A historical perspective of Tian's evolution algebras

A historical perspective of Tian’s evolution algebras Manuel Ceballosa, Ra´ul M. Falc´onb,∗, Juan N´u˜nez-Vald´esc,´ Angel F. Tenoriod aDepartamento de Ingenier´ıa. Universidad Loyola Andaluc´ıa. Seville, Spain. bDepartmento de Matem´atica Aplicada I, Universidad de Sevilla. Seville, Spain. cDepartamento de Geometr´ıa y Topolog´ıa. Universidad de Sevilla. Seville, Spain. dDpto. de Econom´ıa, M´etodos Cuantitativos e Historia Econ´omica. Universidad Pablo de Olavide, Seville, Spain. Abstract Even if it has been less than a decade and a half since Tian introduced his concept of evolution algebras to represent algebraically non-Mendelian rules in Genetics, their study is becoming increasingly widespread mainly due to their applications to many scientific disciplines. In order to facilitate further research on the topic, this paper deals with the past and present research on these kind of algebras, together with the most relevant topics regarding them. Keywords: Evolution Algebras, genetic algebras, historical perspective. 2020 MSC: 05C12, 15-02. 1. Introduction1 Dealing with the chloroplast inheritance in plants, the German botanists2 and geneticists Carl Correns (1864–1933) [32] [33] and Erwin Baur (1875–3 1933) [6] realized the relevance that non-Mendelian rules have in Genetics.4 In spite of this, and unlike Mendelian Genetics, for which an algebraic inter-5 pretation of its rules was already introduced in the 1930’s by Serebrowski [89]6 and Kostitzin [63], it is not until the early 2000s that a similar interpretation7 for non-Mendelian rules was proposed.8 ∗Corresponding author. Email addresses: [email protected] (Manuel Ceballos), [email protected] (Ra´ul M. Falc´on), [email protected] (Juan N´u˜nez-Vald´es), [email protected] (´ Angel F. Tenorio) Preprint submitted to Expositiones Mathematicae July 9, 2021 More specifically, in order to relate Markov chains with the prokaryotic9 cell reproduction, and based on the self-reproduction rules of non-Mendelian10 Genetics, evolution algebras were firstly introduced in the Ph.D. Thesis of Jin11 Jung Paul Tian [92] in 2004; then jointly presented with Petr Vojtechovsky12 [98] in 2006, and later analyzed in greater depth in a book by Tian [93] in13 2008. The dynamic nature of these non-associative algebras is not described14 by a series of identities, unlike many other known non-associative algebras15 such as alternative, Lie, Malcev or Jordan algebras, amongst others. More16 specifically, an evolution algebra Eover a field Kis a finite-dimensional alge-17 bra such that there exists a natural basis {e1, . . . , en}so that ei·ej= 0, for all18 i, j ∈ {1, . . . , n}such that i6=j, and ei·ei=Pkpik ek,for all i∈ {1, . . . , n},19 and some coefficients pik ∈K, which are called the structural constants of20 the evolution algebra E. This multiplication is extended linearly from the21 given multiplication of basis elements. By considering each generator eias an22 allele in Genetics, the structure matrix (pij)i,j encodes somehow the laws of23 inheritance of non-Mendelian Genetics as algebraic properties of the algebra.24 Even if a wide range of mathematicians has dealt with this type of alge-25 bras since the original manuscripts of Tian and Vojtechovsky, the relevance26 of evolution algebras is not still widely known. In order to make easier27 further research on the topic, this paper gathers together the antecedents,28 origin, early stage and later development of this kind of algebras, and shows29 different applications arising from them.30 The paper is organized as follows. Section 2 outlines the origin and de-31 velopment of genetic algebras as primordial antecedents of Tian’s concept of32 evolution algebras. In Section 3 some preliminary concepts and results on33 evolution algebras are indicated. Next, in each subsection of Section 4, a34 brief comment on the main papers published on this topic is given. Further-35 more, Section 5 overviews the relationship among evolution algebras, Graph36 theory, Group theory, Markov chains and Biology. The paper is completed37 with an extensive bibliography, which may be of valuable help to all those38 researchers interested in this topic. Throughout the paper, we follow the39 current notation, which may differ from the original one to which we refer.40 2. Antecedents41 This section deals with the origin of genetic algebras as cornerstone of42 Tian’s concept of evolution algebra. For more details about the historical43 background on genetic algebras, we refer the reader to [12, 105, 67, 84].44 2 In 1866, Gregor Johann Mendel (1822–1884) was the first who made use45 of algebraic symbols to express sexual reproduction laws of inheritance. It46 was done in his original manuscript on experiments in plant hybridization47 [68]. More specifically, given a pair of plant characters Aand a, together48 with the hybrid form Aa in which both characters fuse together, Mendel49 indicated with an expression of the form αA+βAa +γathe ratio α:β:γ50 of numbers of offsprings having each possible plant character A,Aa and a51 in the following generation. In this way, Mendel introduced three principles:52 •The principle of paired factors, which states that characters are con-53 trolled by unit factors that appear in pairs within each individual