Central Extensions of Nilpotent Lie and Leibniz Algebras

This thesis is concerned with the central extensions of nilpotent Lie and Leibniz algebras. The researcher has used the Skjelbred and Sund's method to construct all the 6¡dimensional non-isomorphic Lie algebras over complex numbers. Skjelbred and Sund have published in 1977 their method of constructing all nilpotent Lie algebras of dimension n introducing those algebras of dimension < n, and their automorphism groups. The deployed method in classifying nilpotent Lie algebras for dimension n is necessarily comprised of two main steps. For the frist step, a plausibly redundant list of Lie algebras was confirmed which included all n-dimensional nilpotent Lie algebras. The next step is to discard the isomorphic copies generated from the list of Lie algebras. By then, the n¡dimensional nilpotent Lie algebras are confirmed as central extension of nilpotent Lie algebras for smaller dimensions. As the result, the action of the automorphism group is deployed to drastically decrease the number of isomorphic Lie algebras which may appear in the final list. The maple program is by then deployed to discard the final isomorphic copies from the calculation. Based on the elaborated method, the researcher must and center of Lie algebras then drived algebras. In order to locate a basis for second cohomology group of Lie algebras, the researcher must locate cocycles as well as coboundary. Further, there is the requirement to locate the most appropriate action for finding orbits. By then, every orbit will submit a representative that is illustrated by 0.

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