A historical perspective of Tian’s evolution algebras

Please cite this article as: M. Ceballos, R.M. Falc´ on, J. N´ u˜ nez-Vald´ es et al., A historical perspective of Tian’s evolution algebras, Expositiones Mathematicae (2021), https://doi.org/10.1016/j.exmath.2021.11.004. Available online at www.sciencedirect.com ScienceDirect Expositiones Mathematicae xxx (xxxx) xxx www.elsevier.com/locate/exmath A historical perspective of Tian’s evolution algebras Manuel Ceballosa, Raúl M. Falcónb,∗, Juan Núñez-Valdésc, Ángel F. Tenoriod aDepartamento de Ingeniería. Universidad Loyola Andalucía. Seville, Spain bDepartamento de Matemática Aplicada I. Universidad de Sevilla. Seville, Spain cDepartamento de Geometría y Topología. Universidad de Sevilla. Seville, Spain dDepartamento de Economía, Métodos Cuantitativos e Historia Económica. Universidad Pablo de Olavide, Seville, Spain Received 8 April 2021; received in revised form 17 September 2021; accepted 22 November 2021 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. ©2021TheAuthor(s).PublishedbyElsevierGmbH.ThisisanopenaccessarticleundertheCC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). MSC 2010: primary 05C12; secondary 15-02 Keywords: Evolution algebras; Genetic algebras; Historical perspective 1. Introduction Dealing with the chloroplast inheritance in plants, the German botanists and geneticists Carl Correns (1864–1933) [35,36] and Erwin Baur (1875–1933) [5] realized the relevance that non-Mendelian rules have in Genetics. In spite of this, and unlike Mendelian Genetics, for which an algebraic interpretation of its rules was already introduced in ∗Corresponding author. E-mail addresses: [email protected] (M. Ceballos), [email protected] (R.M. Falc´ on), jnv[email protected] (J. N´ u˜ nez-Vald´ es), [email protected] (´ A.F. Tenorio). https://doi.org/10.1016/j.exmath.2021.11.004 0723-0869/© 2021 The Author(s). Published by Elsevier GmbH. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article as: M. Ceballos, R.M. Falc´ on, J. N´ u˜ nez-Vald´ es et al., A historical perspective of Tian’s evolution algebras, Expositiones Mathematicae (2021), https://doi.org/10.1016/j.exmath.2021.11.004. 2 M. Ceballos, R.M. Falcón, J. Núñez-Valdés et al. / Expo. Math. xxx (xxxx) xxx the 1930s by Serebrowski [105] and Kostitzin [70], it is not until the early 2000s that a similar interpretation for non-Mendelian rules was proposed. More specifically, in order to relate Markov chains with the prokaryotic cell reproduction, and based on the self-reproduction rules of non-Mendelian Genetics, evolution algebras were firstly introduced in the Ph.D. Thesis of Jianjun Paul Tian [108] in 2004; then jointly presented with Petr Vojtechovsky [114] in 2006, and later analyzed in greater depth in a book by Tian [109] in 2008. The dynamic nature of these nonassociative algebras is not described by a series of identities, unlike many other known non-associative algebras such as alternative, Lie, Malcev or Jordan algebras, amongst others. More specifically, an evolution algebra E over a field Kis a finite-dimensional algebra such that there exists a natural basis {e1,...,en}so that ei·ej=0, for all i,j∈ {1,...,n}such that i= j, and ei·ei=∑kpik ek, for all i∈ {1,...,n}, and some coefficients pik ∈K, which are called the structural constants of the evolution algebra E. This multiplication is extended linearly from the given multiplication of basis elements. By considering each generator eias an allele in Genetics, the structure matrix (pi j )i,jencodes somehow the laws of inheritance of non-Mendelian Genetics as algebraic properties of the algebra. Even if a wide range of mathematicians has dealt with this type of algebras since the original manuscripts of Tian and Vojtechovsky, the relevance of evolution algebras is not still widely known. In order to make easier further research on the topic, this paper gathers together the antecedents, origin, early stage and later development of this kind of algebras, and shows different applications arising from them. In any case, we recommend the recent manuscript of Utkir Rozikov [100] for some detailed descriptions on genetic algebras, evolution algebras and quadratic stochastic operators, with particular emphasis in the study of algebraic and probabilistic approaches in population dynamics. See also [96] for a short survey on genetic and evolution algebras, and the expository article written by Tian [111] in 2016. The paper is organized as follows. Section 2outlines the origin and development of genetic algebras as primordial antecedents of Tian’s concept of evolution algebras. In Section 3some preliminary concepts and results on evolution algebras are indicated. Next, in each subsection of Section 4, a brief comment on the main papers published on this topic is given. Furthermore, Section 5overviews the relationship among evolution algebras, Graph theory, Group theory, Markov chains and Biology. The paper is completed with an extensive bibliography, which may be of valuable help to all those researchers interested in this topic. Throughout the paper, we follow the current notation, which may differ from the original one to which we refer. 