or-54 ganism.55 •The principle of dominance, which states that one of the two previous56 factors is dominant and makes its effect in the individual, whereas the57 other one is recessive and does not show its effect unless both characters58 are recessive.59 •The principle of segregation, which states that, during the inheritance60 process, the mentioned pair of factors separate randomly so that the61 offspring receives exactly one factor from each parent.62 Almost forgotten for decades, Mendel’s laws were independently redis-63 covered around 1900 by the Dutch botanist Hugo de Vries (1848–1935) [103],64 who introduced the term mutation and suggested that of gene; the German65 botanist and geneticist Carl Correns (1864–1933) [32], who discovered the66 cell extranuclear inheritance; the Austrian agronomist Erich von Tschermak67 (1871–1962) [101], who introduced the combination of plant characters in or-68 der to improve the efficiency of plant breeding; and the American agronomist69 William Jasper Spillman (1863–1931) [90], whose work was crucial for the70 development of Agricultural Economics. The emergence of all their contribu-71 tions concerning the distribution of characters among offspring in plant hy-72 bridization made fundamental the study and development of Mendel’s laws.73 In 1905, the English biologist William Bateson (1861–1926) introduced74 the term Genetics to describe the study of inheritance processes [5]. Bateson75 was a fervent advocate of Mendel [4], whose principles, even being reborn,76 were questioned at that time. An interesting question in this regard had been77 asked in 1902 by the British statistician George Udny Yule (1871–1951),78 who contemplated [107] that dominant factors could increase indefinitely79 3 in a continuous way throughout generations. One year later, the English80 mathematician and biostatistician Karl Pearson (1857–1936) [83] and the81 American zoologist and geneticist William Ernest Castle (1867–1962) [29]82 established that, under random circumstances, there exists certain stability of83 characters in the inheritance process. Concerning such a stability, the English84 mathematician Godfrey Harold Hardy (1877–1947) [51] asked in 1908 about85 the circumstances under which the distribution of characters in the offspring86 is the same that in the generation before in the absence of disturbing factors.87 The answer was independently given at that moment by Hardy himself88 and the German physician Wilhelm Weinberg (1862–1937) [104], and is cur-89 rently known as the Hardy-Weinberg law. An important limitation of that law90 is the assumption of a potentially infinite population, not taking into account91 the importance of sampling fluctuations in evolutionary processes described92 by finite populations. The first proposal dealing with such a possibility would93 be the so-called Wright-Fisher model based on the original manuscripts of the94 American geneticist Sewall Green Wright (1889–1988) [106] and the British95 statistician and geneticist Ronald Aylmer Fisher (1890–1962) [44].96 In 1923-24, the Soviet mathematician Sergei Natanovich Bernstein (1880-97 1968) [10, 11] introduced the concept of quadratic stochastic operator (QSO)98 as a map V:S→S, where99 S:= {(x1, . . . , xn)∈Rn:xi>0,for all i, and X i xi= 1}, and, for each (x1, . . . , xn)∈S, it is V((x1, . . . , xn)) = (V(x1), . . . , V (xn)),100 where101 V(xk) := X i,j pk ijxixj,for all k∈ {1, . . . , n}. Here, the product xixjis the usual product of real numbers. In addition,102 pk ij ≥0 and pk ij =pk ji, for all i, j, k ∈ {1, . . . , n}, and Pkpk ij = 1, for all103 i, j ∈ {1, . . . , n}. As such, every QSO constitutes an evolutionary operator104 that describes the time evolution or inheritance process of a free population105 with ndifferent genetic types.106 More specifically, each component xiof a given element x= (x1, . . . , xn)∈107 Srepresents the probability that a random individual in the population under108 consideration belongs to the species that is determined by the ith genetic109 type. Hence, the n-tuple xdescribes the distribution of the population with110 respect to the ngenetic types, whereas V(x) describes such a distribution111 4 for the next generation. In particular, each value pk ij =pk ji determines the112 probability that an offspring with genetic type karises from two individuals113 of respective genetic types iand j, without sexual differentiation.114 Based on the possible fluctuation of genetic types throughout subsequent115 generations of an evolutionary process, a main problem in the theory of QSOs116 consists of determining their limit behavior for any given initial distribution117 of genetic types x∈S. That is, the study of the corresponding distribution118 Vm(x) for the mth generation, when mtends to infinity. In this regard,119 Berstein focused in particular on the problem of determining and classifying120 all those QSOs for which a stationary distribution or stable evolution arises121 in only one generation (that is, such that V2=V). Even if Berstein only122 solved this problem for n= 3, it is currently solved for all dimensions (see123 [49, 50]). To this end, it would be crucial the work in 1975 of Philip Holgate124 (1934–1993) [56], who expressed algebraically this problem as follows.125 1. Firstly, Holgate related each