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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 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]

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]

The mitosis process of an eukaryotic cell can be represented by the structure constants of an evolution algebra. Any isotopism of the latter corresponds to a mutation of genotypes of the former. This paper uses Computational Algebraic Geometry to determine the distribution of three-dimensional evolution algebras over any field into isotopism classes and hence, to describe the spectrum of genetic patterns of three distinct genotypes during a mitosis process. Their distribution into isomorphism classes is also determined in case of dealing with algebras having a one-dimensional annihilator. [ABSTRACT FROM AUTHOR]

We classify three dimensional evolution algebras over a field having characteristic different from 2 and in which there are roots of orders 2, 3 and 7. [ABSTRACT FROM AUTHOR]

The type and several invariant subspaces related to the upper annihilating series of finite-dimensional nilpotent evolution algebras are introduced. These invariants can be easily computed from any natural basis. Some families of nilpotent evolution algebras, defined in terms of a nondegenerate, symmetric, bilinear form and some commuting, symmetric, diagonalizable endomorphisms relative to the form, are explicitly constructed. Both the invariants and these families are used to review and complete the classification of nilpotent evolution algebras up to dimension five over algebraically closed fields. [ABSTRACT FROM AUTHOR]

Consider a bisexual population such that the set of females can be partitioned into finitely many different types indexed by { 1 , 2 , … , n } and, similarly, that the male types are indexed by { 1 , 2 , … , ν } . Recently an evolution algebra of bisexual population was introduced by identifying the coefficients of inheritance of a bisexual population as the structure constants of the algebra. In this paper we study constrained evolution algebra of bisexual population in which type “1” of females and males have preference. For such algebras sets of idempotent and absolute nilpotent elements are known. We consider two particular cases of this algebra, giving more constraints on the structural constants of the algebra. By the first our constraint we obtain an n + ν -dimensional algebra with a matrix of structural constants containing only 0 and 1. In the second case we consider n = ν = 2 but with general constraints. In both cases we study dynamical systems generated by the quadratic evolution operators of corresponding constrained algebras. We find all fixed points, limit points and some 2-periodic points of the dynamical systems. Moreover we study several properties of the constrained algebras connecting them to the dynamical systems. We give some biological interpretation of our results. [ABSTRACT FROM AUTHOR]

We study evolution algebras of arbitrary dimension. We analyze in deep the notions of evolution subalgebras, ideals and non-degeneracy and describe the ideals generated by one element and characterize the simple evolution algebras. We also prove the existence and unicity of a direct sum decomposition into irreducible components for every non-degenerate evolution algebra. When the algebra is degenerate, the uniqueness cannot be assured. The graph associated to an evolution algebra (relative to a natural basis) will play a fundamental role to describe the structure of the algebra. Concretely, a non-degenerate evolution algebra is irreducible if and only if the graph is connected. Moreover, when the evolution algebra is finite-dimensional, we give a process (called the fragmentation process) to decompose the algebra into irreducible components. [ABSTRACT FROM AUTHOR]

We consider an evolution algebra which corresponds to a bisexual population with a set of females partitioned into finitely many different types and the males having only one type. We study basic properties of the algebra. This algebra is commutative (and hence flexible), not associative and not necessarily power-associative, in general. We prove that being alternative is equivalent to being associative. We find conditions to be an associative, a fourth power-associative, or a nilpotent algebra. We also prove that if the algebra is not alternative then to be power-associative is equivalent to be Jordan. Moreover it is not unital. In a general case, we describe the full set of idempotent elements and the full set of absolute nilpotent elements. The set of all operators of left (right) multiplications is described. Under some conditions it is proved that the corresponding algebra is centroidal. Moreover the classification of 2-dimensional and some 3-dimensional algebras are obtained. [ABSTRACT FROM AUTHOR]

Abstract: Recently Ladra and Rozikov introduced a notion of evolution algebra of a “chicken” population (EACP). The algebra is given by a rectangular matrix of structural constants. In this paper we introduce a notion of chain of evolution algebras of a “chicken” population (CEACP). The sequence of matrices of the structural constants for this CEACP satisfies an analogue of Chapman–Kolmogorov equation (with a specific multiplication defined for rectangular matrices). We give several examples (time homogeneous, time non-homogeneous, periodic, etc.) of such chains. We construct some periodic 3-dimensional CEACP which contains a continuum set of non-isomorphic EACP and show that the corresponding discrete time CEACP is dense in the set. Moreover we study time depending dynamics of 2 and 3 dimensional CEACP to be isomorphic to a fixed algebra. [Copyright &y& Elsevier]

Abstract: An evolution algebra corresponds to a quadratic matrix A of structural constants. It is known the equivalence between nil, right nilpotent evolution algebras and evolution algebras which are defined by upper triangular matrices A. We establish a criterion for an n-dimensional nilpotent evolution algebra to be with maximal nilpotent index . We give the classification of finite-dimensional complex evolution algebras with maximal nilpotent index. Moreover, for any we construct a wide class of n-dimensional evolution algebras with nilpotent index . We show that nilpotent evolution algebras are not dibaric and establish a criterion for two-dimensional real evolution algebras to be dibaric. [Copyright &y& Elsevier]

Abstract: We introduce a notion of chain of evolution algebras. The sequence of matrices of the structural constants for this chain of evolution algebras satisfies an analogue of Chapman–Kolmogorov equation. We give several examples (time homogenous, time non-homogenous, periodic, etc.) of such chains. For a periodic chain of evolution algebras we construct a continuum set of non-isomorphic evolution algebras and show that the corresponding discrete time chain of evolution algebras is dense in the set. We obtain a criteria for an evolution algebra to be baric and give a concept of a property transition. For several chains of evolution algebras we describe the behavior of the baric property depending on the time. For a chain of evolution algebras given by the matrix of a two-state evolution we define a baric property controller function and under some conditions on this controller we prove that the chain is not baric almost surely (with respect to Lebesgue measure). We also construct examples of the almost surely baric chains of evolution algebras. We show that there are chains of evolution algebras such that if it has a unique (resp. infinitely many) absolute nilpotent element at a fixed time, then it has unique (resp. infinitely many) absolute nilpotent element any time; also there are chains of evolution algebras which have not such property. For an example of two dimensional chain of evolution algebras we give the full set of idempotent elements and show that for some values of parameters the number of idempotent elements does not depend on time, but for other values of parameters there is a critical time such that the chain has only two idempotent elements if time and it has four idempotent elements if time . [Copyright &y& Elsevier]