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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| Format: | Thesis |
| Language: | English English |
| Published: |
2010
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| Subjects: | |
| Online Access: | http://psasir.upm.edu.my/id/eprint/12373/1/IPM_2010_3A.pdf |
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| Summary: | 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. |
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