Classification Of First Class 9-Dimensional Complex Filiform Leibniz Algebras
Faculty: Science Let V be a vector space of dimension n over an algebraically closed ¯eld K (charK=0). Bilinear maps V £ V ! V form a vector space Hom(V V; V ) of dimensional n3, which can be considered together with its natural structure of an a±ne algebraic variety over K and denoted by Algn...
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| Format: | Thesis |
| Language: | English English |
| Published: |
2009
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| Subjects: | |
| Online Access: | http://psasir.upm.edu.my/id/eprint/5700/1/A_FS_2009_6.pdf |
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| Summary: | Faculty: Science
Let V be a vector space of dimension n over an algebraically closed ¯eld
K (charK=0). Bilinear maps V £ V ! V form a vector space Hom(V
V; V ) of dimensional n3, which can be considered together with its natural
structure of an a±ne algebraic variety over K and denoted by Algn(K) »= Kn3 .
An n-dimensional algebra L over K can be considered as an element ¸(L) of
Algn(K) via the bilinear mapping ¸ : L L ! L de¯ning a binary algebraic
operation on L : let fe1; e2; : : : ; eng be a basis of the algebra L: Then the table
of multiplication of L is represented by point (°k
ij) of this a±ne space as follows:
¸(ei; ej) =
Xn
k=1
°k
ijek:
Here °k
ij are called structural constants of L: The linear reductive group GLn(K)
acts on Algn(K) by (g ¤ ¸)(x; y) = g(¸(g¡1(x); g¡1(y)))(\transport of struc-
ture"). Two algebra structures ¸1 and ¸2 on V are isomorphic if and only if
they belong to the same orbit under this action.Recall that an algebra L over a ¯eld K is called a Leibniz algebra if its binary
operation satis¯es the following Leibniz identity:
[x; [y; z]] = [[x; y]; z] ¡ [[x; z]; y];
Leibniz algebras were introduced by J.-L.Loday. (For this reason, they have
also been called \Loday algebras"). A skew-symmetric Leibniz algebra is a Lie
algebra. In this case the Leibniz identity is just the Jacobi identity.
This research is devoted to the classi¯cation problem of Leibn in low dimen-
sional cases. There are two sources to get such a classi¯cation. The ¯rst of
them is naturally graded non Lie ¯liform Leibniz algebras and another one
is naturally graded ¯liform Lie algebras. Here we consider Leibniz algebras
appearing from the naturally graded non Lie ¯liform Leibniz algebras.
It is known that this class of algebras can be split into two subclasses. How-
ever, isomorphisms within each class have not been investigated yet. Recently
U.D.Bekbaev and I.S.Rakhimov suggested an approach to the isomorphism
problem of Leibniz algebras based on algebraic invariants.
This research presents an implementation of this invariant approach in 9-
dimensional case. We give the list of all 9-dimensional non Lie ¯liform Leibniz
algebras arising from the naturally graded non Lie ¯liform Leibniz algebras.
The isomorphism criteria and the list of algebraic invariants will be given. |
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