universiti putra malaysia classification of second-class …

UNIVERSITI PUTRA MALAYSIA

CLASSIFICATION OF SECOND-CLASS 10-DIMENSIONAL COMPLEX FILIFORM LEIBNIZ

ALGEBRAS

SUZILA BINTI MOHD KASIM

IPM 2014 10

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HT UPMCLASSIFICATION OF SECOND-CLASS

10-DIMENSIONAL COMPLEX FILIFORM LEIBNIZ

ALGEBRAS

By


Thesis Submitted to the School of Graduate Studies,Universiti Putra Malaysia, in Ful�lment of the

Requirements for the Degree of Master of Science

April 2014

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COPYRIGHT

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Abstract of thesis presented to the Senate of Universiti Putra Malaysia inful�lment of the requirement for the degree of Master of Science

CLASSIFICATION OF SECOND-CLASS 10-DIMENSIONALCOMPLEX FILIFORM LEIBNIZ ALGEBRAS

By


April 2014

Chair: Professor Isamiddin S. Rakhimov, PhDFaculty: Institute for Mathematical Research

This thesis is concerned on studying the classi�cation problem of a subclass of(n+ 1)-dimensional complex �liform Leibniz algebras. Leibniz algebras that arenon-commutative generalizations of Lie algebras are considered. Leibniz identityand Jacobi identity are equivalent when the multiplication is skew-symmetric.When studying a certain class of algebras, it is important to describe at least thealgebras of lower dimensions up to an isomorphism. For Leibniz algebras, di�-culties arise even when considering nilpotent algebras of dimension greater thanfour. Thus, a special class of nilpotent Leibniz algebras is introduced namely�liform Leibniz algebras. Filiform Leibniz algebras arise from two sources. The�rst source is a naturally graded non-Lie �liform Leibniz algebras and anotherone is a naturally graded �liform Lie algebras.

Naturally graded non-Lie �liform Leibniz algebras contains subclasses FLbn+1

and SLbn+1 . While there is only one subclass obtained from naturally graded �l-iform Lie algebras which is TLbn+1. These three subclasses FLbn+1 , SLbn+1 andTLbn+1 are over a �eld of complex number, C where n+ 1 denotes the dimensionof these subclasses starting with n ≥ 4.

In particular, a method of simpli�cation of the basis transformations of the arbi-trary �liform Leibniz algebras which were obtained from naturally graded non-Lie�liform Leibniz algebras, that allows for the problem of classi�cation of algebrasis reduced to the problem of a description of the structural constants. The inves-tigation of �liform Leibniz algebras which were obtained from naturally gradednon-Lie �liform Leibniz algebras only for subclass SLbn+1 is the subject of thisthesis.

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This research is the continuation of the works on SLbn+1 which have been treatedfor 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-�cation of SLb10 in terms of algebraic invariants. Isomorphism criterion of SLb10is used to split the set of algebras SLb10 into several disjoint subsets. For each ofthese subsets, the classi�cation problem is solved separately. As a result, some ofthem are represented as a union of in�nitely many orbit (parametric families) andothers as single orbits (isolated orbits). Finally, the list of isomorphism classes ofcomplex �liform Leibniz algebras with the table of multiplications are given.

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Abstrak tesis yang dikemukakan kepada Senat Universiti Putra Malaysiasebagai memenuhi keperluan untuk ijazah Sarjana Sains

PENGKELASAN KELAS-KEDUA 10-DIMENSI ALJABARLEIBNIZ FILIFORM KOMPLEKS

Oleh


April 2014

Pengerusi: Profesor Isamiddin S. Rakhimov, PhDFakulti: Institut Penyelidikan Matematik

Tesis ini adalah berkenaan kajian terhadap masalah pengkelasan daripada subke-las (n+ 1)-dimensional aljabar Leibniz �liform kompleks. Aljabar Leibniz yangumum bukan kalis tukar tertib aljabar Lie akan dipertimbangkan. Identiti Leib-niz dan identiti Jacobi adalah setara apabila pendaraban adalah condong-simetri.Ketika mengkaji kelas tertentu aljabar, adalah penting untuk menerangkan al-jabar yang sekurang-kurangnya dimensi yang lebih rendah terhadap satu isomor-�sma. Bagi aljabar Leibniz, timbul kesukaran apabila mempertimbangkan aljabarnilpotent dimensi yang lebih besar daripada empat. Oleh itu, kelas khas aljabarLeibniz nilpotent diperkenalkan iaitu aljabar Leibniz �liform. Aljabar Leibniz�liform diterbitkan daripada dua sumber. Sumber pertama adalah aljabar Leib-niz bukan-Lie �liform tergred secara semula jadi dan satu lagi adalah aljabar Lie�liform tergred secara semula jadi.

Aljabar Leibniz bukan-Lie �liform tergred secara semula jadi mengandungi subkelas-subkelas FLbn+1 dan SLbn+1 . Namun hanya ada satu subkelas yang diperolehdaripada aljabar Lie �liform tergred secara semula jadi iaitu TLbn+1. Ketiga-tiga subkelas FLbn+1 , SLbn+1 dan TLbn+1 berada dalam nombor kompleks , Cdi mana n+ 1 menandakan dimensi subkelas ini bermula dengan n ≥ 4.

Khususnya, kaedah memudahkan perubahan-perubahan dasar terhadap aljabarLeibniz �liform yang diperolehi daripada aljabar Leibniz bukan-Lie �liform ter-gred secara semula jadi membolehkan masalah pengkelasan aljabar dikurangkankepada masalah penerangan mengenai struktur pemalar. Penyiasatan aljabarLeibniz �liform yang diperolehi daripada aljabar Leibniz bukan-Lie �liform ter-gred secara semula jadi adalah subjek tesis ini bagi subkelas SLbn+1 sahaja.