given evolutionary operator Vwith the126 algebra described as127 xy =1 2(V(x+y)−V(x)−V(y)) . (It was called evolution algebra by Joseph Bayara [7, 8], which differs128 from the concept introduced by Tian, the one that is commonly used129 in the current literature and with which this paper deals.)130 2. Then, he introduced the concept of Bernstein algebra as an algebra131 Asuch that x2x2=ω2(x)x2, for all x∈A, where ωis an algebra132 homomorphism from Ainto its base field, which is called the weight133 of the algebra. Recall that an algebra for which one such a non-trivial134 homomorphism exists is called baric [39].135 In 1934, Alexander Pawlowitsch Serebrowski (1884-1938) [89] was the first136 to give an algebraic interpretation of the symbol ×as a mathematical way to137 represent Mendelian inheritance laws. A similar symbolic multiplication was138 independently introduced shortly after by Aleksandrovich Kostitzin (1883-139 1963) [63].140 At the same period, Valery Ivanovich Glivenkov (1896-1940) [46] intro-141 duced the so-called Mendelian algebras for diploid species. Nevertheless, it142 was Ivor Malcolm Haddon Etherington (1908-1994) who, in 1939, introduced143 in Genetics [39] the systematic study of commutative non-associative linear144 algebras, by describing in particular the following three types of algebras:145 5 •Agametic algebra is a finite-dimensional algebra of basis {e1, . . . , en}146 such that ei·ej=Pkpk ijek, where each structural constant pk ij belongs147 to the real interval [0,1] so that Pkpk ij = 1, for all i, j ∈ {1, . . . , n}.148 As such, each value pk ij represents the probability that an arbitrary149 gamete of zygotic type ekderives from an individual of zygotic type150 eiej. As occurs with the QSOs, the probability or progeny distribution151 describes the evolution of the population under consideration. The152 latter constitutes a free population if the inheritance process derives153 from random matings; that is, with absence of sexual differentiation154 and selection.155 •Azygotic algebra is the duplicate of a gametic algebra. That is, the156 former is isomorphic to the set of quadratic forms of the latter.157 •Acopular algebra is the duplicate of a zygotic algebra.158 Each one of the three just described algebras are baric by means of the159 weight ωthat is linearly defined from ω(ei) = 1, for all i∈ {1, . . . , n}. Here,160 {e1, . . . , en}is the basis of the algebra under consideration, and hence,161 ω(ei)ω(ej) = 1 = X k pk ij =X k pk ijω(ek) = ω X k pk ijek!=ω(eiej), for all i, j. Further, Etherington called train algebra any baric algebra A162 with weight ωsuch that the equation of lowest degree relating the principal163 powers of any vector x∈Ahas the form164 xm+c1ω(x)xm−1+. . . +cmω(x)m= 0. Finally, he termed special train algebra any baric algebra Awith weight ω165 such that N= ker(ω) is nilpotent (that is, there exists a positive integer166 m∈Nsuch that Nm= 0) and all the principal powers Ni,with i∈N, are167 ideals of A. In particular, every special train algebra is a train algebra.168 Etherington called genetic algebra any one of the family of gametic, zy-169 gotic and copular algebras and ensured that “all the fundamental genetic170 algebras are special train algebras”. Shortly after, he indicated [40] that this171 statement is only valid for gametic algebras, but not even for zygotic algebras,172 because the duplicate of a special train algebra, although a train algebra, is173 not always a special train algebra. This fact provided serious problems for174 translating some properties and relations from genetic inheritance processes.175 6 In order to avoid the flaws in Etherington’s genetic algebras and give rise176 to a transparent structure theory for them, Richard Donald Schafer (1918–177 2014) [88] provided in 1949 an alternative formal definition that is interme-178 diate between the concepts of train algebra and special train algebra. To this179 end, he made use of the transformation algebra T(A) of a non-associative al-180 gebra A. It is the algebra of all the polynomials with coefficients in the base181 field, whose variables are transformations in the set formed by the identity182 map in A, all the right multiplications Rα:A7→ Asuch that Rα(x) = x α,183 and all the left multiplications Lα:A7→ Asuch that Lα(x) = α x, for all184 α, x ∈A. In particular, the transformation algebra of a baric algebra is baric.185 Further, unlike Etherington, Schafer assumed the commutativity that is186 inherent in any inheritance process and hence, he considered Rα=Lα, for187 all α∈A. Then, he called genetic algebra any commutative baric algebra188 Asuch that all the coefficients of the characteristic function |λI −T|, with189 T∈T(A), only depend on the weight of the algebra. As such, every genetic190 algebra is a train algebra and every special train algebra is a genetic algebra.191 Moreover, unlike special train algebras, the duplicate of a genetic algebra is192 always a genetic algebra.193 In 1971, Harry Gonsh¨or [47] proved that the concept of genetic alge-194 bra proposed by Schafer is equivalent to have a commutative algebra Aof195 basis {e0, e1, . . . , en}such that eiej=Pkck ij ek, for all i, j ∈ {0, . . . , n},196 where c0 00 = 1; ck 0j= 0,for all k < j; and ck ij = 0,for all i, j > 0 and197 k≤max {i, j}. This new