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 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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