2. Antecedents This section deals with the origin of genetic algebras as cornerstone of Tian’s concept of evolution algebra. For more details about the historical background on genetic algebras, we refer the reader to [11,76,99,100,121]. In 1866, Gregor Johann Mendel (1822–1884) was the first who made use of algebraic symbols to express sexual reproduction laws of inheritance. It was done in his original manuscript on experiments in plant hybridization [78]. More specifically, given a pair Please cite this article as: M. Ceballos, R.M. Falc´ on, J. N´ u˜ nez-Vald´ es et al., A historical perspective of Tian’s evolution algebras, Expositiones Mathematicae (2021), https://doi.org/10.1016/j.exmath.2021.11.004. M. Ceballos, R.M. Falcón, J. Núñez-Valdés et al. / Expo. Math. xxx (xxxx) xxx 3 of plant characters Aand a, together with the hybrid form Aa in which both characters fuse together, Mendel indicated with an expression of the form αA+βAa +γathe ratio α:β:γof numbers of offsprings having each possible plant character A,Aa and ain the following generation. In this way, Mendel introduced three principles: •The principle of paired factors, which states that characters are controlled by unit factors that appear in pairs within each individual organism. •The principle of dominance, which states that one of the two previous factors is dominant and makes its effect in the individual, whereas the other one is recessive and does not show its effect unless both characters are recessive. •The principle of segregation, which states that, during the inheritance process, the mentioned pair of factors separate randomly so that the offspring receives exactly one factor from each parent. Almost forgotten for decades, Mendel’s laws were independently rediscovered around 1900 by the Dutch botanist Hugo de Vries (1848–1935) [119], who introduced the term mutation and suggested that of gene; the German botanist and geneticist Carl Correns (1864–1933) [35], who discovered the cell extranuclear inheritance; the Austrian agronomist Erich von Tschermak (1871–1962) [117], who introduced the combination of plant characters in order to improve the efficiency of plant breeding; and the American agronomist William Jasper Spillman (1863–1931) [106], whose work was crucial for the development of Agricultural Economics. The emergence of all their contributions concerning the distribution of characters among offspring in plant hybridization made fundamental the study and development of Mendel’s laws. In 1905, the English biologist William Bateson (1861–1926) introduced the term Genetics to describe the study of inheritance processes [4]. Bateson was a fervent advocate of Mendel [3], whose principles, even being reborn, were questioned at that time. An interesting question in this regard had been asked in 1902 by the British statistician George Udny Yule (1871–1951), who contemplated [123] that dominant factors could increase indefinitely in a continuous way throughout generations. One year later, the English mathematician and biostatistician Karl Pearson (1857–1936) [95] and the American zoologist and geneticist William Ernest Castle (1867–1962) [31] established that, under random circumstances, there exists certain stability of characters in the inheritance process. Concerning such a stability, the English mathematician Godfrey Harold Hardy (1877–1947) [55] asked in 1908 about the circumstances under which the distribution of characters in the offspring is the same that in the generation before in the absence of disturbing factors. The answer was independently given at that moment by Hardy himself and the German physician Wilhelm Weinberg (1862–1937) [120], and is currently known as the Hardy– Weinberg law. An important limitation of that law is the assumption of a potentially infinite population, not taking into account the importance of sampling fluctuations in evolutionary processes described by finite populations. The first proposal dealing with such a possibility would be the so-called Wright–Fisher model based on the original manuscripts of the American geneticist Sewall Green Wright (1889–1988) [122] and the British statistician and geneticist Ronald Aylmer Fisher (1890–1962) [47]. Please cite this article as: M. Ceballos, R.M. Falc´ on, J. N´ u˜ nez-Vald´ es et al., A historical perspective of Tian’s evolution algebras, Expositiones Mathematicae (2021), https://doi.org/10.1016/j.exmath.2021.11.004. 4 M. Ceballos, R.M. Falcón, J. Núñez-Valdés et al. / Expo. Math. xxx (xxxx) xxx In 1923–24, the Soviet mathematician Sergei Natanovich Bernstein (1880–1968) [9,10] introduced the concept of quadratic stochastic operator (QSO) as a map V:S→ S, where S:= {(x1,...,xn)∈Rn:xi>0,for all i,and ∑ i xi=1}, and, for each (x1,...,xn)∈S, it is V((x1,...,xn)) =(V(x1),...,V(xn)), where V(xk):= ∑ i,j pk i j xixj,for all k∈ {1,...,n}. Here, the product xixjis the usual product of real numbers. In addition, pk i j ≥0 and pk i j =pk ji , for all i,j,k∈ {1,...,n}, and ∑kpk i j =1, for all i,j∈ {1,...,n}. As such, every QSO constitutes an evolutionary operator that describes the time evolution or inheritance process of a free population with ndifferent genetic types. More specifically, each component xiof a given element x=(x1,...,xn)∈S represents the probability that a random individual in the population under consideration belongs to the species that is determined by the ith genetic type. Hence, the n-tuple x describes the distribution of the population with respect to the ngenetic types, whereas V(x) describes such a distribution for the next generation. In particular, each value pk i j =pk ji determines the probability that an offspring with genetic type karises from two individuals of respective genetic types iand j, without sexual differentiation. Based on the possible fluctuation of genetic types throughout subsequent generations of an evolutionary process, a main problem in the theory of QSOs consists of determining their limit behavior for any given initial distribution of genetic types x∈S. That is, the study of the corresponding distribution Vm(x) for the mth generation, when mtends to infinity. In this regard, Berstein focused in particular on the problem of determining and classifying all those QSOs for which a stationary distribution or stable evolution arises in only one generation (that is, such that V2=V). Even if Berstein only solved this problem for n=3, it is currently solved for all dimensions (see [52,54]). To this end, it would be crucial the work in 1975 of Philip Holgate (1934–1993) [60], who expressed algebraically this problem as follows. 