Isamiddin S. Rakhimov

ON LIE-LIKE COMPLEX FILIFORM LEIBNIZ ALGEBRAS

In this paper we propose an approach to classifying a subclass of filiform Leibniz algebras. This subclass arises from the naturally graded filiform Lie algebras. We reconcile and simplify the structure constants of such a class. In the arbitrary fixed dimension case an effective algorithm to control the behavior of the structure constants under adapted transformations of basis is presented. In one particular case, the precise formulas for less than 10 dimensions are given. We provide a computer program in Maple that can be used in computations as well.

Classification of a subclass of low-dimensional complex filiform Leibniz algebras

We give a complete classification of a subclass of complex filiform Leibniz algebras obtained from naturally graded non-Lie filiform Leibniz algebras. The isomorphism criteria in terms of invariant functions are given.

On isomorphism classes and invariants of a subclass of low-dimensional complex filiform Leibniz algebras

The article aims to study the classification problem of low-dimensional complex filiform Leibniz algebras. It is known that filiform Leibniz algebras come out from two sources. The first source is a naturally graded non-Lie filiform Leibniz algebra, and another one is a naturally graded filiform Lie algebra. In this article, we classify a subclass of the class of filiform Leibniz algebras appearing from the naturally graded non-Lie filiform Leibniz algebra. We give complete classification and isomorphism criteria in dimensions 5–7. The method of classification is purely algorithmic. The isomorphism criteria are given in terms of invariant functions.

Generalized Derivations in Semiprime Gamma Rings

Let be a 2-torsion-free semiprime -ring satisfying the condition for all , and let be an additive mapping such that for all and for some derivation of . We prove that is a generalized derivation.

ON ISOMORPHISM CRITERIA FOR LEIBNIZ CENTRAL EXTENSIONS OF A LINEAR DEFORMATION OF μn

This paper deals with the classification problems of Leibniz central extensions of linear deformations of a Lie algebra. It is known that any n-dimensional filiform Lie algebra can be represented as a linear deformation of n-dimensional filiform Lie algebra μn given by the brackets [ei, e0] = ei+1, i = 0,1,…,n - 2, in a basis {e0, e1,…,en - 1}. In this paper we consider a linear deformation of μn and its Leibniz central extensions. The resulting algebras are Leibniz algebras, this class is denoted here by Ced(μn). We choose an appropriate basis of Ced(μn) and give general isomorphism criteria. By using the isomorphism criteria, one can classify the class Ced(μn) for any fixed n. Two relevant maple programs are provided.

ON ONE-DIMENSIONAL LEIBNIZ CENTRAL EXTENSIONS OF A FILIFORM LIE ALGEBRA

The paper deals with the classification of Leibniz central extensions of a filiform Lie algebra. We choose a basis with respect to which the multiplication table has a simple form. In low-dimensional cases isomorphism classes of the central extensions are given. In the case of parametric families of orbits, invariant functions (orbit functions) are provided.

Isomorphism classes and invariants of low-dimensional filiform Leibniz algebras

In the paper, we extend the result on classification of a subclass of filiform Leibniz algebras in low dimensions to dimensions seven and eight based on a technique used by Rakhimov and Bekbaev for classification of subclasses which arise from naturally graded non-Lie filiform Leibniz algebras. The class considered here arises from naturally graded filiform Lie algebras. It contains the class of filiform Lie algebras and consequently, by classifying this subclass, we again re-examine the classification result of filiform Lie algebras. The resulting list of filiform Lie algebras is compared with that given by Ancochéa-Bermúdez and Goze in 1988 and by Gómez, Jiménez-Merchán and Khakimdjanov in 1998.

Classification of three dimensional complex Leibniz algebras

The aim of this paper is to complete the classification of three-dimensional complex Leibniz algebras. The description of isomorphism classes of three-dimensional complex Leibniz algebras has been given by Ayupov and Omirov in 1999. However, we found that this list has a little redundancy. In this paper we apply a method which is more elegant and it gives the precise list of isomorphism classes of these algebras. We compare our list with that of Ayupov-Omirov and show the corrections which should be made.

On Prime-Gamma-Near-Rings with Generalized Derivations

Let 𝑁 be a 2-torsion free prime Γ-near-ring with center 𝑍(𝑁). Let (𝑓,𝑑) and (𝑔,ℎ) be two generalized derivations on 𝑁. We prove the following results: (i) if 𝑓([𝑥,𝑦]𝛼)=0 or 𝑓([𝑥,𝑦]𝛼)=±[𝑥,𝑦]𝛼 or 𝑓2(𝑥)∈𝑍(𝑁) for all 𝑥,𝑦∈𝑁, 𝛼∈Γ, then 𝑁 is a commutative Γ-ring. (ii) If 𝑎∈𝑁 and [𝑓(𝑥),𝑎]𝛼=0 for all 𝑥∈𝑁, 𝛼∈Γ, then 𝑑(𝑎)∈𝑍(𝑁). (iii) If (𝑓𝑔,𝑑ℎ) acts as a generalized derivation on 𝑁, then 𝑓=0 or 𝑔=0.

Complete classification of two-dimensional algebras

A complete classification of two-dimensional algebras over algebraically closed fields is provided.