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In the present paper, every evolution algebra is endowed with Banach algebra norm. This together with the description of derivations and automorphisms of nilpotent evolution algebras of maximal nilindex, allows the set $ \exp (\mathop {{\rm Der}}(\mathbf{E})) $ exp ⁡ (Der ⁡ (E)) to be investigated. Moreover $ \exp (\mathop {{\rm Der}}({\mathbf{E}})) $ exp ⁡ (Der ⁡ (E)) is shown to be a normal subgroup of $ \mbox {Aut}(\mathbf{E}) $ Aut (E) , and its corresponding index is calculated. [ABSTRACT FROM AUTHOR]

In this paper, we present a method for finding the absorbing radical of a finite-dimensional evolution algebra. Such a method consists of finding the acyclic vertices of an oriented graph associated with the algebra. The set of generators associated with such vertices turn out to be the generators of the absorption radical. As an application we use the absorption radical to study the decomposability of some degenerate evolution algebras. [ABSTRACT FROM AUTHOR]

In this paper, we give a necessary and sufficient condition for an almost Jordan algebra, which is also called Lie triple algebra to be an evolution algebra. We study the nilpotency, power associativity of such algebras. We specify the necessary and sufficient conditions for such an algebra to be baric. Along the way we show that train evolution algebras which are almost Jordan algebras are of rank at most 5. We prove that any finite-dimensional non-nil-evolution algebra verifying the identity of almost Jordan algebras has at least one non-zero idempotent. Finally, we study the derivations of this class of algebras. [ABSTRACT FROM AUTHOR]

We address Peirce decompositions for evolution algebras at idempotent elements contained in the associative nucleus of the evolution algebras. If the idempotent elements are natural vectors, the requirement of being nuclear is then proved to be equivalent to the evolution algebra to be baric. A description of baric evolution algebras is also provided. [ABSTRACT FROM AUTHOR]

In this paper, we consider n -dimensional nilpotent finite-dimensional evolution algebras with non-maximal index of nilpotency. We give complete invariants of these algebras in order to classify them. Then, we describe their associative enveloping algebras. We also give a positive answer to the question whether a derivation of this nilpotent evolution algebra is possible to be extended to its enveloping algebra. [ABSTRACT FROM AUTHOR]

Evolution algebras are non-associative algebras. In this work we provide an extension of this class of algebras, in a framework of Hilbert spaces, and illustrate the applicability of our approach by discussing a connection with discrete-time Markov chains with infinite countable state space. Specifically, if we associate to each possible state of such a Markov process a generator of the Hilbert evolution algebra structure, then the whole dynamics of the process can be described through consecutive applications of the evolution operator, provided certain boundedness conditions on the transition probabilities hold. [ABSTRACT FROM AUTHOR]

This paper investigates the tensor product of a finite dimensional nilpotent evolution algebra. Some properties that translate from tensor products to factors and vice versa have been investigated, including the index of nilpotency and annihilator. The index of nilpotency of the tensor product of two nilpotent evolution algebras with different indexes of nilpotency is determined. Moreover, we investigate the tensorially decomposable of the 4-dimensional nilpotent evolution algebra. In addition, the decomposable nilpotent evolution algebra with the maximal index of nilpotency has been carried out. [ABSTRACT FROM AUTHOR]

In this paper, we give the necessary and sufficient conditions for an evolution algebra to satisfy a train identity of degree 2 and exponent 3. We show that this class of algebras is a subclass of the Bernstein algebras of order 2. Then, we study the relations existing between these algebras and the Bernstein algebras as well as the power associative algebras. Moreover we give a classification in dimension ≤ 4 of evolution algebras satisfying strictly a train identity of degree 2 and exponent 3. Finally, we describe the derivations and automorphisms of these algebras. [ABSTRACT FROM AUTHOR]

In the present paper, we examined the extendibility of evolution subalgebras generated by idempotents of evolution algebras. The extendibility of the isomorphism of such subalgebras to the entire algebra was investigated. Moreover, the existence of an evolution algebra generated by arbitrary idempotents was also studied. Furthermore, we described the tensor product of algebras generated by arbitrary idempotents and found the conditions of the tensor decomposability of four-dimensional S-evolution algebras. This paper's findings shed light on the field of algebraic structures, particularly in studying evolution algebras. By examining the extendibility of evolution subalgebras generated by idempotents, we provide insights into the structural properties and relationships within these algebras. Understanding the isomorphism of such subalgebras and their extension allows a deeper comprehension of the overall algebraic structure and its behaviour. [ABSTRACT FROM AUTHOR]