1. Firstly, Holgate related each given evolutionary operator Vwith the algebra described as xy =1 2(V(x+y)−V(x)−V(y)). (It was called evolution algebra by Joseph Bayara [6,7], which differs from the concept introduced by Tian, the one that is commonly used in the current literature and with which this paper deals.) 2. Then, he introduced the concept of Bernstein algebra as an algebra Asuch that x2x2=ω2(x)x2, for all x∈A, where ωis an algebra homomorphism from A into its base field, which is called the weight of the algebra. Recall that an algebra for which one such a non-trivial homomorphism exists is called baric [42]. In 1934, Alexander Pawlowitsch Serebrowski (1884–1938) [105] was the first to give an algebraic interpretation of the symbol ×as a mathematical way to represent Mendelian Please cite this article as: M. Ceballos, R.M. Falc´ on, J. N´ u˜ nez-Vald´ es et al., A historical perspective of Tian’s evolution algebras, Expositiones Mathematicae (2021), https://doi.org/10.1016/j.exmath.2021.11.004. M. Ceballos, R.M. Falcón, J. Núñez-Valdés et al. / Expo. Math. xxx (xxxx) xxx 5 inheritance laws. A similar symbolic multiplication was independently introduced shortly after by Aleksandrovich Kostitzin (1883–1963) [70]. At the same period, Valery Ivanovich Glivenkov (1896–1940) [49] introduced the socalled Mendelian algebras for diploid species. Nevertheless, it was Ivor Malcolm Haddon Etherington (1908–1994) who, in 1939, introduced in Genetics [42] the systematic study of commutative non-associative linear algebras, by describing in particular the following three types of algebras: •Agametic algebra is a finite-dimensional algebra of basis {e1,...,en}such that ei·ej=∑kpk i j ek, where each structural constant pk i j belongs to the real interval [0,1] so that ∑kpk i j =1, for all i,j∈ {1,...,n}. As such, each value pk i j represents the probability that an arbitrary gamete of zygotic type ekderives from an individual of zygotic type eiej. As occurs with the QSOs, the probability or progeny distribution describes the evolution of the population under consideration. The latter constitutes a free population if the inheritance process derives from random matings; that is, with absence of sexual differentiation and selection. •Azygotic algebra is the duplicate of a gametic algebra. That is, the former is isomorphic to the set of quadratic forms of the latter. •Acopular algebra is the duplicate of a zygotic algebra. Each one of the three just described algebras are baric by means of the weight ωthat is linearly defined from ω(ei)=1, for all i∈ {1,...,n}. Here, {e1,...,en}is the basis of the algebra under consideration, and hence, ω(ei)ω(ej)=1=∑ k pk i j =∑ k pk i j ω(ek)=ω(∑ k pk i j ek)=ω(eiej), for all i,j. Further, Etherington called train algebra any baric algebra Awith weight ω such that the equation of lowest degree relating the principal powers of any vector x∈A has the form xm+c1ω(x)xm−1+ · · · + cmω(x)m=0. Finally, he termed special train algebra any baric algebra Awith weight ωsuch that N=ker(ω) is nilpotent (that is, there exists a positive integer m∈Nsuch that Nm=0) and all the principal powers Ni, with i∈N, are ideals of A. In particular, every special train algebra is a train algebra. Etherington called genetic algebra any one of the family of gametic, zygotic and copular algebras and ensured that “all the fundamental genetic algebras are special train algebras”. Shortly after, he indicated [43] that this statement is only valid for gametic algebras, but not even for zygotic algebras, because the duplicate of a special train algebra, although a train algebra, is not always a special train algebra. This fact provided serious problems for translating some properties and relations from genetic inheritance processes. In order to avoid the flaws in Etherington’s genetic algebras and give rise to a transparent structure theory for them, Richard Donald Schafer (1918–2014) [104] provided in 1949 an alternative formal definition that is intermediate between the concepts of train algebra and special train algebra. To this end, he made use of the transformation Please cite this article as: M. Ceballos, R.M. Falc´ on, J. N´ u˜ nez-Vald´ es et al., A historical perspective of Tian’s evolution algebras, Expositiones Mathematicae (2021), https://doi.org/10.1016/j.exmath.2021.11.004. 6 M. Ceballos, R.M. Falcón, J. Núñez-Valdés et al. / Expo. Math. xxx (xxxx) xxx algebra T (A) of a non-associative algebra A. It is the algebra of all the polynomials with coefficients in the base field, whose variables are transformations in the set formed by the identity map in A, all the right multiplications Rα:A↦→ Asuch that Rα(x)=xα, and all the left multiplications Lα:A↦→ Asuch that Lα(x)=αx, for all α, x∈A. In particular, the transformation algebra of a baric algebra is baric. Further, unlike Etherington, Schafer assumed the commutativity that is inherent in any inheritance process and hence, he considered Rα=Lα, for all α∈A. Then, he called genetic algebra any commutative baric algebra Asuch that all the coefficients of the characteristic function |λI−T|, with T∈T(A), only depend on the weight of the algebra. As such, every genetic algebra is a train algebra and every special train algebra is a genetic algebra. Moreover, unlike special train algebras, the duplicate of a genetic algebra is always a genetic algebra. In 1971, Harry Gonsh¨ or [50] proved that the concept of genetic algebra proposed by Schafer is equivalent to have a commutative algebra Aof basis {e0,e1,...,en}such that eiej=∑kck i j ek, for