Classification of second-class 10-dimensional complex filiform Leibniz algebras - Mohd Kasim, Suzila (2014)
This thesis is concerned on studying the classification problem of a subclass of (n + 1)-dimensional complex filiform Leibniz algebras. Leibniz algebras that are non-commutative generalizations of Lie algebras are considered. Leibniz identity and Jacobi identity are equivalent when the multiplication is skew-symmetric. When studying a certain class of algebras, it is important to describe at least the algebras of lower dimensions up to an isomorphism. For Leibniz algebras, di - culties arise even when considering nilpotent algebras of dimension greater than four. Thus, a special class of nilpotent Leibniz algebras is introduced namely filiform Leibniz algebras. Filiform Leibniz algebras arise from two sources. The first source is a naturally graded non-Lie filiform Leibniz algebras and another one is a naturally graded filiform Lie algebras. Naturally graded non-Lie filiform Leibniz algebras contains subclasses FLbn+1 and SLbn+1. While there is only one subclass obtained from naturally graded fil- iform Lie algebras which is TLbn+1. These three subclasses FLbn+1, SLbn+1 and TLbn+1 are over a field of complex number, C where n+1 denotes the dimension of these subclasses starting with n>4. In particular, a method of simplification of the basis transformations of the arbi- trary filiform Leibniz algebras which were obtained from naturally graded non-Lie filiform Leibniz algebras, that allows for the problem of classification of algebras is reduced to the problem of a description of the structural constants. The inves- tigation of filiform Leibniz algebras which were obtained from naturally graded non-Lie filiform Leibniz algebras only for subclass SLbn+1 is the subject of this thesis. This research is the continuation of the works on SLbn+1 which have been treated for the cases of n < 9. The main purpose of this thesis is to apply the Rakhimov- Bekbaev approach to classify SLb10. These approach will give a complete classi- fication of SLb10 in terms of algebraic invariants. Isomorphism criterion of SLb10 is used to split the set of algebras SLb10 into several disjoint subsets. For each of these subsets, the classification problem is solved separately. As a result, some of them are represented as a union of infinitely many orbit (parametric families) and others as single orbits (isolated orbits). Finally, the list of isomorphism classes of complex filiform Leibniz algebras with the table of multiplications are given.