definition was proved to have significant relevance198 in Genetics thanks to the genetic meaning that Holgate [55, 57] gave of du-199 plicates and derivations of a genetic algebra. The study of the subspace200 Der(A) of derivations of a genetic algebra Ahas also been dealt with by201 Gonsh¨or [48] himself and also by Roberto Costa [34, 35], Aribano Micali202 and Philippe Revoy [69], and, much more recently, by Rasul Ganikhodzhaev,203 Farrukh Mukhamedov, Abror Pirnapasov and Izzat Qaralleh [45]. Recall in204 this regard that D∈Der(A) if and only if D(xy) = D(x)y+xD(y), for all205 x, y ∈A. As for any algebra, such a subspace constitutes a Lie algebra.206 Holgate also introduced [54] both notions of sex differentiation algebra207 and dibaric algebra. The former characterizes the equal division of offspring208 between both sexes. More specifically, the sex differentiation algebra is a bi-209 dimensional commutative algebra of basis {m, w}such that m2=w2= 0 and210 mw =wm = (m+w)/2. Further, an algebra is called dibaric if it admits211 a homomorphism onto the sex differentiation algebra. Holgate proved in212 particular that, if Ais a dibaric algebra, then its derived algebra A2is a213 7 baric algebra. With the introduction of dibaric algebras, he formalized the214 original idea of Etherington [39] of treating in a separate way male and female215 components of the population.216 Another contribution of Holgate [53] in the theory of genetic algebras217 was the use of isotopisms of algebras in order to represent algebraically the218 mutation of genotypes in the inheritance process. Recall in this regard that219 two n-dimensional algebras Aand A0are said to be isotopic [1] if there exist220 three non-singular linear transformations f,gand hfrom Ato A0such that221 f(u)g(v) = h(uv), for all u, v ∈A. The triple (f, g, h) is called an isotopism222 (an isomorphism, if f=g=h; and an homomorphism, if besides, the223 condition of non-singularity is not imposed) between the algebras Aand A0 224 (see [43] for a recent survey on the theory of isotopisms). Together with225 Tania Campos [24], Holgate proved that certain types of zygotic algebras226 representing chromosome segregation and recombination are isotopic.227 3. Preliminaries on evolution algebras228 In order to make this article as self-contained as possible and facilitate229 a better understanding for the reader, we recall in this section some notions230 and basic results on evolution algebras that were introduced in the original231 manuscripts of Tian and Vojtechovsky [93, 98]. To this end, and from now232 on, let Edenote an n-dimensional evolution algebra over a base field K, with233 natural basis {e1, . . . , en}and structural constants pik ∈K, for all i, k ∈234 {1, . . . , n}. If the base field Kis the real field, then the evolution algebra235 is said to be real. In such a case, it is said to be non-negative if pik ≥0,236 for all i, k ∈ {1, . . . , n}. If, besides, Pkpik = 1, for all i, then the real and237 non-negative evolution algebra is called Markov evolution algebra.238 Every evolution algebra Eis non-associative, commutative, flexible (that239 is, x(yx)=(xy)x, for x, y ∈E); not necessarily power-associative (that is,240 the subalgebra generated by a single element can not be associative); and241 preserved by direct sums. Moreover, evolution algebras are not closed under242 subalgebras. So, a subalgebra of Eis called an evolution subalgebra if the243 former is spanned by a subset of generators of the latter. As such, it is an244 ideal of E. The latter is indecomposable if it is not the direct sum of two245 nonzero ideals; connected if it is not the direct sum of two proper evolution246 subalgebras; irreducible if it has no proper subalgebra; and simple if it has no247 proper evolution subalgebra. Tian proved [93, Theorem 9] that any finite-248 dimensional evolution algebra has a simple evolution subalgebra.249 8 Every evolution algebra Eis uniquely determined by its evolution operator250 L:E7→ E, which is linearly described from251 L(ei) := eiei=X i pikek,for all i∈ {1, . . . , n}. Then, for each positive integer m > 2, it is defined the plenary power252 e[m] i:= Le[m−1] i, where e[0] i:= ei. The operator Lmay be considered as time-step in a discrete-253 time dynamical system, and hence, it describes the dynamical flow of the254 evolutionary process represented by the evolution algebra E. In this regard,255 a generator eiof the evolution algebra Eis called algebraically persistent if256 the evolution subalgebra generated by eiis a simple evolution subalgebra. As257 such, it represents an absorbent state giving rise to a stable evolution. Oth-258 erwise, it is said to be algebraically transient, which gives rise to a transitory259 evolution. The latter would eventually derive to an absorbent state, but only260 after certain generations. Any generator of a simple evolution subalgebra is261 algebraically persistent. In particular, every finite-dimensional evolution al-262 gebra contains an algebraically persistent generator.263 A connected evolution algebra is simple if and only if all its generators264 are algebraically persistent. Tian proved [93, Theorem 11] that, if Eis a265 connected finite-dimensional evolution algebra, then266 E= n0 M j=1 E0,j +B0, where Land + denote, respectively, direct sums of either