all i,j∈ {0,...,n}, where c0 00 =1; ck 0j=0, for all k<j; and ck i j =0, for all i,j>0 and k≤max {i,j}. This new definition was proved to have significant relevance in Genetics thanks to the genetic meaning that Holgate [59,61] gave of duplicates and derivations of a genetic algebra. The study of the subspace Der(A) of derivations of a genetic algebra Ahas also been dealt with by Gonsh¨ or [51] himself and also by Roberto Costa [37,38], Aribano Micali and Philippe Revoy [79], and, much more recently, by Rasul Ganikhodzhaev, Farrukh Mukhamedov, Abror Pirnapasov and Izzat Qaralleh [48]. Recall in this regard that D∈Der(A) if and only if D(xy)=D(x)y+x D(y), for all x,y∈A. As for any algebra, such a subspace constitutes a Lie algebra. Holgate also introduced [58] both notions of sex differentiation algebra and dibaric algebra. The former characterizes the equal division of offspring between both sexes. More specifically, the sex differentiation algebra is a bi-dimensional commutative algebra of basis {m, w}such that m2=w2=0 and mw=wm=(m+w)/2. Further, an algebra is called dibaric if it admits a homomorphism onto the sex differentiation algebra. Holgate proved in particular that, if Ais a dibaric algebra, then its derived algebra A2 is a baric algebra. With the introduction of dibaric algebras, he formalized the original idea of Etherington [42] of treating in a separate way male and female components of the population. Another contribution of Holgate [57] in the theory of genetic algebras was the use of isotopisms of algebras in order to represent algebraically the mutation of genotypes in the inheritance process. Recall in this regard that two n-dimensional algebras Aand A′are said to be isotopic [1] if there exist three non-singular linear transformations f,gand h from Ato A′such that f(u)g(v)=h(uv), for all u, v ∈A. The triple ( f,g,h) is called an isotopism (an isomorphism, if f=g=h; and an homomorphism, if besides, the condition of non-singularity is not imposed) between the algebras Aand A′(see [46] for a recent survey on the theory of isotopisms). Together with Tania Campos [25], Holgate proved that certain types of zygotic algebras representing chromosome segregation and recombination are isotopic. Much more recently, Manuel Ladra, Bakhrom Omirov and Rozikov [72] would introduce the concept of bq-homomorphism as a pair of linear maps fand gfrom the Please cite this article as: M. Ceballos, R.M. Falc´ on, J. N´ u˜ nez-Vald´ es et al., A historical perspective of Tian’s evolution algebras, Expositiones Mathematicae (2021), https://doi.org/10.1016/j.exmath.2021.11.004. M. Ceballos, R.M. Falcón, J. Núñez-Valdés et al. / Expo. Math. xxx (xxxx) xxx 7 algebra to the set of real numbers, so that f(uv)=g(uv)=(f(u)g(v)+f(v)g(u))/2, for all u, v ∈A. (It constitutes an homomorphism whenever f=g.) Then, they proved [72, Theorem 3.7] that an algebra is dibaric if and only if it admits a non-zero bq-homomorphism. 3. Preliminaries on evolution algebras In order to make this article as self-contained as possible and facilitate a better understanding for the reader, we recall in this section some notions and basic results on evolution algebras that were introduced in the original manuscripts of Tian and Vojtechovsky [109,114]. To this end, and from now on, let Edenote an n-dimensional evolution algebra over a base field K, with natural basis {e1,...,en}and structural constants pik ∈K, for all i,k∈ {1,...,n}. If the base field Kis the real field, then the evolution algebra is said to be real. In such a case, it is said to be non-negative if pik ≥0, for all i,k∈ {1,...,n}. If, besides, ∑kpik =1, for all i, then the real and non-negative evolution algebra is called Markov evolution algebra. Every evolution algebra Eis non-associative, commutative, flexible (that is, x(yx)= (xy)x, for x,y∈E); not necessarily power-associative (that is, the subalgebra generated by a single element cannot be associative); and preserved by direct sums. Moreover, evolution algebras are not closed under subalgebras. So, a subalgebra of Eis called an evolution subalgebra if the former is spanned by a subset of generators of the latter. As such, it is an ideal of E. The latter is indecomposable if it is not the direct sum of two nonzero ideals; connected if it is not the direct sum of two proper evolution subalgebras; irreducible if it has no proper subalgebra; and simple if it has no proper evolution subalgebra. Tian proved [109, Theorem 9] that any finite-dimensional evolution algebra has a simple evolution subalgebra. Every evolution algebra Eis uniquely determined by its evolution operator L :E↦→ E, which is linearly described from L(ei):= eiei=∑ i pik ek,for all i∈ {1,...,n}. Then, for each positive integer m>2, it is defined the plenary power e[m] i:= L(e[m−1] i), where e[0] i:= ei. The operator Lmay be considered as time-step in a discrete-time dynamical system, and hence, it describes the dynamical flow of the evolutionary process represented by the evolution algebra E. In this regard, a generator eiof the evolution algebra Eis called algebraically persistent if the evolution subalgebra generated by ei is a simple evolution subalgebra. As such, it represents an absorbent state giving rise to a stable evolution. Otherwise, it is said to be algebraically transient, which gives rise to a transitory evolution. The latter would eventually derive an absorbent state, but only after certain generations. Any generator of a simple evolution subalgebra is algebraically persistent. In particular, every finite-dimensional evolution algebra contains an algebraically persistent generator. Please cite this article as: M. Ceballos, R.M. Falc´ on, J. N´ u˜ nez-Vald´ es et al., A historical perspective of Tian’s evolution algebras, Expositiones Mathematicae (2021), https://doi.org/10.1016/j.exmath.2021.11.004. 