nsuch that En=0.The minimal such number nis called nilpotency index. Aderivation of any algebra Ais a linear operator d:E→ Esuch that d(u·v) =d(u)·v+u·d(v), for all u,v ∈A. Regarding these concepts, the following results on evolution algebras have already been obtained [1] -Inanyn-dimensional evolution algebra, the three following facts are equivalent: being a right nilpotent algebra, being a nil algebra and the structure matrix Acan be written in the following form A=⎛ ⎜ ⎜ ⎜ ⎝ 0a12 a12 ··· a1n 00a13 ···a2n . . .. . .. . .··· . . . 00 0··· 0 ⎞ ⎟ ⎟ ⎟ ⎠ . -Ann-dimensional evolution algebra is nilpotent if and only if its structure matrix can be transformed by the natural basis permutation to the form of the previous result. Moreover, its index of nilpotency is not greater than 2n−1+1. - If the index of nilpotency of a nilpotent evolution algebra is not equal to 2n−1+1,then this one is not greater than 2n−2+1. The reader can check [1,13] for further information. 3 Solvability It is well-known that evolution algebras are, in general, neither associative nor power-associative. Regarding the concept of power of an element, it is usual to work with the so-called principal powers, defined on an element a∈E, recursively, as a1=a, an=an−1·a. However, according to previous section, other powers distinct than the usual one can be considered, such as, for instance, plenary powers, which are defined on an element a∈Eas a[0]=a, a[n]=a[n−1]·a[n−1]. It is already known that these powers satisfy the following property a[n][m] =a[n+m]. 4Page 4 of 8 D. Fernández-Ternero et al. Let us also define, recursively, the following sets E[0]=E, E[n]=E[n−1]·E[n−1]={x·y:x,y∈E[n−1]}. Although it is usual to define E[n]as the subspace generated by the products of the form x·y, with x,y∈E[n−1], we will prove in the following proposition that in our case is somewhat unnecessary, since E[n]has a vector space structure with the definition we have considered. In addition, it is easy to check that E[n][m] =E[n+m]. From here on and unless other conditions are indicated, Ewill denote an evolution algebra, with natural basis B={ei:i∈}and structure matrix A=(aij). We have the following result Proposition 3.1 If L ∈End(E), then E[n]is an evolution subalgebra of E, for all n ≥0. A natural basis of E[n] is {e[n] i:i∈}, for a certain ⊆. In addition, the equalities e[n+1] i=Le[n] i=Ln+1(ei)= k∈ akie[n] k are verified. Proof We prove it by induction on n. For n=0 the result is trivial. Suppose it is satisfied for n−1 and let us see that it also holds for n. In the first place, we will see that E[n]=E[n−1]·E[n−1]has a vector space structure. If x,y∈E[n], then there exist x1,x2,y1,y2∈E[n−1]such that x=x1·x2and y=y1·y2. Since {e[n−1] i:i∈}is a basis of E[n−1], then xi= k∈ xkie[n−1] k,yi= k∈ ykie[n−1] k, for certain scalars xki and yki. Since this basis is natural x=x1·x2= k∈ xk1xk2e[n−1] k·e[n−1] k, y=y1·y2= k∈ yk1yk2e[n−1] k·e[n−1] k. Thus, the sum is x+y= k∈ (xk1xk2+yk1yk2)e[n−1] k·e[n−1] k, which, in turn, can be written as the product  k∈ (xk1xk2+yk1yk2)e[n−1] k· k∈ e[n−1] k. Each factor of this product is an element of E[n−1],sox+y∈E[n]. Furthermore, since E[n−1]is a subalgebra, we have that E[n]⊆E[n−1]. As a consequence, if x,y∈E[n], then x,y∈E[n−1]and hence x·y∈E[n]. This proves that E[n]is a subalgebra. In addition, from the above reasoning it follows that {e[n−1] i·e[n−1] i:i∈}={e[n] i:i∈}is a generator set of E[n], so there exists ⊆⊆such that {e[n] i:i∈}is a basis. Furthermore, this is a natural basis, since if i= j e[n] i·e[n] j=Le[n−1] i·Le[n−1] j=Le[n−1] i·e[n−1] j=L(0)=0. Dealing with the Resolubility of Evolution Algebras Page 5 of 8 4 Finally e[n+1] i=e[n] i·e[n] i=Le[n−1] i·Le[n−1] i=Le[n−1] i·e[n−1] i=Le[n] i, Le[n] i=LLn(ei)=Ln+1(ei), Ln+1(ei)=Ln(L(ei))=Ln k∈ akiek= k∈ aki Ln(ek)= k∈ akie[n] k.  