subalgebras or lin-267 ear subspaces; each E0,j is a simple evolution subalgebra so that E0,j ∩E0,j0=268 {0}, whenever j6=j0; and B0is a linear subspace spanned by algebraically269 transient generators, which is called transient space. Even if B0is not an270 evolution subalgebra in general, it may be endowed of evolution structure271 from Eonce the multiplication is restricted within B0. The previous decom-272 position process can inductively be repeated until no transient space arises.273 That is, there exist an integer m > 0 and a subset {n1, . . . , nm} ⊂ Nsuch274 that275 E= m X i=0 ni M j=1 Ei,j, 9 As for any algebra, the space of derivations of a given algebra is a Lie484 algebra that constitutes a good approximation to its automorphism group485 and hence, to its algebraic structure. The system of equations describing by486 this space was given by Tian himself [93]. More specifically, if dis a derivation487 of the evolution algebra Ethat is described so that d(ei) = Pkdikek, with488 dik ∈K, for all i, then it must be489 pikdji +pjkdij = 0,for all i, j, k such that i6=j, and490 X k pikdkj = 2pijdii,for all i, j. It was not until 2013 that Camacho, G´omez, Omirov and Turdibaev [21]491 studied formally the space of derivations of n-dimensional complex evolu-492 tion algebras depending on the rank of certain matrices. In particular, they493 proved that such a subspace is zero for evolution algebras having a non-494 singular structure matrix [21, Theorem 2.1]. They also described the space495 of derivations of those n-dimensional evolution algebras having a structure496 matrix of rank n−1.497 Much more recently, Paula Cadavid, Mary Luz Rodi˜no and Pablo M.498 Rodr´ıguez [20] described the space of derivations of those evolution algebras499 that are uniquely associated with finite simple and connected graphs of order500 n≥3 according to Tian’s proposal (see Subsection 5.1). It was observed that501 each one of these spaces of derivations depends on the twin partition of the set502 of vertices of the corresponding graph. (Recall here that two vertices within a503 graph are called twins if their neighborhoods coincide.) In particular, if every504 part in this twin partition contains at most two vertices, then the space of505 derivations is only formed by the null map [20, Theorem 2.3]. The space of506 derivations in case of existing some part within this twin partition with at507 least three vertices was also characterized [20, Theorem 2.6]. It is remarkable508 the fact that the authors dealt with examples of finite-dimensional evolution509 algebras having structure matrices of any rank.510 The same authors, together with Cabrera, have recently explored [15] a511 similar extended approach by considering to this end the non-zero entries512 of the structure matrix of the evolution algebra under consideration and its513 associated directed graph, which had previously been described by Cabrera,514 Siles and Velasco [17]. By making use of this approach, the authors have515 characterized the derivations of non-degenerate irreducible three-dimensional516 evolution algebras.517 16 Let us finish this subsection with a proposal of further work on the topic518 under consideration. Similarly to the relationship between the subset of519 derivations of an algebra and its automorphism group, there also exists a re-520 lationship between its autotopism group (that is, the group of isotopisms521 preserving the algebra Aunder consideration) and its subset of ternary522 derivations. This terminology was introduced for general algebras by Clara523 Jim´enez-Gestal and Jos´e Mar´ıa P´erez-Izquierdo [61]. More specifically, a524 ternary derivation of an algebra Ais any triple (d1, d2, d3) of endomorphisms525 of the algebra such that526 d1(xy) = d2(x)y+xd3(y), for all x, y ∈A. (See [3] for a comprehensive motivation of this concept.)527 The following question arises from the important role that isotopisms play528 as algebraic representations of mutations in Genetics.529 Problem 4.1. Characterize the subspace of ternary derivations of any given530 finite-dimensional evolution algebra and establish the relationship with its531 autotopism group.532 4.7. Classification of evolution algebras533 In the previoius subsections, the distribution into isomorphism classes of534 different types of evolution algebras has been outlined. Let us indicate here535 some more results concerning the classification of finite-dimensional evolution536 algebras.537 In 2013, Casas, Ladra, Omirov and Rozikov distributed [27, Theorem 4.1]538 all the two-dimensional evolution algebras over the complex field into six non-539 isomorphic classes. Nevertheless, Cabrera, Siles and Velasco [18] realized that540 this classification did not consider the evolution algebra with natural basis541 {e1, e2}verifying e2 1=e2and e2 2=e1. It was observed when the authors542 classified those three-dimensional evolution algebras whose derived algebra543 has dimension two and having annihilator of dimension one [18, Theorem544 3.5].545 In 2014, Murodov distributed [73, Theorem 1] into isomorphism classes546 the set of two-dimensional evolution algebras over the real field. More re-547 cently, the classification over a general field was independently achieved by548 ´ Oscar J. Falc´on, Ra´ul M. Falc´on and Juan N´u˜nez [41], and