Sharifah Kartini Said Husain

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.

Application of modified complex Tremblay operator

In this paper, we introduce a new fractional integral operator defined by modified fractional derivative Tremblay operator of analytic functions and show that the univalence of this integral operator is preserved under certain sufficient conditions in complex domain ℂ.

Centroids and derivations of associative algebras

In this paper we study centroids and derivations of associative algebras. Using a classification result of associative algebras, we describe the centroids and derivations of low-dimensional associative algebras. We review some properties of the centroids in the light of associative algebras and use these properties to categorize the algebras into having small and not small centroids. We also give a list of low-dimensional characteristically nilpotent associative algebras.

ISOMORPHISM CLASSES AND INVARIANTS FOR A SUBCLASS OF NINE-DIMENSIONAL FILIFORM LEIBNIZ ALGEBRAS

The paper concerns the classification problem of a subclass of nilpotent Leibniz algebras. This subclass arises from naturally graded non Lie filiform Leibniz algebras. An invariant-theoretic approach to the classification problem of this subclass has been suggested by Rakhimov and Bekbaev. This approach provides to solve the problems with minimum difficulty. Implementing this approach, it is given complete list of isomorphism classes for a subclass of filiform Leibniz algebras and appropriate invariants in dimension nine.

Combinatorial structures associated with low dimensional second class of non-Lie filiform Leibniz algebra

In this talk we propose a graphical representation of some classes of Leibniz algebras. With each of algebra from these classes we associate a graph. This assignment enables us to reformulate some structural properties of the Leibniz algebras in terms of some conditions for the graphs. The talk focuses on a so-called filiform Leibniz algebras. It is well-known that this class is split into three subclasses called first, second and third class denoted, in dimension n over a field K, by FLbn(K), SLbn(K) and TLbn(K), respectively. In this article, we concern more on the combinatorial structures associated with SLbn(K) in low-dimensions.

Contractions of low dimensional complex associative algebras

Contraction is one of the most important concepts that play an important role from the mathematical and physical point of view. In this work, the contractions of complex associative algebras are considered. We focus on the variety A2(ℂ) that consisting of all associative algebras of dimension two over the complex numbers ℂ (including nonunital). Various contractions criteria are collected and new criteria are proposed to test the possible existence of contraction for each pair of associative algebras. As a result, we prove that the variety A2(ℂ) has three irreducible components, two of dimension 2 and one of dimension 4.

On semi ring bornologies

Our main focus in this work is to introduce new structure bornological semi rings. This generalizes the theory of algebraic semi rings from the algebraic setting to the framework of bornological sets. We give basic properties for this new structure. As well as, We study the fundamental construction of bornological semi ring as product, inductive limits and projective limits and their extensions on bornological semi ring. Additionally, we introduce the category of bornological semi rings and study product and pullback (fiber product) in the category of bornological semi rings.

Isomorphism classes of 10-dimensional filiform Leibniz algebras

This paper implements Rakhimov-Bekbaev approach to present a complete list of isomorphism classes of a subclass of complex filiform Leibniz algebras obtained from naturally graded non-Lie filiform Leibniz algebra. This class is split into two subclasses. In this paper we shall consider the second class which is denoted by SLb n in dimension n. The isomorphism criteria in terms of invariant functions for SLb 10 are presented. We represent SLb10as a union of subsets and show that some of these subsets are represented as union of infinitely many orbits (a set of isomorphic to each other algebras) while others are represented as just a single orbit. In former case we give invariant functions to distinguish the orbits, while for the latter case the representatives of the single orbits are provided. As a result, we give the list of isomorphism classes with the table of multiplications.

On isomorphism classes of a subclass of filiform leibniz algebras in dimension 10

Some classes of filiform Leibniz algebras arising from the naturally graded non Lie filiform Leibniz algebras were introduced by Ayupov and Omirov in 2006. One of them is the first class of filiform Leibniz algebras and has been denoted by FLBn in fixed dimension n.Rakhimov and Bekbaev have suggested an approach of classifying FLBn based on algebraic invariants method. In terms of classification, it has been classified up to dimension 9 and the main purpose of this paper is to use method of invariants and apply to dimension 10. To complete this goal, we set up an isomorphism criterion, here we introduced some new functions to avoid big expressions involved in isomorphism criterion. From the criterion some lists of disjoint subsets from FLB10 are obtained. These disjoint subsets are represented as a union of parametric family of orbits or just single orbits. For parametric cases, the invariants will be given.