An important observation is that from the equality e[n] i2 = k∈ akie[n] k we cannot deduce, in general, that the structure matrix associated to the evolution subalgebra E[n], with respect to the natural basis {e[n] i:i∈},isAor a submatrix of A, since is contained in . In fact, this last inclusion is strict as long as Lis not an automorphism. In the case where  k∈\ akie[n] k=0, we will have that the structure matrix is the submatrix of Aformed by the rows and columns whose indexes are in . We show next an example of this assertion. Example 3.2 Let us consider the evolution algebra Ewith basis {e1,e2,e3}and structure matrix A=⎛ ⎝ a−aλ a−aλ 001 ⎞ ⎠, where aand λare non-zero scalars. It can be verified that L(ei·ej)=L(ei)·L(ej),soL∈End(E). Since e[1] 1=−e[1] 2=a(e1+e2)and e[1] 3=λ(e1+e2)+e3, then E[1]=e[1] 1,e[1] 3. Since e[1] 12 =ae[1] 1+ae[1] 2=0, e[1] 32 =λe[1] 1+λe[1] 2+e[1] 3=e[1] 3, then the structure matrix of the evolution subalgebra E[1]is 00 01 , which is not a submatrix of matrix A. Moreover, we have that E[n]is an evolution ideal of E[n−1]:ifx∈E[n]⊆E[n−1]and y∈E[n−1], then x·y∈E[n]. In particular, E[1]is an evolution ideal of E. In the following example it can be seen that, in general, E[n]is not an ideal, for n>1. Moreover, in general, the evolution subalgebra E[n]does not have the extension property. Example 3.3 Let us consider the evolution algebra Ewith basis {e1,e2,e3}and structure matrix A=⎛ ⎝ a−aλ a−aλ 001 ⎞ ⎠, 4Page 6 of 8 D. Fernández-Ternero et al. where aand λare non-zero scalars. We have previously seen that in this case L∈End(E). Since A2=⎛ ⎝ 00λ 00λ 001 ⎞ ⎠ and e[2] i=L2(ei), then e[2] 1=e[2] 2=0, e[2] 3=λ(e1+e2)+e3and E[2]=e[2] 3. Let us see that E[2]is not an ideal. To do this we consider x=e1∈E, y=λ(e1+e2)+e3∈E[2]. Then x·y=λe2 1=λa(e1+e2)/∈E[2], from which it follows that E[2]is not an ideal. Let us now consider E[1]. Since e[1] 1=−e[1] 2=a(e1+e2)and e[1] 3=λ(e1+e2)+e3, then E[1]=e[1] 1,e[1] 3. Let us see that E[1]does not have the extension property. By contradiction, suppose there exists x=3 i=1xiei being linearly independent of e[1] 1and e[1] 3and such that x·e[1] 1=x·e[1] 3=0. Since x·e[1] 1=0, then a(x1e2 1+x2e2 2)=0, from which a2(x1−x2)(e1+e2)=0, so x1=x2. Since x·e[1] 3=0, then 0=λ(x1e2 1+x2e2 2)+x3e2 3=x3e2 3=λx3(e1+e2)+x3e3, so x3=0. Accordingly, x=x1(e1+e2), which is linearly dependent of e[1] 1, and so a contradiction is reached. Corollary 3.4 Let L ∈End(E)be the evolution operator of E.Let n be a natural number and ⊆a subset such that {e[n] i:i∈}is a natural basis of E[n]. Let Lbe the evolution operator of E[n]associated to this basis. Then, L=L|E[n]and L∈End(E[n]). Proof To prove that L=L|E[n], it suffices to see that this equality is satisfied by the elements of the natural basis {e[n] i:i∈} Le[n] i=e[n] i2 =e[n+1] i=Le[n] i. Furthermore, since L(x·y)=L(x·y)=L(x)·L(y)=L(x)·L(y), for all x,y∈E[n], it follows that L∈End(E[n]). A question that arises naturally is whether the sequence of subalgebras E=E[0]⊇E[1]⊇···⊇ E[n]⊇··· stabilizes and, if so, when does happen it. Regarding it, if Eis a finite dimensional evolution algebra, we obtain the following result Proposition 3.5 Let E be an evloution algebra with structure matrix of rank n −k. Then 1. If E[i]=E[i+1], then E[i+r]=E[i], for all r ≥0. 2. k =0if and only if E[r]=E, for all r ≥0. 3. If k >0, then E[n−k+1+r]=E[n−k+1], for all r ≥0. Dealing with the Resolubility of Evolution Algebras Page 7 of 8 4 Proof 1. Let us prove by induction on r.Forr=0 and r=1 is trivial. Suppose the result holds for r. Then E[i+r+1]=E[i+r]·E[i+r]=E[i]·E[i]=E[i+1]=E[i]. That is, the result holds for r+1. 2. E[1]is generated by {e2 i:i=1,...,n}and these vectors are the colums of A,sok=0(Ahas maximum rank) if and only if E[1]=E=E[0]. By the previous case, this is equivalent to E[r]=E, for all r≥0. 3. Since E[1]=Eis generated by {e2 i:i=1,...,n}, then dim(E[1])=n−k. Due to dimensionality, the equality E[i]=E[i+1]will be obtained for some i≤n−k+1. The result follows from case 1.  