8 M. Ceballos, R.M. Falcón, J. Núñez-Valdés et al. / Expo. Math. xxx (xxxx) xxx A connected evolution algebra is simple if and only if all its generators are algebraically persistent. Tian proved [109, Theorem 11] that, if Eis a connected finitedimensional evolution algebra, then E= n0 ⨁ j=1 E0,j+B0, where ⨁and +denote, respectively, direct sums of either subalgebras or linear subspaces; each E0,jis a simple evolution subalgebra so that E0,j∩E0,j′= {0}, whenever j= j′; and B0is a linear subspace spanned by algebraically transient generators, which is called transient space. Even if B0is not an evolution subalgebra in general, it may be endowed of evolution structure from Eonce the multiplication is restricted within B0. The previous decomposition process can inductively be repeated until no transient space arises. That is, there exist an integer m>0 and a subset {n1,...,nm} ⊂ Nsuch that E= m ∑ i=0 ni ⨁ j=1 Ei,j, where ∑denote direct sum of linear subspaces and each Ei,jis a simple evolution algebra so that Ei,j∩Ei,j′= {0}, whenever j= j′. Each subset {Ei,1,...,Ei,ni} constitutes the ith level of a hierarchical structure describing the dynamical flow of E, which enabled Tian [111] to ensure that “simple evolution algebras are the basic blocks for building general evolution algebras”. Tian dealt in particular with the spectrum of the evolution operator Lat the 0th level of this hierarchy, and set out the following problem. Problem 3.1 ([109]).Study the spectrum of the evolution operator of an evolution algebra at the ith level of the hierarchical structure of the latter, for each i>1. Furthermore, study the spectrum theory concerning plenary powers of evolution algebras. Furthermore, the just described hierarchical structure may also be used to classify evolution algebras. More specifically, evolution algebras can be distributed according to the number of levels of its related hierarchical structure and the number of simple evolution subalgebras at each level. Every evolution algebra is homomorphic to a unique evolution algebra, which is called its skeleton-shape, whose hierarchy has only onedimensional evolution algebras in all its levels [109, Theorem 15]. In particular, evolution algebras within each class of the mentioned distribution are uniquely determined up to their skeleton-shape homomorphism. Another method for classifying evolution algebras is based on the period of their generators. In this regard, a generator eiis said to occur in x=∑n i=1αiei∈Eif αi= 0. It is denoted ei≺x. The period of a generator eiis defined as di:= g.c.d.{log2m:ei≺e[m] i}. If {log2m:ei≺e[m] i}= ∅, then di= ∞. Further, if di=1, then the generator eiis called aperiodic. Otherwise, it is called periodic. If the evolution algebra is non-negative simple, then all its generators have the same period [109, Theorem 7]. As such, nonnegative simple evolution algebras may be classified as either periodic or aperiodic. In the Please cite this article as: M. Ceballos, R.M. Falc´ on, J. N´ u˜ nez-Vald´ es et al., A historical perspective of Tian’s evolution algebras, Expositiones Mathematicae (2021), https://doi.org/10.1016/j.exmath.2021.11.004. M. Ceballos, R.M. Falcón, J. Núñez-Valdés et al. / Expo. Math. xxx (xxxx) xxx 9 first case, if d>1 is the period of all the generators of a non-negative simple evolution algebra E, then these generators can be partitioned into ddisjoint classes C0,...,Cd−1 such that, if ∆idenotes the linear subspace spanned by Ci, for all i∈ {0,...,d−1}, then L(∆i)⊆∆i+1 mod d, and E= d−1 ⨁ i=0 ∆i. Based on the already mentioned fact that simple evolution algebras may be seen as the basic blocks on the hierarchical structure of any evolution algebra, Tian proposed to generalize the previous results on non-negative simple evolution algebras. In particular, he set out the following problem. Problem 3.2 ([111]).Delve into the general study of simple evolution algebras, and not only non-negative ones. Characterize them and prove the existence of transitive occurrence relations. To this end, study in particular simple evolution algebras over finite fields. Furthermore, it is also required to delve into the study of simple evolution algebras at higher levels of the hierarchy of evolution algebras, and not only at the 0th level. 4. Developing the theory of evolution algebras This section deals with some of the most important results concerning the mathematical fundamentals of the theory of evolution algebras. Again, let Edenote from now on an n-dimensional evolution algebra over a base field K, with natural basis {e1,...,en} and structural constants pik ∈K, for all i,k∈ {1,...,n}. 4.1. Nilpotent evolution algebras In 2013, based on the non-necessary associativity of evolution algebras, Jos´ e Manuel Casas, Ladra, Omirov and Rozikov [28] distinguished among nil evolution algebras, right nilpotent evolution algebras and nilpotent evolution algebras. More specifically, for each vector x∈Eand each positive integer i>1, let x⟨i⟩:= x⟨i−1⟩x, where x⟨1⟩:= x. Then, the vector xis nil if there exists a positive integer nsuch that x⟨n⟩=0. The evolution algebra Eis nil if all its elements are nil. Its nilindex is the minimum positive integer n such that x⟨n⟩=0, for all x∈E. Further, the evolution algebra Eis right nilpotent if E⟨n⟩=0, for some positive integer n, where E⟨1⟩:= Eand E⟨k+1⟩:= E⟨k⟩E, for every positive integer k>1. Finally, the evolution algebra Eis nilpotent if En=0, for some positive integer n, where E1:= Eand Ek:= ∑k−1 i=1EiEk−i, for every positive integer k>1. The index of (right) nilpotency is defined similarly to the nilindex. This distinction is not necessary in case of dealing with finite-dimensional complex evolution algebras. The mentioned authors proved [28, Theorem 3.1] the equivalence of right nilpotency and nility for finite-dimensional evolution algebras. Their equivalence with nilpotency was shortly after proved by Luisa Mar´ ıa Camacho, Jos´ e Ram´ on G´ omez, Omirov and Rustam Turdibaev [23, Corollary 4.6] in case of dealing with finitedimensional complex evolution algebras. Their structure matrices are all of them upper Please cite this article as: M. Ceballos, R.M. Falc´ on, J. N´ u˜ nez-Vald´ es et al., A historical perspective of Tian’s evolution algebras, Expositiones Mathematicae (2021), https://doi.org/10.1016/j.exmath.2021.11.004. 