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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 investigate in detail the local and 2-local derivations of the ow of quantum genetic Lotka-Volterra algebra belonging toM2(C). It aims to reveal how the different values of the parameter affect the different characteristic properties of these derivations. Our findings furnish structural details of the FQGLV-A algebras and develop the gap in actualizing local and 2-local derivations. We show that each local derivation is a derivation, and a set with 2-local derivations is a set of derivations. Further, we investigate the characterization of the Rota-Baxter operators in one of the considered scenarios with derivations to get better insights into the role and consequences of the algebraic structure of FQGLV-A. [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]

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 present paper deals with a class of ξ (a) -quadratic stochastic operators, referred to as QSOs, on a two-dimensional simplex. It investigates the algebraic properties of the genetic algebras associated with ξ (a) -QSOs. Namely, the associativity, characters and derivations of genetic algebras are studied. Moreover, the dynamics of these operators are also explored. Specifically, we focus on a particular partition that results in nine classes, which are further reduced to three nonconjugate classes. Each class gives rise to a genetic algebra denoted as A i , and it is shown that these algebras are isomorphic. The investigation then delves into analyzing various algebraic properties within these genetic algebras, such as associativity, characters, and derivations. The conditions for associativity and character behavior are provided. Furthermore, a comprehensive analysis of the dynamic behavior of these operators is conducted. [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]

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]

The current study provides a new class of ζss -QSO defined on 2D simplex and classifies it into 18 non-conjugate (isomorphic) classes. This classification is based on their conjugacy and the remuneration of coordinates. The current study also examines the limiting points associated with the behavior of trajectories for four classes defined on 2D simplex. [ABSTRACT FROM AUTHOR]

The present paper is devoted to genetic Volterra algebras. We first study characters of such algebras. We fully describe associative genetic Volterra algebras, in this case all derivations are trivial. In general setting, i.e., when the algebra is not associative, we provide a sufficient condition to get trivial derivation on generic Volterra algebras. Furthermore, we describe all derivations of three dimensional generic Volterra algebras, which allowed us to prove that any local derivation is a derivation of the algebra. [ABSTRACT FROM AUTHOR]

A quadratic stochastic operator (QSO) describes the time evolution of different species in biology. The main problem with regard to a nonlinear operator is to study its behavior. This has not been studied in depth; even QSOs, which are the simplest nonlinear operators, have not been studied thoroughly. This paper investigates the global behavior of an operator taken from ξ(s)-QSO when the parameter a = 1/2. Moreover, we study the local behavior of this operator at each value of a, where 0 < a < 1. [ABSTRACT FROM AUTHOR]