The starting point of this work is the fact that the class of evolution algebras over a fixed field is closed under tensor product. We prove that, under certain conditions, the tensor product is an evolution algebra if and only if every factor is an evolution algebra. Another issue arises about the inheritance of properties from the tensor product to the factors and conversely. For instance, nondegeneracy, irreducibility, perfectness and simplicity are investigated. The four-dimensional case is illustrative and useful to contrast conjectures, so we achieve a complete classification of four-dimensional perfect evolution algebras emerging as tensor product of two-dimensional ones. We find that there are four-dimensional evolution algebras that are the tensor product of two nonevolution algebras. [ABSTRACT FROM AUTHOR]

A chain of evolution algebras (CEA) is an uncountable family (depending on time) of evolution algebras on the field of real numbers. The matrix of structural constants of a CEA satisfies the Chapman-Kolmogorov equation. In this paper, we consider three CEAs of three-dimensional real evolution algebras. These CEAs depend on several (non-zero) functions defined on the set of time. For each chain we give a full classification (up to isomorphism) of the algebras depending on the time-parameter. We find concrete functions ensuring that the corresponding CEA contains all possible three-dimensional evolution algebras. [ABSTRACT FROM AUTHOR]

We consider an evolution algebra identifying the coefficients of SIS–SIR worm propagation models as the structure constants of the algebra. The basic properties of this algebra are studied. We prove that it is a commutative (and hence flexible), not associative and baric algebra. We describe the full set of idempotent elements and the full set of absolute nilpotent elements. We find all the fixed points of the dynamical systems. We also study several properties of the algebra connecting them to dynamical systems. [ABSTRACT FROM AUTHOR]

The space of derivations of finite dimensional evolution algebras associated to graphs over a field with characteristic zero has been completely characterized in the literature. In this work we generalize that characterization by describing the derivations of this class of algebras for fields of any characteristic. [ABSTRACT FROM AUTHOR]

Evolution algebras are a special class of nonassociative algebras exhibiting connections with various fields of mathematics. Hilbert evolution algebras generalize the concept in the framework of Hilbert spaces. This allows us to deal with a wide class of infinite-dimensional spaces. We study Hilbert evolution algebras associated to a graph. Inspired by the definitions of evolution algebras we define the Hilbert evolution algebra that is associated to a given graph and the Hilbert evolution algebra that is associated to the symmetric random walk on a graph. For a given graph, we provide the conditions for these structures to be or not to be isomorphic. Our definitions and results extend to the graphs with infinitely many vertices. We also develop a similar theory for the evolution algebras associated to finite graphs. [ABSTRACT FROM AUTHOR]

In the present paper, we examined the extendibility of evolution subalgebras generated by idempotents of evolution algebras. The extendibility of the isomorphism of such subalgebras to the entire algebra was investigated. Moreover, the existence of an evolution algebra generated by arbitrary idempotents was also studied. Furthermore, we described the tensor product of algebras generated by arbitrary idempotents and found the conditions of the tensor decomposability of four-dimensional S-evolution algebras. This paper’s findings shed light on the field of algebraic structures, particularly in studying evolution algebras. By examining the extendibility of evolution subalgebras generated by idempotents, we provide insights into the structural properties and relationships within these algebras. Understanding the isomorphism of such subalgebras and their extension allows a deeper comprehension of the overall algebraic structure and its behaviour.