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Kajian ini adalah kesinambungan kerja-kerja pada SLbn+1 yang dikaji bagi kes-kes n < 9. Tujuan utama tesis ini adalah untuk mengaplikasi pendekatanRakhimov-Bekbaev untuk mengkelaskan SLb10 . Pendekatan ini akan memberikanpengkelasan lengkap SLb10 dalam sebutan aljabar tak varian. Kriteria isomor-�sma SLb10 digunakan untuk memisahkan set aljabar SLb10 kepada beberapasubset terpisah. Bagi setiap subset ini, masalah pengkelasan ini diselesaikan se-cara berasingan. Hasilnya, sebahagian daripada subset diwakili sebagai kesatuanorbit tak terhingga banyaknya (keluarga parametrik) dan selebihnya sebagai orbittunggal (orbit diasingkan). Akhirnya, senarai kelas isomor�sma aljabar Leibniz�liform kompleks dengan jadual pendaraban diberikan.

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ACKNOWLEDGEMENTS

First and foremost, all praises are to the almighty ALLAH for His blessings andmercy that enable me to learn and complete this thesis.

I am sincerely grateful to Professor Dr. Isamiddin S. Rakhimov, my supervi-sor, for giving me the opportunity to work under his supervision. He has beenvery generous and patient to contribute his valuable time to the numerous check-ing and rechecking sessions. I highly appreciate his assistance and commitmentin preparation and completion of this thesis.

I want to thank my supervisory committee Dr Sharifah Kartini Said Husain forher helpful guidance. This thesis will not be possible without the contributionfrom her. I would also like to thank group of peoples in Algebraist Group UPMfor their collaborations.

Many thanks to Professor Dato' Dr. Kamel Ari�n Mohd Atan, the Directorof Institute for Mathematical Research (INSPEM) for his encouragements.

My master study was supported by UPM scholarship (Graduate Research Fel-lowship) and also a scholarship from Ministry of Higher Education (MyMaster).

Last but not least, I would like to express my gratitute to my beloved mother(Nekmah Ngah Derani), my late father (Mohd Kasim bin Alang Mahat), mysisters and brothers for their love, support, understanding and encouragements.

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This thesis was submitted to the Senate of Universiti Putra Malaysia and has beenaccepted as ful�lment of the requirement for the degree of Master of Science.

The members of the Supervisory Committee were as follows:

Isamiddin S. Rakhimov, PhDProfessorInstitute for Mathematical ResearchUniversiti Putra Malaysia(Chairman)

Sharifah Kartini Said Husain, PhDLecturerInstitute for Mathematical ResearchUniversiti Putra Malaysia(Member)

BUJANG KIM HUAT, PhDProfessor and DeanSchool of Graduate StudiesUniversiti Putra Malaysia

Date:

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DECLARATION

Declaration by graduate student

I hereby con�rm that:

• this thesis is my original work;

• quotations, illustrations and citations have been duly referenced;

• this thesis has not been submitted previously or concurrently for any otherdegree at any other institutions;

• intellectual property from the thesis and copyright of thesis are fully-ownedby Universiti Putra Malaysia, as according to the Universiti Putra Malaysia(Research) Rules 2012;

• written permission must be obtained from supervisor and the o�ce ofDeputy Vice-Chancellor (Research and Innovation) before thesis is pub-lished (in the form of written, printed or in electronic form) includingbooks, journals, modules, proceedings, popular writings, seminar papers,manuscripts, posters, reports, lecture notes, learning modules or any othermaterials as stated in the Universiti Putra Malaysia (Research) Rules 2012;

• there is no plagiarism or data falsi�cation/fabrication in the thesis, andscholarly integrity is upheld as according to the Universiti Putra Malaysia(Graduate Studies) Rules 2003 (Revision 2012-2013) and the Universiti Pu-tra Malaysia (Research) Rules 2012. The thesis has undergone plagiarismdetection software.

Signature: Date:

Name and Matric No.: Suzila binti Mohd Kasim (GS31371)

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Declaration by Members of Supervisory Committee

This is to con�rm that:

• the research conducted and the writing of this thesis was under our super-vision;

• supervision responsibilities as stated in the Universiti Putra Malaysia (Grad-uate Studies) Rules 2003 (Revision 2012-2013) are adhered to.

Signature: Date:Name ofChairman ofSupervisoryCommittee: Isamiddin S. Rakhimov, PhD

Signature: Date:Name ofMemmber ofSupervisoryCommittee: Sharifah Kartini Said Husain, PhD

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TABLE OF CONTENTS

Page

ABSTRACT i

ABSTRAK iii

ACKNOWLEDGEMENTS v

APPROVAL vi

DECLARATION viii

LIST OF FIGURES xii

LIST OF ABBREVIATIONS xiii

CHAPTER

1 INTRODUCTION 11.0 Basic Concepts 11.1 Literature Review 2

1.1.1 Relations on Leibniz Algebras 31.2 Research Objectives 41.3 Scope of Research 51.4 Research Methodology 51.5 Outline of Contents 6

2 CONCEPT OF FILIFORM LEIBNIZ ALGEBRAS 72.0 Introduction 72.1 Naturally Graded Filiform Leibniz Algebras 72.2 Adapted Base Change and Isomorphism Criteria for SLbn+1 92.3 Conclusion 15

3 CLASSIFICATION METHOD OF SLb10 163.0 Introduction 163.1 Isomorphism Criterion for SLb10 163.2 Speci�cation of Disjoint Subsets 18

3.2.1 Notations in SLba10

18

3.2.2 Notations in SLbb10

183.3 Description of Orbits in Parametric Families 513.4 Description of Isolated Orbits 813.5 Conclusion 86

4 RESULTS ON CLASSIFICATION OF SLb10 874.0 Introduction 874.1 The Representatives of SLb10 874.2 Conclusion 934.3 94

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5 CONCLUSION 955.1 Summary 955.2 Recommendations For Future Works 95

