Classification of non-lie complex filiform Leibniz algebras for low dimensions

The thesis is concerned with the structural properties of Leibniz algebras. These algebras satisfy certain identity that was suggested by J.-L.Loday (1993). When he used the tensor product instead of external product in the definition of the n-th cochain, in order to prove the differential property defined on cochains,it suffices to replace the anticommutativity and Jacobi identity by the Leibniz identity. The algebras satisfying the Leibniz identity are called Leibniz algebras. We will investigate filiform Leibniz algebras. It is known that filiform Leibniz algebras arise from two sources. The first source is a naturally graded non-Lie filiform Leibniz algebras and the other one is a naturally graded filiform Lie algebras. Here we consider the class of filiform Leibniz algebras arising from the naturally graded non-Lie filiform Leibniz algebra. In 2001, Ayupov and Omirov divided this class into two subclasses. However,isomorphism problems within these classes are yet to be investigated. The classes in dimension n over a field k are denoted by FLeibn(k) and SLeibn(k), respectively. Bekbaev and Rakhimov (2006) suggested an algebraic approach to the description of isomorphism classes of filiform Leibniz algebras in terms of algebraic invariants. The main purpose of this thesis is to apply this method and find the complete classification and invariants of low dimensional complex filiform Leibniz algebras. The main result is the complete classification of complex filiform Leibniz algebras arising from the naturally graded non-Lie filiform Leibniz algebras from dimensions 5 to 8.

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