In this paper, we describe absolute nilpotent and some idempotent elements of an n -dimensional evolution algebra corresponding to two permutations and we decompose such algebras to the direct sum of evolution algebras corresponding to cycles of the permutations. [ABSTRACT FROM AUTHOR]

In this paper, new advances concerning the link between evolution algebras and (pseudo)digraph are shown. Some important elements that can be read from the (pseudo)digraph that is associated with an evolution algebra are studied. Moreover, several results concerning solvability, nilpotency, and the preservation of them under the graph union operation are proved. To complement the theoretical study, an algorithmic method has been implemented. This is devoted to computing the nilpotency index of a nilpotent evolution algebra using its associated digraph. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new invariant for a non-degenerate evolution algebra, which consists of an ordered sequence of evolution algebras of lower dimension, belonging all of them to a specific family. We use this invariant to propose a method to classify non-degenerate evolution algebras, and we apply it up to dimension 3. We also use it to describe the derivations of some families of evolution algebras and the degenerations of the variety of evolution algebras with square not greater than 1. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the link between evolution algebras and (pseudo)digraphs. We study some theoretical properties about this association and determine the properties of the (pseudo)digraphs associated with each type of evolution algebras. We also analyze the isomorphism classes for each configuration associated with these algebras providing a new method to classify them, and we compare our results with the current classifications of two‐ and three‐dimensional evolution algebras. In order to complement the theoretical study, we have designed and performed the implementation of an algorithm, which constructs and draws the (pseudo)digraph associated with a given evolution algebra and another procedure to study the solvability of a given evolution algebra. [ABSTRACT FROM AUTHOR]

This paper deals with the concept of hierarchy of algebras and graphs. In the case of algebras, the work constitutes a generalization to any algebra of the concept of hierarchy that Tian gave for a particular type of them, evolution algebras, via concepts of occurrence and persistence. This new hierarchy proves to be invariant under isomorphism of algebras, which leads to a necessary condition for two generic algebras to be isomorphic. Furthermore, the task of how to effectively obtain the hierarchy of an algebra is also discussed, arriving to the association of a certain type of graphs to generic algebras, which leads us to introduce new concepts, based in hierarchy, in graph theory. [ABSTRACT FROM AUTHOR]

In this paper, we introduce Volterra evolution algebras which are evolution algebras whose structural matrices are described by skew symmetric matrices. A main result of the present paper gives a connection between such kind of algebras with ergodicities of Volterra quadratic stochastic operators. Furthermore, some of the properties of the considered algebras such as nilpotency, derivations have been studied as well. [ABSTRACT FROM AUTHOR]

Evolution algebras are non-associative algebras inspired from biological phenomena, with applications to or connections with different mathematical fields. There are two natural ways to define an evolution algebra associated to a given graph. While one takes into account only the adjacencies of the graph, the other includes probabilities related to the symmetric random walk on the same graph. In this work we state new properties related to the relation between these algebras, which is one of the open problems in the interplay between evolution algebras and graphs. On the one hand, we show that for any graph both algebras are strongly isotopic. On the other hand, we provide conditions under which these algebras are or are not isomorphic. For the case of finite non-singular graphs we provide a complete description of the problem, while for the case of finite singular graphs we state a conjecture supported by examples and partial results. The case of graphs with an infinite number of vertices is also discussed. As a sideline of our work, we revisit a result existing in the literature about the identification of the automorphism group of an evolution algebra, and we give an improved version of it. [ABSTRACT FROM AUTHOR]

In this work we approach three-dimensional evolution algebras from certain constructions performed on two-dimensional algebras. More precisely, we provide four different constructions producing three-dimensional evolution algebras from two-dimensional algebras. Also we introduce two parameters, the annihilator stabilizing index and the socle stabilizing index, which are useful tools in the classification theory of these algebras. Finally, we use moduli sets as a convenient way to describe isomorphism classes of algebras. [ABSTRACT FROM AUTHOR]

We associate a square to any two-dimensional evolution algebra. This geometric object is uniquely determined, does not depend on the basis and describes the structure and the behavior of the algebra. We determine the identities of degrees at most four, as well as derivations and automorphisms. We look at the group of automorphisms as an algebraic group, getting in this form a new algebraic invariant. The study of associative representations of evolution algebras is also started and we get faithful representations for most two-dimensional evolution algebras. In some cases, we prove that faithful commutative and associative representations do not exist, giving rise to the class of what could be termed as "exceptional" evolution algebras (in the sense of not admitting a monomorphism to an associative algebra with deformed product). [ABSTRACT FROM AUTHOR]