by Maria Inez549 Cardoso, Daniel Gon¸calves, Dolores Mart´ın, C´andido Mart´ın and Siles [25].550 17 In [41], it was also observed that the distribution of evolution algebras into551 isotopism classes is uniquely related with the mutation of alleles in non-552 Mendelian Genetics. In that reference, it was obtained the distribution into553 four isotopism classes of all two-dimensional evolution algebras, whatever the554 base field is.555 In 2017, Cabrera, Siles and Velasco [18] classified into 116 distinct types556 the set of three-dimensional evolution algebras over any field of characteristic557 distinct from two and in which there are roots of orders two, three and seven.558 Their results agreed with those ones of Elduque and Labra [37] concerning the559 classification of indecomposable nilpotent evolution algebras of dimension up560 to five over any algebraically closed fields of characteristic distinct from two.561 More recently, based on an alternative approach, Mar´ıa Eugenia Celorrio and562 Velasco [31, Theorem 11] classified the mentioned 116 types of algebras into563 14 non-isomorphic types,564 In 2018, the distribution of three-dimensional evolution algebras having565 one-dimensional annihilator was determined in [42, Theorem 4.7], where it566 was also established [42, Corollary 3.3] the distribution of three-dimensional567 evolution algebras into isotopism classes, whatever the base field is. This568 last classification enables one to describe the spectrum of genetic patterns of569 three distinct genotypes during a mitosis process.570 Also in 2018, Anvar N. Imomkulov [58] introduced an evolution operator571 for evolution algebras. Both the set of fixed points and the Jacobian matrix572 of this operator were studied for two-dimensional evolution algebras. In par-573 ticular, it was established the distribution into isomorphism classes of those574 two-dimensional evolution algebras having this Jacobian matrix as structure575 matrix. The distribution of the three-dimensional case was also dealt with576 by the same author [59], who proposed as further work the following research577 problem.578 Problem 4.2. Study possible approximations of finite-dimensional algebras579 by means of evolution algebras.580 A first approach of this problem was considered by Imomkulov himself,581 together with Rozikov, who determined [60] the relationship among twoand582 three-dimensional Leibniz algebras and evolution algebras.583 18 5. Relationship to other topics584 This section outlines the relationship among evolution algebras, Graph585 theory, Group theory, Markov chains and Biology.586 5.1. Graph theory587 In his original manuscript, Tian [93] realized that every finite graph G=588 (V, E) may be associated with an evolution algebra A(G) having the vertex589 set V={e1, . . . , en}as its natural basis, and such that590 eiei=X ek∈Γ(ei) ek, for every positive integer i≤n, where Γ(ei) is the neighbourhood of the ver-591 tex eiin the graph G. In this way, isomorphic graphs give rise to isomorphic592 evolution algebras [93, Theorem 41]. Moreover, if593 Lm(ei) = n X j=1 pikek, where Lis the evolution operator associated to A(G), then pik coincides with594 the total number of paths of length mbetween both vertices eiand ejin the595 graph G[93, Theorem 43].596 In order to deal with the reciprocal, Tian [93] introduced the concept of597 graphicable algebra as an evolution algebra of natural basis V={e1, . . . , en}598 such that599 eiei=X ek∈Vi ek, for all positive integer i≤n, where Vi⊆V. Even if evolution algebras are600 not graphicable in general, every graphicable algebra is uniquely associated601 to a graph such that Vi= Γ(ei). In this way, two isomorphic graphicable602 algebras give rise to isomorphic graphs [93, Theorem 42]. In particular, Tian603 introduced cycle algebras,path algebras and complete algebras as graphicable604 algebras giving rise, respectively, to cycles, paths and complete graphs. In605 a similar way, N´u˜nez, Mar´ıa Luisa Rodr´ıguez-Ar´evalo and Mar´ıa Trinidad606 Villar [75] dealt with complete tripartite and n-partite, star, friendship, wheel607 and snark graphicable algebras. These authors also proved [75, Theorem608 3.8] the existence of graphicable subalgebras whithin a graphicable algebra.609 19 Further, N´u˜nez, Marithania Silvero and Villar dealt [76] with the particular610 case of graphicable algebras for which ek∈Viif and only if ei∈Vk, and611 ei6∈ Vi, for every pair of positive integers i, k ≤nsuch that i6=k. They612 called S-graphicable algebras this type of algebras.613 Based on both notions of evolution algebra arising from a graph and614 graphicable algebra, Tian remarked [93] that the ‘intrinsic and coherent re-615 lation of evolution algebras with graph theory allows to analyze graphs alge-616 braically [...] and graph theory may be used as a tool to study non-associative617 algebras”. It is so that he asked whether “every statement or problem in618 graph theory can be translated into the language of evolution algebras”. He619 put particular interest in the possible relationship of evolution algebras with620 random graphs and networks, random walks on graphs, weighted graphs and621 directed graphs. Furthermore, Tian also asked for those evolution algebras622 arising from finite graphs whose associated Ihara-Selberg zeta