TY  - THES
T1  - Classification Of Low Dimensional Nilpotent Leibniz Algebras Using Central Extensions

A1  - Langari, Seyed Jalal
LA  - English
LA  - English
YR  - 2010
UL  - http://discoverylib.upm.edu.my/discovery/Record/oai:psasir.upm.edu.my:12373
AB  - 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.
KW  - Nilpotent Lie groups
KW  - Lie algebras
ER  -