REFERENCES/BIBLIOGRAPHY 96

APPENDICES 98A.1 The Program 99

A.1.1 Isomorphism Criteria 99B.1 Classi�cation of SLba10 101

B.2 Classi�cation of SLbb10 113

BIODATA OF STUDENT 133

LIST OF PUBLICATIONS 134

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LIST OF FIGURES

Figure Page

B.1 Graph tree of SLba10 112

B.2 Graph tree of SLbb10 129

B.3 Graph tree of SLbb10 (continued from Figure B.2) 130

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LIST OF ABBREVIATIONS

C The �eld of complex numbersC∗ The �eld of nonzero complex numbersLbn+1 The class of (n+ 1)-dimensional �liform

Leibniz algebrasLBn The class of n-dimensional Leibniz algebrasSLbn+1 Second class of (n+ 1)-dimensional �liform

Leibniz algebrasSLba10 Second class of 10-dimensional �liform Leibniz

algebras (β3 6= 0)

SLbb10 Second class of 10-dimensional �liform Leibnizalgebras (β3 = 0)

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CHAPTER 1

INTRODUCTION

The aim of this thesis is to classify a subclass of complex �liform Leibniz alge-bras. The complex �liform Leibniz algebras are arising from two sources, natu-rally graded non-Lie �liform Leibniz algebras and naturally graded �liform Liealgebras. The class of �liform Leibniz algebras derived from naturally gradednon-Lie �liform Leibniz algebras is separated into two subclasses. There are �rstclass �liform Leibniz algebras, FLbn+1 and second class �liform Leibniz algebras,SLbn+1 in dimension (n+ 1), respectively.

An algebraic approach is focused to solve isomorphism problem in terms of al-gebraic invariants (invariant functions) in the classi�cation of second class 10-dimensional complex �liform Leibniz algebras, SLb10 . By using this approach,a complete classi�cation of SLb10 is obtained and the isomorphism criterion aregiven.

This chapter is organized by reviewing basic concepts of algebras and Leibnizalgebras brie�y and next some literature reviews are described and followed by afew research objectives, scope of research and research methodology. Outline ofcontents of the thesis is also provided.

1.0 Basic Concepts

Let V be a vector space of dimension n over an algebraically closed �eld K(charK = 0). The set of bilinear maps V × V → V form a vector spaceHom (V ⊗ V, V ) of dimension n3, which can be considered together with its nat-ural structure of an a�ne algebraic variety over K and be denoted by Algn (K) ∼=Kn3 . An n-dimensional algebra L over K can be considered as an element L ofAlgn (K) via bilinear mapping L⊗ L→ L de�ning a binary algebraic operationon L: let {e0 , e1 , e2 , ..., en} be a basis of the algebra L. The table of multiplicationof L is represented by point

(γkij

)as follows:

[ei, ej

]=

n∑k=1

γkijek , i, j = 1, 2, . . . , n.

Here γkijare called structural constants of L. The linear reductive group GLn (K)

acts on Algn (K) is said to be "transport of structure" with

(g ∗ λ) (x, y) = g(λ(g−1 (x) , g−1 (y)

)).

Two algebras L1 and L2 on V are isomorphic if and only if they belong to thesame orbit under action of transport of structure.

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Recall that an algebra L over a �eld K is called a Leibniz algebra if it satis�esthe following Leibniz identity:

[x, [y, z]] = [[x, y] , z]− [[x, z] , y] , (1.1)

where [·, ·] denotes the multiplication in L.

If a Leibniz algebra has the property of antisymmetricity

[x, y] = − [y, x] ,

then the Leibniz identity is simpli�ed into the Jacobi identity:

[[x, y] , z] + [[y, z] , x] + [[z, x] , y] = 0.

Therefore Leibniz algebras are generalization of Lie algebras.

Let LBn (K) be a subvariety of Algn (K) consisting of all n-dimensional Leibnizalgebras over K. It is stable under the above mentioned action of GLn (K). Asa subset of Algn (K), the set LBn (K) is speci�ed by the following polynomial

equalities with respect to the structural constants γkij:

γljkγmil− γl

ijγmlk

+ γlikγmlj

= 0, i, j, k, l,m = 1, 2, . . . , n. (1.2)

The solution by this system of equations in the set LBn (K) with respect to

γkijhas been done for low-dimensional cases of some classes of algebra (n ≤ 3).

The complexity of the computations increases much with increasing of dimensionto classify LBn (K) for any �xed n. Hence, one considers some subclasses ofLBn (K) to be classi�ed.

1.1 Literature Review

Many papers concern with the study of Lie algebras case. According to thestructural theory of Lie algebras a �nite-dimensional Lie algebra is written asa semidirect sum of its semisimple subalgebra and the solvable radical (Levi'stheorem). The semisimple part is a direct sum of simple Lie algebras whichare completely classi�ed in �fties of the last century. At the same period, theessential progress has been made in the solvable part by Malcev [20]. Malcev [20]reduced the problem of classi�cation of solvable Lie algebras to the classi�cationof nilpotent Lie algebras. Later all the classi�cation results have been related tothe nilpotent part.

The theory of Lie algebras is one of the most developed branches of modernalgebra. It has been deeply investigated for many years. The investigation of theproperties of Lie algebras leads to the generalization of Leibniz algebras.

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Leibniz algebra was introduced by Loday [17] as a non-commutative generaliza-tion of Lie algebra. It is well-known that the Leibniz identity and the classicalJacobi identity are equivalent, when the multiplication is skew-symmetric. Leib-niz algebras inherit on important property of Lie algebras which is the rightmultiplication operator of a Leibniz algebra is a derivation. Many results on Liealgebras have been extended to Leibniz algebras.

1.1.1 Relations on Leibniz Algebras

1. (Co)homological problems

The (co)homology theory, representations and related problems of Leibnizalgebras were studied by Loday and Pirashvili [19] and Frabetti [13]. Prob-lems which related to the group theoretical realizations of Leibniz algebraswere studied by Kinyon et al. [16]. A good survey about classi�cation isgiven by Loday et al. [18]. In 1998, Mikhalev et al. [21] discussed thesolution of the noncommutative analogue of the Jacobian conjecture in thea�rmative for free Leibniz algebras, in the spirit of the corresponding re-sults of Reutenauer [28], Shpilrain [29] and Umirbaev [31].