In the case of Ebeing a finite dimensional evolution algebra, the following result is easily obtained as an immediate consequence of the fact that E[n]is generated by {e[n] i:i=1,...,n}and that e[n] i=Ln(ei) Proposition 3.6 Let L ∈End(E)be the evolution operator of E. Then, E[n]=0if and only if An=0.Asa consequence, E is solvable if and only if A is a nilpotent matrix. Moreover, the solvability index of E and the nilpotency index of A are the same.  However, the following examples show that this result does not hold if L/∈End(E). Example 3.7 Let us consider the evolution algebra Ewith basis {e1,e2,e3}and structure matrix A=⎛ ⎝ 1−21 −12−1 1−21 ⎞ ⎠. On the one hand, it can be verified that An=4n−1A,soAis not nilpotent. On the other hand e2 1·e2 1=(e1−e2+e3)·(e1−e2+e3)=e2 1+e2 2+e2 3=0, e2 2·e2 2=−2e2 1·−2e2 1=4e2 1·e2 1=0, e2 3·e2 3=e2 1·e2 1=0, e2 1·e2 2=e2 1·−2e2 1=−2(e2 1·e2 1)=0, e2 1·e2 3=e2 1·e2 1=0, e2 2·e2 3=−2e2 1·e2 1=−2(e2 1·e2 1)=0. Since E[1]is generated by {e2 1,e2 2,e2 3}, then E[2]=0, therefore Eis solvable. Example 3.8 Let us consider the evolution algebra Ewith basis {e1,e2,e3}and structure matrix A=⎛ ⎝ 5−32 15 −96 10 −64 ⎞ ⎠. On the one hand, it can be verified that A2=0, so Ais nilpotent. On the other hand e2 2·e2 3=(−3e1−9e2−6e3)·(2e1+6e2+4e3)=−6e2 1−54e2 2−24e2 3= =−6(5e1+15e2+10e3)−54(−3e1−9e2−6e3)−24(2e1+6e2+4e3)= =84e1+252e2+168e3=−28 e2 2=42 e2 3. 4Page 8 of 8 D. Fernández-Ternero et al. Therefore (e2 2·e2 3)[1]=−28 e2 2·42 e2 3=−28 ·42 e2 2·e2 3=(−28)2·42 e2 2=−28 ·422e2 3. Let us see by induction that (e2 2·e2 3)[n]=(−28)2n422n−1e2 2=(−28)2n−1422ne2 3, from which it follows that Eis not solvable. Indeed, let suppose the result holds for nand let us see that it holds for n+1 (e2 2·e2 3)[n+1]=(e2 2·e2 3)[n]·(e2 2·e2 3)[n]=(−28)2n422n−1e2 2·(−28)2n−1422ne2 3= =(−28)2n+1−1422n+1−1e2 2·e2 3=(−28)2n+1422n+1−1e2 2= =(−28)2n+1−1422n+1e2 3. Funding Information Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Alsarayreh, A., Qaralleh, I., Ahmad, M.Z.: Derivation of three-dimensional evolution algebra. JP J. Algebra, Number Theory Appl. 39(4), 25–44 (2017) 2. Cadavid, P., Rodiño, M.L., Rodríguez, P.M.: The connection between evolution algebras, random walks and graphs. J. Algebra Appl. 19(2), 2050023 (2020) 3. Casas, J.M., Ladra, M., Omirov, B.A., Rozikov, U.A.: On nilpotent index and dibaricity of evolution algebras. Linear Algebra Appl. 43(9), 90–105 (2013) 4. Dzhumadil’daev, A., Omirov, B.A., Rozikov, U.A.: Constrained evolution algebras and dynamical systems of a bisexual population. Linear Algebra Appl. 496, 351–380 (2016) 5. Elduque, A., Labra, A.: Evolution algebras and graphs. J. Algebra Appl. 14(7), 1550103 (2015) 6. Falcón, O.J., Falcón, R.M., Núñez, J.: Classification of asexual diploid organisms by means of strongly isotopic evolution algebras defined over any field. J. Algebra 472, 573–593 (2017) 7. Labra, A., Ladra, M., Rozikov, U.A.: An evolution algebra in population genetics. Linear Algebra Appl. 457, 348–362 (2014) 8. López-Blázquez, F., Núñez-Valdés, J., Recacha, S., Villar-Liñán, M.T.: Connecting Statistics, Probability, Algebra and Discrete Mathematics. J. Theor. Probab. (2021). arXiv identifier arXiv:2104.02470 9. Núñez, J., Silvero, M., Villar, M.T.: Mathematical tools for the future: Graph Theory and graphicable algebras. Appl. Math. Comput. 219(11), 6113–6125 (2013) 10. Núñez, J., Rodríguez-Arévalo, M.L., Villar, M.T.: Certain particular families of graphicable algebras. Appl. Math. Comput. 246(1), 416–425 (2014) 11. Omirov, B.A., Rozikov, U.A.: On subalgebras of an evolution algebra of a chicken population. J. Algebra Relat. Topics 5(2), 13–24 (2017) 12. Tian, J.P.: Evolution algebra theory. Thesis (Ph.D.)- University of California, Riverside. (2004) 13. Tian, J.P.: Evolution algebras and their applications. Lecture Notes in Mathematics 1921. Berlin: Springer-Verlag (2008) 14. Tian, J.P., Vojtechovsky, P.: Mathematical concepts of evolution algebras in non-Mendelian genetics. Quasigroups Relat. Syst. 14(1), 111–122 (2006) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.