function satisfy623 the Sunada’s analogue of Riemann hypothesis [91].624 In 2011, Rozikov and Tian [86] described an alternative for relating an625 evolution algebra to a given finite and simple graph by means of state spaces626 endowed with Gibbs measures. For connected graphs, they determined the627 hierarchical structure of these algebras, together with the dimension and628 number of oneand four-dimensional evolution subalgebras [86, Theorem 3.2].629 Moreover, all the evolution algebras arising from the same finite and simple630 graph, but defined by different Gibbs measures, are pairwise isomorphic [86,631 Theorem 3.3]. The authors established as further open problem the study of632 the hierarchical structure of evolution algebras associated to graphs that are633 either finite, but not connected; or countable and connected.634 In 2015, Elduque and Labra [36] defined the directed graph attached to an635 n-dimensional evolution algebra of structure matrix (pik)i,k with respect to a636 given natural basis as the graph of set of vertices V={1, . . . , n}and set of637 arcs A={(i, k)∈V×V:pik 6= 0}. If each arc (i, k)∈Ais labeled as pik,638 then one obtains the directed weighted graph attached to E. If Ann(E) = {0},639 then its directed graph is connected if and only if Edoes not split into a640 direct sum of simple ideals [36, Proposition 2.8]. If the annihilator is not641 trivial, then this decomposition is not possible if and only if its directed642 graph is connected for every natural basis [36, Proposition 2.10] (see also [17,643 Proposition 5.4]). Further, Eis nil if and only if its attached directed graph644 contains no oriented cycles [36, Theorem 3.4]. These attached (weighted)645 directed graphs were also used to distribute into isomorphism classes the set646 of fourand five-dimensional nilpotent evolution algebras [37].647 20 In 2020, Cadavid, Rodi˜no and Rodr´ıguez [19] described the evolution648 algebra induced by the random walk on a graph and studied its relationship649 with the evolution algebra determined by the same graph according to Tian’s650 proposal [93]. Recall here that the random walk on a graph of set of vertices651 Vand adjacency matrix (aij)i,j is a discrete Markov chain with state space652 Vsuch that the transition probability of moving from state ito state jis653 pij =aij Pk∈Vaik . The mentioned authors described the evolution algebra induced by this ran-654 dom walk as the evolution algebra of natural basis {e1, . . . , en}and structure655 matrix (pij)i,j. Then, they studied under which conditions one may ensure656 that this evolution algebra is isomorphic to Tian’s evolution algebra associ-657 ated to the graph under consideration.658 Also in 2020, Manuel Ceballos, N´u˜nez and ´ Angel Tenorio [30] described659 the directed weighted pseudograph associated to an n-dimensional evolution660 algebra of structure matrix (pik)i,k as the pseudograph resulting after adding661 a loop on each vertex iof the Elduque and Labra’s directed weighted graph662 attached to E, whenever pii 6= 0. This loop is then labeled as pii. The663 distribution into isomorphism classes of these pseudodigraphs enabled them664 to establish a new classification of evolution algebras.665 5.2. Group theory666 Tian [93] associated any given group (G, ◦) having a finite set of genera-667 tors Swith the evolution algebra described over a base field Kso that, for668 each g∈G, one has669 g·g=X e∈S keg·e, (1) for some ke∈K. Particularly, Tian set out the following question.670 Problem 5.1 ([93]). How can the properties of a group be translated to the671 corresponding evolution algebra?672 5.3. Markov chains673 Tian himself [93] realized that every Markov chain is related to a Markov674 evolution algebra whose structural constants coincide with the transition675 probabilities of the former and whose basis constitutes the state space of the676 Markov process.677 21 As such, generators of Markov evolution algebras represent states of678 stochastic processes described by Markov chains, which may, therefore, be679 studied by means of the theory of evolution algebras. In particular, Markov680 chains may be classified according to the hierarchies of their corresponding681 evolution algebras. Tian proved that a Markov chain is irreducible if and682 only if its related evolution algebra is simple [93, Theorem 18], and that a683 subset of state space of a Markov chain is probabilistic closed if and only if it684 generates an evolution subalgebra [93, Theorem 17]. In addition, probabilis-685 tic periods in Markov chains are uniquely related to periods of generators in686 evolution algebras [93, Proposition 10].687 Tian also introduced the possibility of dealing with continuous evolution688 algebras, once structural constants are replaced by differential functions de-689 pending on a variable. If the latter represents the time, then continuous690 evolution algebras are uniquely related to continuous-time Markov chains691 [99]. Tian thought that the study of continuous evolution algebras “will be692 very interesting because they have a kind of semi-Lie group structure”.693 In 2011, as a generalization of Tian’s idea, Casas and Rozikov [28] re-694 alized that every Markov chain is associated to a chain of n-dimensional695 evolution algebras {A[s,t]: 0 ≤s≤t}depending on two time