2. Deformations and contractions

Deformation theory of Leibniz algebras and its related physical applica-tions, was initiated by Fialowski et al. [12]. The algebraic variety of four-dimensional complex nilpotent Leibniz algebras was suggested by Albeverioet al. [2].

3. Structural problems

In Leibniz algebras, the analogue of Levi's theorem was proved by Barnes[5]. He showed that any �nite-dimensional complex Leibniz algebra is de-composed into a semidirect sum of the solvable radical and a semisimpleLie algebra. The semisimple part is composed by simple Lie algebras andthe main issue in the classi�cation problem of �nite-dimensional complexLeibniz algebras is to study the solvable part. Thus solvable Leibniz alge-bras still play a central role and have been recently studied by Canete et al.[6] and Casas et al. [7] which shows the importance of the consideration oftheir nilradicals in Leibniz algebras in dimensions less than 5. Casas et al.[8, 9] also devoted to the study of solvable Leibniz algebras by consideringtheir nilradicals. The notion of simple Leibniz algebra was suggested byDzhumadil'daev et al. [11], which obtained some results concerning specialcases of simple Leibniz algebras.The problems investigating Cartan subal-gebras were studied by Omirov [22]. Albeverio et al. [1] studied on Cartansubalgebras and solvability of Leibniz algebras.

The classi�cation, up to isomorphism, of any class of algebras is a funda-mental and very di�cult problem. It is one of the �rst problem that oneencounters when trying to understand the structure of a member of thisclass of algebras. Due to result by Ayupov et al. [4], there are some struc-tural results concerning nilpotency of Leibniz algebras and its multiplication

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in the year 2001. Ayupov et al. [4] described the so called complex nonLie �liform Leibniz algebras. In particular case, some equivalent conditionsfor Leibniz algebra to be �liform and classi�cation of the set of nilpotentLeibniz algebras containing an algebra of maximal nilindex are given aswell. Naturally graded complex Leibniz algebras also is characterized. Inthe year 2006, the classi�cation of 4-dimensional nilpotent complex Leibnizalgebras was studied by Albeverio et al. [3].

Gomez et al. [15] devised a method of simplication of the basis transfor-mation of the arbitrary �liform Leibniz algebras obtained from naturallygraded non Lie �liform Leibniz algebras. It was considered those trans-formations that change the structural constants in the description of such�liform Leibniz algebras. In 2011, an approach classifying a class of �liformLeibniz algebras in terms of algebraic invariants was developed by Rakhi-mov et al. [24]. The method is applicable to any �xed dimensional cases of�liform Leibniz algebras.

The results of algebraic invariants method can be performed in low dimen-sional cases of complex �liform Leibniz algebras were devoted by Rakhimovet al. [26, 27]. Subclasses of �liform Leibniz algebras appearing from natu-rally graded non Lie �liform Leibniz algebras in dimensions �ve until eightare classi�ed named as �rst class and second class. Classi�cation problemfor dimension nine is presented by Sozan et al. [30] for �rst class case while,Deraman et al. [10] treated second class case.

Especially useful for this thesis is the work by Rakhimov et al. [24] whichclassi�ed �liform Leibniz algebras in more invariant method. This approachis extended for the classi�cation of second class 10-dimensional �liform Leib-niz algebra over �eld K = C.

1.2 Research Objectives

In this thesis, the classi�cation problem of a class of complex �liform Leibniz alge-bras arising from naturally graded non-Lie �liform Leibniz algebras is considered.It is known that this class of �liform Leibniz algebras are split into two subclasses.Each of them are denoted as FLbn+1 and SLbn+1, in dimension (n+ 1), respec-tively. The investigation of classi�cation problem of SLb10 for dimension 10 isfollowed by some research objectives:

(1) To study the algebraic and geometric properties of second class complex�liform Leibniz algebras.

(2) To create isomorphism criterion for second class 10-dimensional complex�liform Leibniz algebras in terms of invariant functions.

(3) To �nd the list of isomorphism classes of second class 10-dimensional com-plex �liform Leibniz algebras.

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1.3 Scope of Research

The classi�cation of a class of algebras corresponds to a �ber of this class, thatbeing the isomorphism classes. Classi�cation or algebraic classi�cation means thedetermination of the types of isomorphic algebras.

This research can be bene�cially applied in quantum mechanics and geomet-rical constructions as well as may triggers in fundamental physic and appliedengineering.

1.4 Research Methodology

There are two sources to get classi�cation of complex �liform Leibniz algebras.The �rst of them is naturally graded non-Lie complex �liform Leibniz algebrasand the second one is naturally graded �liform Lie algebras. (n+ 1)-dimensionalcomplex �liform Leibniz algebras appearing from naturally graded non Lie �liformLeibniz algebras is split into two subclasses denoted by FLbn+1 and SLbn+1

respectively. Classi�cation of SLbn+1 is only considered in this thesis particularlyfor n = 9. Here some methods on classifying SLb10 are presented.

(1) Isomorphism criteria for SLb10

An isomorphism criterion for subclass SLb10 is provided where it is ex-pressed by system of equalities. The polynomial equalities are simpli�edinto the simplest form.

(2) Speci�cation of a union of disjoint subsets

The isomorphism criterion of SLb10 is split into several disjoint subsets.For each subset, the classi�cation problem is considered separately. Someof these subsets will represented as union parametric families of orbits andothers are stated as single orbits.

(3) Algebraic invariant functions and basis change

System of algebraic invariant functions are presented in parametric fam-ilies of orbits. While in single orbits case, the basis change are leading tothe representatives of these orbits.