16 M. Ceballos, R.M. Falcón, J. Núñez-Valdés et al. / Expo. Math. xxx (xxxx) xxx In order to deal with the reciprocal, Tian [109] introduced the concept of graphicable algebra as an evolution algebra of natural basis V= {e1,...,en}such that eiei=∑ ek∈Vi ek, for all positive integer i≤n, where Vi⊆V. Even if evolution algebras are not graphicable in general, every graphicable algebra is uniquely associated to a graph such that Vi=Γ(ei). In this way, two isomorphic graphicable algebras give rise to isomorphic graphs [109, Theorem 42]. In particular, Tian introduced cycle algebras,path algebras and complete algebras as graphicable algebras giving rise, respectively, to cycles, paths and complete graphs. In a similar way, N´ u˜ nez, Mar´ ıa Luisa Rodr´ ıguez-Ar´ evalo and Mar´ ıa Trinidad Villar [86] dealt with complete tripartite and n-partite, star, friendship, wheel and snark graphicable algebras. These authors also proved [86, Theorem 3.8] the existence of graphicable subalgebras within a graphicable algebra. Further, N´ u˜ nez, Marithania Silvero and Villar dealt [87] with the particular case of graphicable algebras for which ek∈Vi if and only if ei∈Vk, and ei∈ Vi, for every pair of positive integers i,k≤nsuch that i= k. They called S-graphicable algebras this type of algebras. Based on both notions of evolution algebra arising from a graph and graphicable algebra, Tian remarked [109] that the ‘intrinsic and coherent relation of evolution algebras with graph theory allows to analyze graphs algebraically [...] and graph theory may be used as a tool to study non-associative algebras”. It is so that he asked whether “every statement or problem in graph theory can be translated into the language of evolution algebras”. He put particular interest in the possible relationship of evolution algebras with random graphs and networks, random walks on graphs, weighted graphs and directed graphs. Furthermore, Tian also asked for those evolution algebras arising from finite graphs whose associated Ihara–Selberg zeta function satisfies the Sunada’s analogue of Riemann hypothesis [107]. In 2011, Rozikov and Tian [102] described an alternative for relating an evolution algebra to a given finite and simple graph by means of state spaces endowed with Gibbs measures. For connected graphs, they determined the hierarchical structure of these algebras, together with the dimension and number of one- and four-dimensional evolution subalgebras [102, Theorem 3.2]. Moreover, all the evolution algebras arising from the same finite and simple graph, but defined by different Gibbs measures, are pairwise isomorphic [102, Theorem 3.3]. The authors established as further open problem the study of the hierarchical structure of evolution algebras associated to graphs that are either finite, but not connected; or countable and connected. In 2015, Elduque and Labra [39] defined the directed graph attached to an ndimensional evolution algebra of structure matrix (pik)i,kwith respect to a given natural basis as the graph of set of vertices V= {1,...,n}and set of arcs A= {(i,k)∈ V×V:pik = 0}. If each arc (i,k)∈Ais labeled as pik, then one obtains the directed weighted graph attached to E. If Ann(E)= {0}, then its directed graph is connected if and only if Edoes not split into a direct sum of simple ideals [39, Proposition 2.8]. If the annihilator is not trivial, then this decomposition is not possible if and only if its directed graph is connected for every natural basis [39, Proposition 2.10] (see also Please cite this article as: M. Ceballos, R.M. Falc´ on, J. N´ u˜ nez-Vald´ es et al., A historical perspective of Tian’s evolution algebras, Expositiones Mathematicae (2021), https://doi.org/10.1016/j.exmath.2021.11.004. M. Ceballos, R.M. Falcón, J. Núñez-Valdés et al. / Expo. Math. xxx (xxxx) xxx 17 [27, Proposition 5.4]). Further, Eis nil if and only if its attached directed graph contains no oriented cycles [39, Theorem 3.4]. These attached (weighted) directed graphs were also used to distribute into isomorphism classes the set of four- and five-dimensional nilpotent evolution algebras [40]. In 2020, Cadavid, Rodi˜ no and Rodr´ ıguez [20] described the evolution algebra induced by the random walk on a graph and studied its relationship with the evolution algebra determined by the same graph according to Tian’s proposal [109]. Recall here that the random walk on a graph of set of vertices Vand adjacency matrix (ai j )i,jis a discrete Markov chain with state space Vsuch that the transition probability of moving from state ito state jis pi j =ai j ∑k∈Vaik . The mentioned authors described the evolution algebra induced by this random walk as the evolution algebra of natural basis {e1,...,en}and structure matrix (pi j )i,j. Then, they studied under which conditions one may ensure that this evolution algebra is isomorphic to Tian’s evolution algebra associated to the graph under consideration. More recently, the same authors have proved [21, Theorem 2.2] that both algebras are indeed strongly isotopic. In addition, they have provided conditions [21, Theorem 2.3] under which these algebras are isomorphic or not. Also in 2020, Manuel Ceballos, N´ u˜ nez and ´ Angel Tenorio [32] described the directed weighted pseudograph associated to an n-dimensional evolution algebra of structure matrix (pik )i,kas the pseudograph resulting after adding a loop on each vertex iof the Elduque and Labra’s directed weighted graph attached to E, whenever pii = 0. This loop is then labeled as pii . The distribution into isomorphism classes of these pseudodigraphs enabled them to establish a new classification of evolution algebras. Furthermore, Rafael Gonz´ alez and N´ u˜ nez [53] translated into the algebraic language some basic concepts and results concerning the directed graphs associated to an evolution algebra. In particular, based on the adjacency of graphs, they introduced the notions of adjacency, walk, trail, circuit, path and cycle of an evolution algebra. In addition, strongly and weakly connected evolution algebras were introduced as the algebraic equivalences of the same concepts in graph theory. It enabled the authors to introduce the notions of distance, girth, circumference, eccentricity, center, radio, diameter and geodesic of an evolution algebra, together with the concepts of Eulerian and Hamiltonian evolution algebras. In 2021, Qaralleh and Mukhamedov [98] have introduced the concept of Volterra evolution algebra as an evolution algebra whose structural matrix is described by skew symmetric matrices. This type of algebras are not nilpotent. In particular, they have proved [98, Theorem 6.5] that the directed weighted graphs associated to two Volterra evolution algebras are isomorphic if and only if the mentioned algebras are isomorphic. 5.2. Group theory Tian [109] associated any given group (G,◦) having a finite set of generators S with the evolution algebra described over a base field Kso that, for each g∈G, one Please cite this article as: M. Ceballos, R.M. Falc´ on, J. N´ u˜ nez-Vald´ es et al., A historical perspective of Tian’s evolution algebras, Expositiones Mathematicae (2021), https://doi.org/10.1016/j.exmath.2021.11.004. 18 M. Ceballos, R.M. Falcón, J. Núñez-Valdés et al. / Expo. Math. xxx (xxxx) xxx has g·g=∑ e∈S keg◦e,(1) for some ke∈K. Particularly, Tian set out the following question. Problem 5.1 ([109]).How can the properties of a group be translated to the corresponding evolution algebra? 5.3. Markov chains Tian himself [109] realized that every Markov chain is related to a Markov evolution algebra whose structural constants coincide with the transition probabilities of the former and whose basis constitutes the state space of the Markov process. As such, generators of Markov evolution algebras represent states of stochastic processes described by Markov chains, which may, therefore, be studied by means of the theory of evolution algebras. In particular, Markov chains may be classified according to the hierarchies of their corresponding evolution algebras. Tian proved that a Markov chain is irreducible if and only if its related evolution algebra is simple [109, Theorem 18], and that a subset of state space of a Markov chain is probabilistic closed if and only if it generates an evolution subalgebra [109, Theorem 17]. In addition, probabilistic periods in Markov chains are uniquely related to periods of generators in evolution algebras [109, Proposition 10]. Tian also introduced the possibility of dealing with continuous evolution algebras, once structural constants are replaced by differential functions depending on a variable. If the latter represents the time, then continuous evolution algebras are uniquely related to continuous-time Markov chains [115]. Tian thought that the study of continuous evolution algebras “will be very interesting because they have a kind of semi-Lie group structure”. In 2011, as a generalization of Tian’s idea, Casas and Rozikov [30] realized that every Markov chain is associated to a chain of n-dimensional evolution algebras {A[s,t]:0≤ s≤t}depending on two time parameters sand t. Their structure matrices M[s,t]are all of them stochastic and satisfy that M[s,t]=M[s,τ]M[τ,t] for all s< τ < t. This chain represents a continuous time dynamical system that is an evolution algebra in each fixed time. If all the matrices of the chain coincide, then they constitute the same evolution algebra that Tian associated to a Markov chain. Furthermore, if both time parameters sand tare reduced to only one of them, then the chain is called time-homogeneous and is associated to a time-homogeneous Markov chain. The authors also dealt with chains of evolution algebras for which the stochastic condition is removed. In 2013, Rozikov and Mudorov [101] constructed 25 distinct examples of chains of two-dimensional evolution algebras, for which they studied their baricity, nilpotency and idempotency. (Some new examples have been recently described by Ladra and Murodov [71].) In 2020, Imomkulov and Velasco [67] got a description depending Please cite this article as: M. Ceballos, R.M. Falc´ on, J. N´ u˜ nez-Vald´ es et al., A historical perspective of Tian’s evolution algebras, Expositiones Mathematicae (2021), https://doi.org/10.1016/j.exmath.2021.11.004. M. Ceballos, R.M. Falcón, J. Núñez-Valdés et al. / Expo. Math. xxx (xxxx) xxx 19 on rank of chains of three-dimensional evolution algebras. And then, for these chains, Imomkulov [64] studied the behavior of both the baric property and the set of absolute nilpotent elements, and also the time-depending dynamics of the set of idempotent elements. The general case was already considered in 2015, when Omirov, Rozikov and Kaisar Tulenbayev [88] dealt with real chains of n-dimensional evolution algebras corresponding to a permutation of nnumbers and showed that one such a chain is trivial if and only if the permutation has no fixed points [88, Proposition 1]. (The study of evolution algebras arising from permutations was introduced by Narkuziev [85], who described a nilpotency condition on this type of algebras.) Moreover, since every trivial chain of evolution algebras is a chain of nilpotent evolution algebras [88, Proposition 2], they constructed chains of three-dimensional evolution algebras. They also dealt with the construction of arbitrary-dimensional symmetric chains of evolution algebras. In 2017, Ladra and Rozikov [73,74] described non-homogeneous continuous time Markov chains. More recently, Irene Paniello [94] has delved into the algebraic structure of Markov evolution algebras for both the discrete-time and the continuous-time arising from standard stochastic semigroups. 