Algorithmic method to obtain combinatorial structures associated with Leibniz algebras (original) (raw)
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Given a finite-dimensional Leibniz algebra with certain basis, we show how to associate such a Lie algebra with a combinatorial structure of dimension 2. In some particular cases, this structure can be reduced to a digraph or a pseudodigraph. In this paper, we study some theoretical properties about this association and we determine the type of Leibniz algebra associated with each of these structures having no more than 3 vertices.
Combinatorial structures associated with finite-dimensional Leibniz algebras
This paper is the second in a series giving the foundations for associating Leibniz algebras with combinatorial structures of dimension 2. On this occasion, we characterize Leibniz algebras of dimension greater than 3 associated with (psuedo)digraphs and, consequently, we also study the converse, proving some properties about (pseudo)digraphs associated with Leibniz algebras. Finally, we analyze the nilpotency and solvability for Leibniz algebras associated with (pseudo)digraphs.
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This paper studies the link between isomorphic digraphs and isomorphic Leibniz algebras, determining in detail this fact when using (psuedo)digraphs of 2 and 3 vertices associated with Leibniz algebras according to their isomorphism classes. Moreover, we give the complete list with all the combinatorial structures of 3 vertices associated with Leibniz algebras, studying their isomorphism classes. We also compare our results with the current classifications of 2-and 3-dimensional Leibniz algebras. Finally, we introduce and implement the algorithmic procedure used for our goals and devoted to decide if a given combinatorial structure is associated or not with a Leibniz algebra.
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In this paper, we study the structure and properties of those n-dimensional Lie algebras associated with either summed structures of complete graphs or some families of digraphs, having into consideration that all these combinatorial structures are made up of n vertices. Our main goal is to obtain criteria determining when a Lie algebra is associated with some of combinatorial structures considered in this paper, as well as to study the properties of those structures in order to use them as a tool for classifying the types of Lie algebras associated with them.
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This paper shows a characterization of digraphs of three vertices associated with Lie algebras, as well as determining the list of isomorphism classes for Lie algebras associated with these digraphs. Additionally, we introduce and implement two algorithmic procedures related to this study: the first is devoted to draw, if exists, the digraph associated with a given Lie algebra; whereas the
Algorithm for Testing the Leibniz Algebra Structure
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Given a basis of a vector space V over a field K and a multiplication table which defines a bilinear map on V , we develop a computer program on Mathematica which checks if the bilinear map satisfies the Leibniz identity, that is, if the multiplication table endows V with a Leibniz algebra structure.
On the description of Leibniz algebras with nilindex nā3
Acta Mathematica Hungarica, 2011
In this paper we present the classification of a subclass of naturally graded Leibniz algebras. These n-dimensional Leibniz algebras have the characteristic sequence equal to (n ā 3, 3). For this purpose we use the software M athematica.
Combinatorial structures associated with Lie algebras of finite dimension
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Given a Lie algebra of finite dimension, with a selected basis of it, we show in this paper that it is possible to associate it with a combinatorial structure, of dimension 2, in general. In some particular cases, this structure is reduced to a weighted graph. We characterize such graphs, according to they have 3-cycles or not.
On Classification of Finite Dimensional Complex Filiform Leibniz Algebras (PartI)
The paper is devoted to classification problem of finite dimensional complex none Lie filiform Leibniz algebras. Actually, the observations show there are two resources to get classification of filiform Leibniz algebras. The first of them is naturally graded none Lie filiform Leibniz algebras and the another one is naturally graded filiform Lie algebras. Using the first resource we get two disjoint classes of filiform Leibniz algebras [10]. The present paper deals with the second of the above two classes, the first class has been considered in . The algebraic classification here means to specify the representatives of the orbits, whereas the geometric classification is the problem of finding generic structural constants in the sense of algebraic geometry. Our main effort in this paper is the algebraic classification. We suggest here an algebraic method based on invariants. Utilizing this method for any given low dimensional case all filiform Leibniz algebras can be classified. Moreover, the results can be used for geometric classification of orbits of such algebras.
Description of some classes of Leibniz algebras
Linear Algebra and its Applications, 2012
In this paper we describe the isomorphism classes of finitedimensional complex Leibniz algebras whose quotient algebra with respect to the ideal generated by squares is isomorphic to the direct sum of three-dimensional simple Lie algebra sl 2 and a threedimensional solvable ideal. We choose a basis of the isomorphism classes' representatives and give explicit multiplication tables. (I.M. Rikhsiboev), khabror@mail.ru (A.Kh. Khudoyberdiyev), iqboli@gmail.com (I.A. Karimjanov).