(4) Adapted linear transformations and elementary basis changing

According to the table of multiplication of SLb10 , the action of the reductivelinear group, GL10 on SLb10 is reduced to the action of a subgroup, consistof only transformations sending adapted basis to adapted basis. Then, onede�nes the so called elementary basis changing and it is proved that anadapted transformation can be represented as a product of the elementarybasis changing.

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(5) List of non-isomorphic classes

The list of non-isomorphic classes of SLb10 are given according to the tableof multiplications.

1.5 Outline of Contents

The thesis consists of �ve chapters. Now, we brie�y mention the layout of thethesis.

Chapter 1 introduces the basic concepts of Leibniz algebras.

Chapter 2 describes a method of classi�cation of �liform Leibniz algebras. Herewe introduce de�nition of nilpotent and �liform Leibniz algebras and followed bynaturally graded non-Lie �liform Leibniz algebras. Basic concepts about adaptedbase change, adapted transformations and isomorphism criteria for �liform Leib-niz algebras arising from naturally graded non-Lie �liform Leibniz algebras arediscussed.

Chapter 3 gives a complete classi�cation of SLb10 which is including all pos-sible list of isomorphism classes and invariants of SLb10 containing parametricfamilies of orbits and single orbits.

Chapter 4 presents the main result of the classi�cation. All of the represen-tatives in SLb10 joined with the table of multiplication of each algebra are givenin this chapter.

Chapter 5 contains some summaries on results of the thesis and suggests a fewproblem for future work in these areas.

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BIBLIOGRAPHY

[1] Albeverio, S., Ayupov, S. A. and Omirov, B. A. (2006). On Cartan subalge-bras, weight spaces and criterion of solvability of �nite dimensional Leibnizalgebras. Revista Mathematica Complutence, 19 (1): 183�195.

[2] Albeverio, S., Omirov, B. A. and Rakhimov, I. S. (2005). Varieties of nilpo-tent complex Leibniz algebras of dimension less than �ve. Communicationin Algebra, 33: 1575�1585.

[3] Albeverio, S., Omirov, B. A. and Rakhimov, I. S. (2006). Classi�cation of 4-dimensional Complex Leibniz algebras. Extracta Mathematicae, 21 (3): 197�210.

[4] Ayupov, S. A. and Omirov, B. A. (2001). On some classes of nilpotent Leibnizalgebras. Siberian Math. J., 42 (1): 18�29.

[5] Barnes, D. W. (2012). On Levi's theorem for Leibniz algebras. Bull. Aust.Math. Soc., 86 (2): 184�185.

[6] Canete, E. M. and Khudoyberdiyev, A. K. (2013). The classi�cation of 4-dimensional Leibniz algebras. Linear Algebra and its Applications, 439 (1):273�288.

[7] Casas, J. M., Insua, M. A., Ladra, M. and Ladra, S. (2012). An algorithm forthe classi�cation of 3-dimensional complex Leibniz algebras. Linear Algebraand its Application, 436: 3747�3756.

[8] Casas, J. M., Ladra, M., Omirov, B. A. and Karimjanov, I. A. (2013). Classi-�cation of solvable Leibniz algebras with naturally graded �liform nilradical.Linear Algebra and its Applications, 438 (7): 2973�3000.

[9] Casas, J. M., Ladra, M., Omirov, B. A. and Karimjanov, I. A. (2013). Clas-si�cation of solvable Leibniz algebras with null-�liform nilradical. Linear andMultilinear Algebra, 61 (6): 758�774.

[10] Deraman, F., Rakhimov, I. S. and Said Husain, S. K. (2012). Isomorphismclasses and invariants for a subclass of nine-dimensional �liform Leibniz al-gebras. AIP Conference Proceedings, 1450: 326�331.

[11] Dzhumadil'daev, A. and Abdykassymova, S. (2001). Leibniz algebras in char-acteristic p. C.R. Acad. Sci., Paris, 332: 1047�1052. Serie I.

[12] Fialowski, A., Mandal, A. and Mukherjee, G. (2009). Versal deformationsof Leibniz algebras. Journal of K-theory: K-theory and its Applications toAlgebra, Geometry, and Topology, 3: 327�358.

[13] Frabetti, A. (1998). Leibniz homology of dialgebras of matrices. Journal Pureand Applied Algebra, 129: 123�141.

[14] Gomez, J. R. and Khakimdjanov, Y. (1998). Low dimensional �liform Liealgebras. Journal of Pure and Applied Algebra, 130: 133�158.

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[15] Gomez, J. R. and Omirov, B. A. (2012). On classi�cation of complex �liformLeibniz algebras. Available at arXiv:math/0612735v2 [math.RA] .

[16] Kinyon, M. K. and Weinstein, A. (2001). Leibniz algebras, Courant al-gebroids, and multiplications on reductive homogeneous spaces. AmericanJournal of Mathematics, 123 (3): 525�550.

[17] Loday, J. L. (1993). Une version non commutative des algebres de Lie: lesalgebres de Leibniz. Ens. Math., 39: 269�293.

[18] Loday, J. L., Frabetti, A., Chapoton, F. and Goichot, F. (2001). Dialgebrasand related operads. Springer-Verlag, IV.

[19] Loday, J. L. and Pirashvili, T. (1993). Universal enveloping algebras of Leib-niz algebras and (co)homology. Math. Ann., 296: 139�158.

[20] Malcev, A. (1945). On solvable Lie algebras. Amer. Math. Soc. Transl., 9:228�262.

[21] Mikhalev, A. A. and Umirbaev, U. U. (1998). Subalgebras of free Leibnizalgebras. Communication Algebra, 26: 435�446.

[22] Omirov, B. A. (2006). Conjugacy of Cartan subalgebras of complex �nite-dimensional Leibniz algebras. Journal of Algebra, 302: 887�896.

[23] Omirov, B. A. and Rakhimov, I. S. (2009). On Lie-like complex �liformLeibniz algebras. Bull. Aust. Math. Soc., 79: 391�404.