We study the space of derivations for some finite-dimensional evolution algebras, depending on the twin partition of an associated directed graph. For evolution algebras with a twin-free associated graph, we prove that the space of derivations is zero. For the remaining families of evolution algebras, we obtain sufficient conditions under which the study of such a space can be simplified. We accomplish this task by identifying the null entries of the respective derivation matrix. Our results suggest how strongly the associated graph's structure impacts in the characterization of derivations for a given evolution algebra. Therefore, our approach constitutes an alternative to the recent developments in the research of this subject. As an illustration of the applicability of our results, we provide some examples and we exhibit the classification of the derivations for non-degenerate irreducible three-dimensional evolution algebras. [ABSTRACT FROM AUTHOR]

The affine group scheme of automorphisms of an evolution algebra E with E 2 = E is shown to lie in an exact sequence 1 → D → A u t (E) → S , where D , diagonalizable, and S , constant, depend solely on the directed graph associated to E . As a consequence, the Lie algebra of derivations D e r (E) (with E 2 = E ) is shown to be trivial if the characteristic of the ground field is 0 or 2, and to be abelian, with a precise description, otherwise. [ABSTRACT FROM AUTHOR]

As a system of abstract algebra, evolution algebras are commutative and non-associative algebras. There is no deep structure theorem for general non-associative algebras. However, there are deep structure theorem and classification theorem for evolution algebras because it has been introduced concepts of dynamical systems to evolution algebras. Recently, in [25], it has been studied some properties of nilpotent evolution algebra with maximal index (dim E² = dim E - 1). This paper is devoted to studying nilpotent finite-dimensional evolution algebras E with dim E² = dim E - 2. We describe Lie algebras related to the evolution of algebras. Moreover, this result allowed us to characterize all local and 2-local derivations of the considered evolution algebras. All automorphisms and local automorphisms of the nilpotent evolution algebras are found. [ABSTRACT FROM AUTHOR]

This article is devoted to n-dimensional nilpotent finite-dimensional evolution algebras E with maximal index of nilpotency. We are able to find full invariants of these algebras which allows us to classify them. Furthermore, we explicitly describe its associative enveloping algebras. Moreover, we give an affirmative answer to the question: is it possible to extend a given derivation of the nilpotent evolution algebra to its enveloping algebra. [ABSTRACT FROM AUTHOR]

It is well-known that the space of derivations of n-dimensional evolution algebras with non-singular matrices is zero. On the other hand, the space of derivations of evolution algebras with matrices of rank n−1 have also been completely described in the literature. In this work, we provide a complete description of the space of derivations of evolution algebras associated to graphs, depending on the twin partition of the graph. For graphs without twin classes with at least three elements, we prove that the space of derivations of the associated evolution algebra is zero. Moreover, we describe the spaces of derivations for evolution algebras associated to the remaining families of finite graphs. It is worth pointing out that our analysis includes examples of finite dimensional evolution algebras with matrices of any rank. [ABSTRACT FROM AUTHOR]

It is known that any multiplication of a finite dimensional algebra is determined by a matrix of structural constants. In general, this is a cubic matrix. Difficulty of investigation of an algebra depends on the cubic matrix. Such a cubic matrix defines a quadratic mapping called an evolution operator. In the case of evolution algebras, the cubic matrix consists many zeros allowing to reduce it to a square matrix. In this paper for any finite dimensional algebra we construct a family of evolution algebras corresponding to Jacobian of the evolution operator at a point of the algebra. We obtain some results answering how properties of an algebra depends on the properties of the corresponding family of evolution algebras. Moreover, we consider evolution algebras corresponding to 2 and 3-dimensional nilpotent Leibniz algebras. We prove that such evolution algebras are nilpotent too. Also we classify such evolution algebras. [ABSTRACT FROM AUTHOR]

Evolution algebras are a new type of non-associative algebras which are inspired from biological phenomena. A special class of such algebras, called Markov evolution algebras, is strongly related to the theory of discrete time Markov chains. The winning of this relation is that many results coming from Probability Theory may be stated in the context of Abstract Algebra. In this paper, we explore the connection between evolution algebras, random walks and graphs. More precisely, we study the relationships between the evolution algebra induced by a random walk on a graph and the evolution algebra determined by the same graph. Given that any Markov chain may be seen as a random walk on a graph, we believe that our results may add a new landscape in the study of Markov evolution algebras. [ABSTRACT FROM AUTHOR]

