Classification Of Low Dimensional Nilpotent Leibniz Algebras Using Central Extensions

This thesis is concerned with the classification of low dimensional nilpotent Leibniz algebras by central extensions over complex numbers. Leibniz algebras introduced by J.-L. Loday (1993) are non-antisymmetric generalizations of Lie algebras. There is a cohomology theory for these algebraic obje...

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Main Author:	Langari, Seyed Jalal
Format:	Thesis
Language:	English English
Published:	2010
Subjects:	Nilpotent Lie groups Lie algebras
Online Access:	http://psasir.upm.edu.my/id/eprint/12373/1/IPM_2010_3A.pdf
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id	my-upm-ir.12373
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spelling	my-upm-ir.123732013-05-27T07:51:54Z Classification Of Low Dimensional Nilpotent Leibniz Algebras Using Central Extensions 2010-07 Langari, Seyed Jalal This thesis is concerned with the classification of low dimensional nilpotent Leibniz algebras by central extensions over complex numbers. Leibniz algebras introduced by J.-L. Loday (1993) are non-antisymmetric generalizations of Lie algebras. There is a cohomology theory for these algebraic objects whose properties are similar to those of the classical Chevalley-Eilenberg cohomology theory for Lie algebras. The central extensions of Lie algebras play a central role in the classification theory of Lie algebras. We know that if a Leibniz algebra L satisfies the additional identity [x; x] = 0; x E L, then the Leibniz identity is equivalent to the Jacobi identity [[x; y]; z] + [[y; z]; x] + [[z; x]; y] = 0 8x; y; z E L: Hence, Lie algebras are particular cases of Leibniz algebras.In 1978 Skjelbred and Sund reduced the classification of nilpotent Lie algebras in a given dimension to the study of orbits under the action of a group on the space of second degree cohomology of a smaller Lie algebra with coefficients in a trivial module. The main purpose of this thesis is to establish elementary properties of central extensions of nilpotent Leibniz algebras and apply the Skjelbred-Sund's method to classify them in low dimensional cases. A complete classification of three and four dimensional nilpotent Leibniz algebras is provided in chapters 3 and 4. In particular, Leibniz central extensions of Heisenberg algebras Hn is provided in chapter 4. Chapter 5 concerns with application of the Skjelbred and Sund's method to the classification of filiform Leibniz algebras in dimension 5. Chapter 6 contains the conclusion and some proposed future directions. Nilpotent Lie groups Lie algebras 2010-07 Thesis http://psasir.upm.edu.my/id/eprint/12373/ http://psasir.upm.edu.my/id/eprint/12373/1/IPM_2010_3A.pdf application/pdf en public phd doctoral Universiti Putra Malaysia Nilpotent Lie groups Lie algebras Institute For Mathematical Research English
institution	Universiti Putra Malaysia
collection	PSAS Institutional Repository
language	English English
topic	Nilpotent Lie groups Lie algebras
spellingShingle	Nilpotent Lie groups Lie algebras Langari, Seyed Jalal Classification Of Low Dimensional Nilpotent Leibniz Algebras Using Central Extensions
description	This thesis is concerned with the classification of low dimensional nilpotent Leibniz algebras by central extensions over complex numbers. Leibniz algebras introduced by J.-L. Loday (1993) are non-antisymmetric generalizations of Lie algebras. There is a cohomology theory for these algebraic objects whose properties are similar to those of the classical Chevalley-Eilenberg cohomology theory for Lie algebras. The central extensions of Lie algebras play a central role in the classification theory of Lie algebras. We know that if a Leibniz algebra L satisfies the additional identity [x; x] = 0; x E L, then the Leibniz identity is equivalent to the Jacobi identity [[x; y]; z] + [[y; z]; x] + [[z; x]; y] = 0 8x; y; z E L: Hence, Lie algebras are particular cases of Leibniz algebras.In 1978 Skjelbred and Sund reduced the classification of nilpotent Lie algebras in a given dimension to the study of orbits under the action of a group on the space of second degree cohomology of a smaller Lie algebra with coefficients in a trivial module. The main purpose of this thesis is to establish elementary properties of central extensions of nilpotent Leibniz algebras and apply the Skjelbred-Sund's method to classify them in low dimensional cases. A complete classification of three and four dimensional nilpotent Leibniz algebras is provided in chapters 3 and 4. In particular, Leibniz central extensions of Heisenberg algebras Hn is provided in chapter 4. Chapter 5 concerns with application of the Skjelbred and Sund's method to the classification of filiform Leibniz algebras in dimension 5. Chapter 6 contains the conclusion and some proposed future directions.
format	Thesis
qualification_name	Doctor of Philosophy (PhD.)
qualification_level	Doctorate
author	Langari, Seyed Jalal
author_facet	Langari, Seyed Jalal
author_sort	Langari, Seyed Jalal
title	Classification Of Low Dimensional Nilpotent Leibniz Algebras Using Central Extensions
title_short	Classification Of Low Dimensional Nilpotent Leibniz Algebras Using Central Extensions
title_full	Classification Of Low Dimensional Nilpotent Leibniz Algebras Using Central Extensions
title_fullStr	Classification Of Low Dimensional Nilpotent Leibniz Algebras Using Central Extensions
title_full_unstemmed	Classification Of Low Dimensional Nilpotent Leibniz Algebras Using Central Extensions
title_sort	classification of low dimensional nilpotent leibniz algebras using central extensions
granting_institution	Universiti Putra Malaysia
granting_department	Institute For Mathematical Research
publishDate	2010
url	http://psasir.upm.edu.my/id/eprint/12373/1/IPM_2010_3A.pdf
_version_	1747811359769231360

Classification Of Low Dimensional Nilpotent Leibniz Algebras Using Central Extensions

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