[24] Rakhimov, I. S. and Bekbaev, U. D. (2010). On isomorphisms and invariantsof �nite dimensional complex �liform Leibniz algebras. Communications inAlgebra, 38 (12): 4705�4738.

[25] Rakhimov, I. S. and Hassan, M. A. (2011). On low-dimensional �liform Leib-niz algebras and their invariants. Bull. Malaysian Mathematical Science So-ciety (2), 34 (3): 475�485.

[26] Rakhimov, I. S. and Said Husain, S. K. (2011). Classi�cation of a subclassof low-dimensional complex �liform Leibniz algebras. Linear and MultilinearAlgebra, 59 (3): 339�354.

[27] Rakhimov, I. S. and Said Husain, S. K. (2011). On isomorphism classes andinvariants of a subclass of low-dimensional complex �liform Leibniz algebras.Linear and Multilinear Algebra, 59 (2): 205�220.

[28] Reutenauer, C. (1992). Applications of a noncommutative Jacobian matrix.Journal of Pure and Applied Algebra , 77: 169�181.

[29] Shpilran, V. (1993). On generators of L/R2 Lie algebras. Proceedings of theAmerican Mathematical Society, 119: 1039�1043.

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[30] Sozan, J., Rakhimov, I. S. and Mohd Atan, K. A. (2010). Classi�cation of�rst class of complex �liform Leibniz algebras. Lambert Academic Publish-ing.

[31] Umirbaev, U. U. (1993). Partial derivatives and endomorphisms of somerelatively free Lie algebras. Sibirskii Matematicheskii Zhurnal, 34 (6): 179�188.

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Procedure

F := proc (n) local β, t, s,Γ;

ifn ≤ 3 then return 0; end if;

for t from 3 tondo

if t = 3 then q:=0; else q := 1; end if;

βt :=BDγ

At+

1

At−1

(Dbt −

t−1∑k=3

((1∏i=1

(k − i)Ak−2Bbt+2−k

)+∏2

i=1 (k − i)2

Ak−3B2t∑

i[1]=k+2

bt+3−i1bi1+1−k

+

(∏3i=1 (k − i)

6Ak−4B3

t∑i2=k+3

i2∑i1=k+3

bt+3−i2bi2+3−i1bi1−k

+

∏4i=1 (k − i)

24Ak−5B4

t∑i3=k+4 i3∑

i2=k+4

i2∑i1=k+4

bt+3−i3bi3+3−i2bi2+3−i1bi1−1−k

βk

;

end do;

for s from 3 tondo

print (simplify (βs)) ;

end do;

Γ :=1

An−2

(D

A

)2

γ,

print (simplify (Γ)) ;

end proc;

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APPENDIX B

Classi�cation Procedures

This part contains some procedures of classi�cation of subclass SLb10 . It is

known that SLb10 is divided into two cases: (i) SLba10and (ii) SLbb

10. Section B.1

presents classi�cation procedures in SLba10

while Section B.2 gives a few methods

of classi�cation of subclass SLbb10.

B.1 Classi�cation of SLba10

The isomorphism criteria for SLba10 can be expressed as follows:

Λ′3 =

1

A

(D

A

)Λ3, f7

(Λ′)

=1

A8

(D

A

)4

f7 (Λ) , (B.1)

Λ′4 = 0, f8

(Λ′)

=1

A10

(D

A

)5

f8 (Λ) ,

Λ′5 =

1

A4

(D

A

)2

Λ5, f9

(Λ′)

=1

A12

(D

A

)6

f9 (Λ) ,

f6

(Λ′)

=1

A6

(D

A

)3

f6 (Λ) , Γ′

=1

A7

(D

A

)2

Γ.

Under condition Λ3 6= 0, the system (B.1) is isomorphic to the following system:

Λ′3 = 1,

f7(Λ′)

Λ′43

=1

A4

f7 (Λ)

Λ43

, (B.2)

Λ′4 = 0,

f8(Λ′)

Λ′53

=1

A5

f8 (Λ)

Λ53

,

Λ′5

Λ′23

=1

A2

Λ5

Λ23

,f9(Λ′)

Λ′63

=1

A6

f9 (Λ)

Λ63

,

f6(Λ′)

Λ′33

=1

A3

f6 (Λ)

Λ33

,Γ′

Λ′23

=1

A5

Γ

Λ23

.

It shows that SLba10 = F1⋃F2, where F1 =

{L (Λ) ∈ SLba10 : Λ3 6= 0,Λ5 6= 0

}and F2 =

{L (Λ) ∈ SLba10 : Λ3 6= 0,Λ5 = 0

}.

(i) F1 ={L (Λ) ∈ SLba10 : Λ3 6= 0,Λ5 6= 0

}.

(a) F1 = U1⋃F3 :

• U1 ={L (Λ) ∈ SLba10 : Λ3 6= 0,Λ5 6= 0, f6 (Λ) 6= 0

}.

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By reducing Λ′3 = 1 and Λ

′4 = 0, then we obtain A =

f6(Λ)Λ3Λ5

through

conditions Λ5 6= 0 and f6 (Λ) 6= 0 in (B.2). Automatically, the values of

B = Λ4A2Λ3

3Λ5and D = A2

Λ3are obtained. Subset U1 brings the following

system of equations:

Λ′3 = 1, (B.3)

Λ′4 = 0,

Λ′5

Λ′23

=

(Λ3Λ5

f6 (Λ)

)2 Λ5

Λ23

,

f6(Λ′)

Λ′33

=

(Λ3Λ5

f6 (Λ)

)3 f6 (Λ)

Λ33

,

f7(Λ′)

Λ′43

=

(Λ3Λ5

f6 (Λ)

)4 f7 (Λ)

Λ43

,

f8(Λ′)

Λ′53

=

(Λ3Λ5

f6 (Λ)

)5 f8 (Λ)

Λ53

,

f9(Λ′)

Λ′63

=

(Λ3Λ5

f6 (Λ)

)6 f9 (Λ)

Λ63

,

Γ′

Λ′23

=

(Λ3Λ5

f6 (Λ)

)5 Γ

Λ23

.