5.4. Biology Since the original manuscripts of Tian and Vojtechovsky, evolution algebras have been implemented in Biology to represent different aspects of non-Mendelian inheritance (particularly, uniparental inheritance) in an algebraic way. Thus, Tian himself [110] made use of these algebras to analyze how the homoplasmy of a cell population (represented by persistent generators) can derive from an heteroplasmy one (represented by transient generators). This is useful, for instance, to study mitochondrial disorders and mutations in tissues of patients. He also implemented evolution algebras to describe genetically dynamical patterns that provide information about the asexual reproduction of Phytophthora infectants causing the late blight of tomatoes and potatoes. An algebraic concept as nilpotency represents the extinction of original genetic types after certain generations. Continuous-time dynamical systems of mosquito populations were studied by Rozikov and Velasco [103]. Mosquito populations were already been dealt with from an algebraic point of view in 2011 by Junliang Lu and Jia Li [75], who proposed a discrete time structured model of such a population. On their own, Rozikov and Velasco considered a discrete-time dynamical system described by an evolution algebra with two fixed points, which become saddle points under some conditions on the parameters of the system. Its structure matrix is equal to the Jacobian of the QSO at a fixed point. Another aspect that Tian [109] realized was the possible use of evolution algebras on coalescent theory [113]. That is, on genetic evolution reversely over time. The study of backwards evolution of Mendelian genetic systems by means of coalgebras had been previously introduced in 2004 by Tian and Bai-Lian Li [112]. In 2019, Paniello [92] introduced the concept of evolution coalgebra in order to model backwards evolution of Non-Mendelian genetic systems. She established the connection between such coalgebras and evolution algebras, by focusing in particular on the cases of genetic realizations. An Please cite this article as: M. Ceballos, R.M. Falc´ on, J. N´ u˜ nez-Vald´ es et al., A historical perspective of Tian’s evolution algebras, Expositiones Mathematicae (2021), https://doi.org/10.1016/j.exmath.2021.11.004. 20 M. Ceballos, R.M. Falcón, J. Núñez-Valdés et al. / Expo. Math. xxx (xxxx) xxx illustrative example of this approach was given by Paniello herself [93] in case of dealing with chicken populations. In 2020, Miguel Bustamante, Mellon and Velasco [13,14] described necessary and sufficient conditions for a given algebra to be an evolution algebra. They proved that this problem is equivalent to the simultaneous diagonalization via congruence of a given set of matrices. Based on this fact, they realized that arbitrarily small perturbations of classical genetic algebras representing Mendelian and auto-tetraploid inheritance (which are not evolution algebras) may give rise to evolution algebras. The algebraic representations of sexual and asexual inheritance by means of genetic and evolution algebras seem, therefore, to be closer than previously thought. A further study of baric and evolution algebras was established by the authors as a first stage to better understand this relationship. In any case, notice that Mukhamedov and Qaralleh already set out the following question in 2014. Problem 5.2 ([82]).Is there a transformation of a given genetic algebra to some evolution algebra? They gave an affirmative answer in case of dealing with two-dimensional genetic algebras (see [82, Theorem 4.1]). The resulting transformation enabled them to provide necessary conditions on the structure matrix of these genetic algebras for ensuring the existence of non-trivial derivations (see [82, Theorem 5.1]). 6. Conclusion This paper has dealt with the past, origin, development and possible further work of the theory of evolution algebras introduced by Tian [108] in 2004, with particular emphasis in the relationship that this theory has with a wide amount of distinct branches, not only in Mathematics, but with other areas of research. As such, the paper aims to be a starting point for all those researchers interested in this topic. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The authors want to express their gratitude to the anonymous referee for the comprehensive reading of the paper and his/her pertinent comments and suggestions, which helped improve the manuscript. Falc´ on’s work is partially supported by the research project FQM-016 from Junta de Andaluc´ ıa, Spain. Please cite this article as: M. Ceballos, R.M. Falc´ on, J. N´ u˜ nez-Vald´ es et al., A historical perspective of Tian’s evolution algebras, Expositiones Mathematicae (2021), https://doi.org/10.1016/j.exmath.2021.11.004. M. Ceballos, R.M. Falcón, J. Núñez-Valdés et al. / Expo. Math. xxx (xxxx) xxx 21 References [1] A.A. Albert, Non-associative algebras: I. Fundamental concepts and isotopy, Ann. Math. 43 (1942) 685–707. [2] A. Alsarayreh, I. Qaralleh, M.Z. 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