Degenerations of low-dimensional complex Liebniz algebras

Non-commutative analog of Lie algebras are Leibniz algebras. One of the important course of study is the degenerations of Leibniz algebras. Degenerations (or formerly known as contractions) were effectively applied to a wide range of physical and mathematical points of view. This thesis focuses on the degenerations of low-dimensional Leibniz algebras over the field of complex numbers particularly in the algebraic description of the varieties of three-dimensional complex Leibniz algebras and five-dimensional complex filiform Leibniz algebras arising from naturally gradaed non-Lie Leibniz algebras. The first part of this thesis describe the basic concepts and definitions of structural theory of Leibniz algebras and its degenerations. From the classification list, calculation of invariance arguments are collected. As a result, degenerations of algebras have been constructed by using algebraic invariants. The second part of this thesis concentrates on finding some essential degenerations of an arbitrary pair of the algebras of the same dimensions. Existence of degeneration matrices, gt is needed in order to prove the degenerations. For non degeneration case, it is enough to provide certain reasons to reject the degenerations. The last part of this thesis gives the orbit closure, rigid algebras and irreducible components of an affine algebraic variety of three-dimensional complex Leibniz algebras.

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