New parameters are represented as λ1, λ2, λ3, λ4 and λ5 to imply that

Λ′5 =

Λ35

f26 (Λ)= λ1, f6

(Λ′)

=Λ35

f26 (Λ)= λ1, f7

(Λ′)

=f7(Λ)Λ4

5

f46 (Λ)= λ2,

f8(Λ′)

=f8(Λ)Λ5

5

f56 (Λ)= λ3, f9

(Λ′)

=f9(Λ)Λ6

5

f66 (Λ)= λ4 and Γ′ =

Λ33Λ5

5Γ

f56 (Λ)= λ5.

Thus all algebras from U1 are isomorphic to L (1, 0, λ1, λ1, λ2, λ3, λ4, λ5) forλ1 ∈ C∗ and λ2, λ3, λ4, λ5 ∈ C.

• F3 ={L (Λ) ∈ SLba10 : Λ3 6= 0,Λ5 6= 0, f6 (Λ) = 0

}.

Through the conditions in F3, the system of equations in (B.2) becomes asfollows:

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Λ′3 = 1,

f7(Λ′)

Λ′43

=1

A4

f7 (Λ)

Λ43

, (B.4)

Λ′4 = 0,

f8(Λ′)

Λ′53

=1

A5

f8 (Λ)

Λ53

,

Λ′5

Λ′23

=1

A2

Λ5

Λ23

,f9(Λ′)

Λ′63

=1

A6

f9 (Λ)

Λ63

,

f6(Λ′)

Λ′33

= 0,Γ′

Λ′23

=1

A5

Γ

Λ23

.

We can rewrite F3 as F3 = U2⋃F4, where

U2 ={L (Λ) ∈ SLba10 : Λ3 6= 0,Λ5 6= 0, f6 (Λ) = 0, f8 (Λ) 6= 0

}and

F4 ={L (Λ) ∈ SLba10 : Λ3 6= 0,Λ5 6= 0, f6 (Λ) = 0, f8 (Λ) = 0

}.

(b) F3 = U2⋃F4 :

• U2 ={L (Λ) ∈ SLba10 : Λ3 6= 0,Λ5 6= 0, f6 (Λ) = 0, f8 (Λ) 6= 0

}.

By reducingΛ′5

Λ′23

= 1A2

Λ5

Λ23and

f8(Λ′)Λ′53

= 1A5

f8(Λ)

Λ53

in (B.4), then we obtain

A =f8(Λ)

Λ3Λ25. The values of B and D appear as B = Λ4A

2Λ33Λ5

and D = A2

Λ3.

Then by using value 1A =

Λ3Λ25

f8(Λ), the system of equalities for U2 is as follows:

Λ′3 = 1, (B.5)

Λ′4 = 0,

Λ′5

Λ′23

=

(Λ3Λ2

5

f8 (Λ)

)2Λ5

Λ23

,

f6(Λ′)

Λ′33

= 0,

f7(Λ′)

Λ′43

=

(Λ3Λ2

5

f8 (Λ)

)4f7 (Λ)

Λ43

,

f8(Λ′)

Λ′53

=

(Λ3Λ2

5

f8 (Λ)

)5f8 (Λ)

Λ53

,

f9(Λ′)

Λ′63

=

(Λ3Λ2

5

f8 (Λ)

)6f9 (Λ)

Λ63

,

Γ′

Λ′23

=

(Λ3Λ2

5

f8 (Λ)

)5Γ

Λ23

.

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New parameters are represented as λ1, λ2, λ3 and λ4 to imply that

Λ′5 =

Λ55

f28 (Λ)= λ1, f7

(Λ′)

=Λ85f7(Λ)

f48 (Λ)= λ2, f8

(Λ′)

=Λ105

f48 (Λ)= λ2

1,

f9(Λ′)

=f9(Λ)Λ12

5

f68 (Λ)= λ3 and Γ′ =

Λ33Λ10

5 Γ

f58 (Λ)= λ4. Thus all algebras

from U2 are isomorphic to L(1, 0, λ1, 0, λ2, λ

21, λ3, λ4

)for λ1 ∈ C∗ and

λ2, λ3, λ4 ∈ C.

• F4 ={L (Λ) ∈ SLba10 : Λ3 6= 0,Λ5 6= 0, f6 (Λ) = 0, f8 (Λ) = 0

}.

The system of equalities in F4 is as below:

Λ′3 = 1, (B.6)

Λ′4 = 0,

Λ′5

Λ′23

=1

A2

Λ5

Λ23

,

f6(Λ′)

Λ′33

= 0,

f7(Λ′)

Λ′43

=1

A4

f7 (Λ)

Λ43

,

f8(Λ′)

Λ′53

= 0,

f9(Λ′)

Λ′63

=1

A6

f9 (Λ)

Λ63

,

Γ′

Λ′23

=1

A5

Γ

Λ23

.

The subset of F4 can be written as F4 = U3⋃U4 where

U3 ={L (Λ) ∈ SLba10 : Λ3 6= 0,Λ5 6= 0, f6 (Λ) = 0, f8 (Λ) = 0,Γ 6= 0

}and

U4 ={L (Λ) ∈ SLba10 : Λ3 6= 0,Λ5 6= 0, f6 (Λ) = 0, f8 (Λ) = 0,Γ = 0

}.

(c) For F4 = U3⋃U4 :

• U3 ={L (Λ) ∈ SLba10 : Λ3 6= 0,Λ5 6= 0, f6 (Λ) = 0, f8 (Λ) = 0,Γ 6= 0

}By reducing

Λ′5

Λ′23

= 1A2

Λ5

Λ23and Γ′

Λ′23

= 1A5

ΓΛ23in (B.6), then we obtain

A =Λ23Γ

Λ25. The values of B and D appear as B = Λ4A

2Λ33Λ5

and D = A2

Λ3.

