Citation
Mohd Kasim, Suzila
(2014)
Classification of secondclass 10dimensional complex filiform Leibniz algebras.
Masters thesis, Universiti Putra Malaysia.
Abstract
This thesis is concerned on studying the classification problem of a subclass of
(n + 1)dimensional complex filiform Leibniz algebras. Leibniz algebras that are
noncommutative generalizations of Lie algebras are considered. Leibniz identity
and Jacobi identity are equivalent when the multiplication is skewsymmetric.
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 nonLie filiform Leibniz algebras and another
one is a naturally graded filiform Lie algebras.
Naturally graded nonLie 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 nonLie
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
nonLie 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.
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