This paper is devoted to the nilpotent finite-dimensional evolution algebras E with dim E 2 = dim E − 1. We describe the Lie algebra of derivations of these algebras. Moreover, in terms of these Lie algebras, we fully construct nilpotent evolution algebra with maximal index of nilpotency. Furthermore, this result allowed us fully characterize all local and 2-local derivations of the considered evolution algebras. Besides, all automorphisms and local automorphisms of these algebras are found. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce S-evolution algebras and investigate their solvability, simplicity, and semisimplicity. The structure of enveloping algebras has been carried out through the attached graph of S-evolution algebras. Moreover, we introduce the concept of E-linear derivation of S-evolution algebras, and prove such derivations can be extended to their enveloping algebras under certain conditions.

In this paper we classify a family of three-dimensional real evolution algebras. We also consider an evolution operator for an evolution algebra and find fixed points of this operator for two and three-dimensional cases. Then we construct an evolution algebra, the matrix of structural constants of which is Jacobian of the evolution operator at a fixed point. We study isomorphism between these evolution algebras. [ABSTRACT FROM AUTHOR]

Abstract With the aim of finding useful tools and invariants to classify finite dimensional evolution algebras, we introduce and study the notion of a basic ideal. Every n -dimensional perfect evolution algebra has a maximal basic ideal I which is unique except when the dimension of I is n-1. An application of our results leads to the description of the four dimensional perfect non-simple evolution algebras over a field with mild restrictions. [ABSTRACT FROM AUTHOR]

Recently, by A. Elduque and A. Labra a new technique and a type of an evolution algebra are introduced. Several nilpotent evolution algebras defined in terms of bilinear forms and symmetric endomorphisms are constructed. The technique then used for the classification of the nilpotent evolution algebras up to dimension five. In this article, we develop this technique for high dimensional evolution algebras. We construct nilpotent evolution algebras of any type. Moreover, we show that, except the cases considered by Elduque and Labra, this construction of nilpotent evolution algebras does not give all possible nilpotent evolution algebras. [ABSTRACT FROM AUTHOR]

In this paper we study subalgebras of complex finite dimensional evolution algebras. We obtain the classification of nilpotent evolution algebras whose any subalgebra is an evolution subalgebra with a basis which can be extended to a natural basis of algebra. Moreover, we formulate three conjectures related to the description of such non-nilpotent algebras. [ABSTRACT FROM AUTHOR]

Evolution algebras are non-associative algebras that describe non-Mendelian hereditary processes and have connections with many other areas. In this paper, we obtain necessary and sufficient conditions for a given algebra A to be an evolution algebra. We prove that the problem is equivalent to the so-called SDC problem, that is, the simultaneous diagonalisation via congruence of a given set of matrices. More precisely we show that an n-dimensional algebra A is an evolution algebra if and only if a certain set of n symmetric n×n matrices {M1,…,Mn} describing the product of A are SDC. We apply this characterisation to show that while certain classical genetic algebras (representing Mendelian and auto-tetraploid inheritance) are not themselves evolution algebras, arbitrarily small perturbations of these are evolution algebras. This is intringuing, as evolution algebras model asexual reproduction, unlike the classical ones.

In this paper, we merge two theories: that of pulse processes on weighted digraphs and that of evolution algebras. We enrich both of them. In fact, we obtain new results in the theory of pulse processes thanks to the new algebraic tool that we introduce in its framework, also extending the theory of evolution algebras, as well as its applications.

Any gene mutation during the mitotic cell cycle of a eukaryotic cell can be algebraically represented by an isotopism of the evolution algebra describing the genetic pattern of the inheritance process. We identify any such pattern with a total-colored graph so that any isotopism of the former is uniquely related to an isomorphism of the latter. This enables us to develop some results on graph theory in the context of the molecular processes that occur during the S-phase of a mitotic cell cycle. In particular, each monochromatic subset of edges is identified with a mutation or regulatory mechanism that relates any two statuses of the genotypes of a pair of chromatids.