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

Pre-derivations and description of non-strongly nilpotent filiform Leibniz algebras

summary:In this paper we give the description of some non-strongly nilpotent Leibniz algebras. We pay our attention to the subclass of nilpotent Leibniz algebras, which is called filiform. Note that the set of filiform Leibniz algebras of fixed dimension can be decomposed into three disjoint families. We describe the pre-derivations of filiform Leibniz algebras for the first and second families and determine those algebras in the first two classes of filiform Leibniz algebras that are non-strongly nilpotent

Pameran Reka Cipta, Penyelidikan dan Inovasi (PRPI) 2009

PRPI 2009 kini telah memasuki tahun penganjurannya yang ke-7. Pameran penyelidikan di UPM telah bermula sejak tahun 1997 semasa Exhibition & Seminar Harnessing for Industry Advantage. Pada tahun 2002, Pameran Reka Cipta dan Penyelidikan (PRP) buat pertama kali telah diadakan dengan menggunakan konsep pertandingan hasil projek penyelidikan yang telah dijalankan oleh para penyelidik UPM. Kejayaan penganjuran PRP 2002 telah merintis usaha untuk menjadikannya sebagai aktiviti tahunan UPM dan ianya terus berkembang sejajar dengan nama baharunya yang ditukar kepada Pameran Reka Cipta, Penyelidikan dan Inovasi yang bermula penganjurannya pada tahun 2005. Sebagai kesinambungan daripada kejayaan penganjuran PRPI 2006, 2007 dan 2008 yang lalu dan status UPM sebagai salah sebuah Universiti Penyelidikan, PRPI 2009 kali ini yang merupakan pameran penyelidikan yang terbesar di UPM terus dilaksanakan dengan aspirasi dan semangat yang lebih jitu. Pameran ini juga menjadi pelantar kepada para penyelidik untuk mengenengahkan hasil penyelidikan yang dijalankan dan penemuan baharu kepada umum. Di samping itu ianya juga menjadi penanda aras terhadap kualiti sesuatu projek penyelidikan bagi melayakkan para penyelidik UPM untuk menyertai pameran di peringkat kebangsaan dan seterusnya antarabangsa. Adalah diharapkan pelaksanaan PRPI 2009 ini akan dapat menyemarakkan budaya penyelidikan di kalangan staf dan juga pelajar UPM sekaligus menjadikan UPM sebagai Universiti Penyelidikan yang cemerlang di negara ini

Classification of second-class 10-dimensional complex filiform Leibniz algebras

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