Derivations of some classes of Leibniz algebras

Let L be an algebra over a field K. It is called a Leibniz algebra if its the bilinear binary operation [ ; ] satisfies the following Leibniz identity: (x, [y, z]) = ([x, y]), z – ([x, z]), y , ∀ x, y, z ∈ L. The thesis deals with derivations of Liebniz algebras introduced by Loday in 1993....

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Main Author:	Ahmed I, Al-Nashri Al-Hossain
Format:	Thesis
Language:	English
Published:	2013
Subjects:	Mathematics Algebra
Online Access:	http://psasir.upm.edu.my/id/eprint/67527/1/FS%202013%2098%20IR.pdf
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spelling	my-upm-ir.675272019-03-14T02:16:31Z Derivations of some classes of Leibniz algebras 2013-05 Ahmed I, Al-Nashri Al-Hossain Let L be an algebra over a field K. It is called a Leibniz algebra if its the bilinear binary operation [ ; ] satisfies the following Leibniz identity: (x, [y, z]) = ([x, y]), z – ([x, z]), y , ∀ x, y, z ∈ L. The thesis deals with derivations of Liebniz algebras introduced by Loday in 1993. Leibniz algebra is a generalization of Lie algebra. The class of Leibniz algebras in dimensional two and three has been classified by Loday and Omirov, respectively. Starting from dimensions four and above there are classifications of subclasses (in dimension four nilpotent case and in dimensions (5-8) there are classifications of filiform Leibniz algebras). According to the classification results above, in the thesis the derivations of each of the classes are given. It is known that the class of filiform Leibniz algebras in dimension n is split into three disjoint sub classes denoted by F Lbn, SLbn and T Lbn. The main results of the research are follows: • An algorithm to find the derivations of any class of algebras is given, which is implemented to low-dimensional cases of Leibniz algebras. • Descriptions of derivations of low-dimensional Leibniz algebras (in dimensions two and three this description is given for all Leibniz algebras, in dimension four the derivations of nilpotent part are given and in dimensions 5 -8 the derivations of filiform Leibniz algebras are found). In each of the cases the Leibniz algebras which are characteristically nilpotent are identified. • The derivations of naturally graded Leibniz algebras is described and the derivation algebra dimensions are provided. • For the classes FLbn, SLbn and T Lbn estimations of dimensions of the derivation algebras, depending on n, are given . • A computer program for finding the derivations of algebras is provided. Mathematics Algebra 2013-05 Thesis http://psasir.upm.edu.my/id/eprint/67527/ http://psasir.upm.edu.my/id/eprint/67527/1/FS%202013%2098%20IR.pdf text en public doctoral Universiti Putra Malaysia Mathematics Algebra
institution	Universiti Putra Malaysia
collection	PSAS Institutional Repository
language	English
topic	Mathematics Algebra
spellingShingle	Mathematics Algebra Ahmed I, Al-Nashri Al-Hossain Derivations of some classes of Leibniz algebras
description	Let L be an algebra over a field K. It is called a Leibniz algebra if its the bilinear binary operation [ ; ] satisfies the following Leibniz identity: (x, [y, z]) = ([x, y]), z – ([x, z]), y , ∀ x, y, z ∈ L. The thesis deals with derivations of Liebniz algebras introduced by Loday in 1993. Leibniz algebra is a generalization of Lie algebra. The class of Leibniz algebras in dimensional two and three has been classified by Loday and Omirov, respectively. Starting from dimensions four and above there are classifications of subclasses (in dimension four nilpotent case and in dimensions (5-8) there are classifications of filiform Leibniz algebras). According to the classification results above, in the thesis the derivations of each of the classes are given. It is known that the class of filiform Leibniz algebras in dimension n is split into three disjoint sub classes denoted by F Lbn, SLbn and T Lbn. The main results of the research are follows: • An algorithm to find the derivations of any class of algebras is given, which is implemented to low-dimensional cases of Leibniz algebras. • Descriptions of derivations of low-dimensional Leibniz algebras (in dimensions two and three this description is given for all Leibniz algebras, in dimension four the derivations of nilpotent part are given and in dimensions 5 -8 the derivations of filiform Leibniz algebras are found). In each of the cases the Leibniz algebras which are characteristically nilpotent are identified. • The derivations of naturally graded Leibniz algebras is described and the derivation algebra dimensions are provided. • For the classes FLbn, SLbn and T Lbn estimations of dimensions of the derivation algebras, depending on n, are given . • A computer program for finding the derivations of algebras is provided.
format	Thesis
qualification_level	Doctorate
author	Ahmed I, Al-Nashri Al-Hossain
author_facet	Ahmed I, Al-Nashri Al-Hossain
author_sort	Ahmed I, Al-Nashri Al-Hossain
title	Derivations of some classes of Leibniz algebras
title_short	Derivations of some classes of Leibniz algebras
title_full	Derivations of some classes of Leibniz algebras
title_fullStr	Derivations of some classes of Leibniz algebras
title_full_unstemmed	Derivations of some classes of Leibniz algebras
title_sort	derivations of some classes of leibniz algebras
granting_institution	Universiti Putra Malaysia
publishDate	2013
url	http://psasir.upm.edu.my/id/eprint/67527/1/FS%202013%2098%20IR.pdf
_version_	1747812480117112832

Derivations of some classes of Leibniz algebras

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