parameters s696 and t. Their structure matrices M[s,t]are all of them stochastic and sat-697 isfy that M[s,t]=M[s,τ]M[τ,t], for all s < τ < t. This chain represents a698 continuous time dynamical system that is an evolution algebra in each fixed699 time. If all the matrices of the chain coincide, then they constitute the same700 evolution algebra that Tian associated to a Markov chain. Furthermore, if701 both time parameters sand tare reduced to only one of them, then the chain702 is called time-homogeneous and is associated to a time-homogeneous Markov703 chain. The authors also dealt with chains of evolution algebras for which the704 stochastic condition is removed.705 In 2013, Rozikov and Mudorov [85] constructed 25 distinct examples of706 chains of two-dimensional evolution algebras, for which they studied their707 baricity, nilpotency and idempotency. In 2015, Omirov, Rozikov and Kaisar708 Tulenbayev [77] dealt with real chains of n-dimensional evolution algebras709 corresponding to a permutation of nnumbers and showed that one such710 a chain is trivial if and only if the permutation has no fixed points [77,711 Proposition 1]. Moreover, since every trivial chain of evolution algebras is712 a chain of nilpotent evolution algebras [77, Proposition 2], they constructed713 chains of three-dimensional evolution algebras. They also dealt with the714 construction of arbitrary-dimensional symmetric chains of evolution algebras.715 22 In 2017, Ladra and Rozikov [64, 65] described non-homogeneous continu-716 ous time Markov chains. More recently, Irene Paniello [82] has delved into the717 algebraic structure of Markov evolution algebras for both the discrete-time718 and the continuous-time arising from standard stochastic semigroups.719 5.4. Biology720 Since the original manuscripts of Tian and Vojtechovsky, evolution al-721 gebras have been implemented in Biology to represent different aspects of722 non-Mendelian inheritance (particularly, uniparental inheritance) in an alge-723 braic way. Thus, Tian himself [94] made use of these algebras to analyze how724 the homoplasmy of a cell population (represented by persistent generators)725 can derive from an heteroplasmy one (represented by transient generators).726 This is useful, for instance, to study mitochondrial disorders and mutations727 in tissues of patients. He also implemented evolution algebras to describe728 genetically dynamical patterns that provide information about the asexual729 reproduction of Phytophthora infectans causing the late blight of tomatoes730 and potatoes. An algebraic concept as nilpotency represents the extinction731 of original genetic types after certain generations.732 Continuous-time dynamical systems of mosquito populations were studied733 by Rozikov and Velasco [87]. Mosquito populations were already been dealt734 with from an algebraic point of view in 2011 by Junliang Lu and Jia Li [66],735 who proposed a discrete time structured model of such a population. On736 their own, Rozikov and Velasco considered a discrete-time dynamical system737 described by an evolution algebra with two fixed points, which become saddle738 points under some conditions on the parameters of the system. Its structure739 matrix is equal to the Jacobian of the QSO at a fixed point.740 Another aspect that Tian [93] realized was the possible use of evolution741 algebras on coalescent theory [97]. That is, on genetic evolution reversely742 over time. The study of backwards evolution of Mendelian genetic systems743 by means of coalgebras had been previously introduced in 2004 by Tian and744 Bai-Lian Li [96]. In 2019, Paniello [81] introduced the concept of evolution745 coalgebra in order to model backwards evolution of Non-Mendelian genetic746 systems. She established the connection between such coalgebras and evolu-747 tion algebras, by focusing in particular on the cases of genetic realizations.748 In 2020, Miguel Bustamante, Mellon and Velasco [14] described necessary749 and sufficient conditions for a given algebra to be an evolution algebra. They750 proved that this problem is equivalent to the simultaneous diagonalization751 via congruence of a given set of matrices. Based on this fact, they realized752 23 that arbitrarily small perturbations of classical genetic algebras representing753 Mendelian and auto-tetraploid inheritance (which are not evolution algebras)754 may give rise to evolution algebras. The algebraic representations of sexual755 and asexual inheritance by means of genetic and evolution algebras seem,756 therefore, to be closer than previously thought. A further study of baric757 and evolution algebras was established by the authors as a first stage to758 better understand this relationship. In any case, notice that Mukhamedov759 and Qaralleh already set out the following question in 2014.760 Problem 5.2 ([70]). Is there a transformation of a given genetic algebra to761 some evolution algebra?762 They gave an affirmative answer in case of dealing with two-dimensional763 genetic algebras (see [70, Theorem 4.1]). The resulting transformation en-764 abled them to provide necessary conditions on the structure matrix of these765 genetic algebras for ensuring the existence of non-trivial derivations (see [70,766 Theorem 5.1]).767 6. 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