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(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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