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Λ25

Λ23Γ, the system of equalities for U3 is as follows:

Λ′3 = 1, (B.7)

Λ′4 = 0,

Λ′5

Λ′23

=

(Λ2

5

Λ23Γ

)2Λ5

Λ23

,

f6(Λ′)

Λ′33

= 0,

f7(Λ′)

Λ′43

=

(Λ2

5

Λ23Γ

)4f7 (Λ)

Λ43

,

f8(Λ′)

Λ′53

= 0,

f9(Λ′)

Λ′63

=

(Λ2

5

Λ23Γ

)6f9 (Λ)

Λ63

,

Γ′

Λ′23

=

(Λ2

5

Λ23Γ

)5Γ

Λ23

.

New parameters are represented as λ1, λ2 and λ3 to imply that

Λ′5 =

Λ55

Λ63Γ2 = λ1, f7

(Λ′)

=Λ85f7(Λ)

Λ123 Γ4 = λ2, f9

(Λ′)

=f9(Λ)Λ12

5

Λ183 Γ6 = λ3

and Γ6 =Λ105

Λ123 Γ4 = λ1. Thus all algebras from U3 are isomorphic to

L(1, 0, λ1, 0, λ2, 0, λ3, λ

21

)for λ1 ∈ C∗ and λ2, λ3 ∈ C.

• U4 ={L (Λ) ∈ SLba10 :

Λ3 6= 0,Λ5 6= 0, f6 (Λ) = 0, f8 (Λ) = 0,Γ = 0} .

By reducingΛ′5

Λ′23

= 1A2

Λ5

Λ23in (B.6), then we obtain A2 = Λ5

Λ23. The values of

B and D appear as B = Λ4A2Λ3

3Λ5and D = A2

Λ3.

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Then by using value 1A2 =

Λ23

Λ5, the system of equalities for U4 is as follows:

Λ′3 = 1, (B.8)

Λ′4 = 0,

Λ′5

Λ′23

=

(Λ2

3

Λ5

)Λ5

Λ23

,

f6(Λ′)

Λ′33

= 0,

f7(Λ′)

Λ′43

=

(Λ2

3

Λ5

)2f7 (Λ)

Λ43

,

f8(Λ′)

Λ′53

= 0,

f9(Λ′)

Λ′63

=

(Λ2

3

Λ5

)3f9 (Λ)

Λ63

,

Γ′

Λ′23

= 0.

New parameters are represented as λ1 and λ2 to imply that Λ′5 = 1,

f7(Λ′)

=f7(Λ)

Λ25

= λ1, f9(Λ′)

=f9(Λ)

Λ35

= λ2. Thus all algebras from

U4 are isomorphic to L (1, 0, 1, 0, λ1, 0, λ2, 0) for λ1, λ2 ∈ C.

(ii) F2 ={L (Λ) ∈ SLba10 : Λ3 6= 0,Λ5 = 0

}The subset F2 brings the following system of equations:

Λ′3 = 1,

f7(Λ′)

Λ′43

=1

A4

f7 (Λ)

Λ43

, (B.9)

Λ′4 = 0,

f8(Λ′)

Λ′53

=1

A5

f8 (Λ)

Λ53

,

Λ′5

Λ′23

= 0,f9(Λ′)

Λ′63

=1

A6

f9 (Λ)

Λ63

,

f6(Λ′)

Λ′33

=1

A3

f6 (Λ)

Λ33

,Γ′

Λ′23

=1

A5

Γ

Λ23

.

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(a) F2 = F5⋃F6 :

• F5 ={L (Λ) ∈ SLba10 : Λ3 6= 0,Λ5 = 0, f6 (Λ) 6= 0

}The subset F5 brings the following system of equations:

Λ′3 = 1,

f7(Λ′)

Λ′43

=1

A4

f7 (Λ)

Λ43

, (B.10)

Λ′4 = 0,

f8(Λ′)

Λ′53

=1

A5

f8 (Λ)

Λ53

,

Λ′5

Λ′23

= 0,f9(Λ′)

Λ′63

=1

A6

f9 (Λ)

Λ63

,

f6(Λ′)

Λ′33

=1

A3

f6 (Λ)

Λ33

,Γ′

Λ′23

=1

A5

Γ

Λ23

.

• F6 ={L (Λ) ∈ SLba10 : Λ3 6= 0,Λ5 = 0, f6 (Λ) = 0

}The system of equations of subset F6 is as following:

Λ′3 = 1,

f7(Λ′)

Λ′43

=1

A4

f7 (Λ)

Λ43

, (B.11)

Λ′4 = 0,

f8(Λ′)

Λ′53

=1

A5

f8 (Λ)

Λ53

,

Λ′5

Λ′23

= 0,f9(Λ′)

Λ′63

=1

A6

f9 (Λ)

Λ63

,

f6(Λ′)

Λ′33

= 0,Γ′

Λ′23

=1

A5

Γ

Λ23

.

(b) F5 = U5⋃F7 :

• U5 ={L (Λ) ∈ SLba10 : Λ3 6= 0,Λ5 = 0, f6 (Λ) 6= 0, f7 (Λ) 6= 0

}By reducing

f6(Λ′)Λ′33

= 1A3

f6(Λ)

Λ33

andf7(Λ′)

Λ′43

= 1A4

f7(Λ)

Λ43

in (B.10), then we

obtain A =f7(Λ)

Λ3f6(Λ). The values of B and D appear as B = Λ4A

2Λ33Λ5

and

D = A2

Λ3. Then by using value 1

A =Λ3f6(Λ)f7(Λ)

, the system of equalities for U5

is as follows:

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Λ′3 = 1, (B.12)

Λ′4 = 0,

Λ′5

Λ′23

= 0,

f6(Λ′)

Λ′33

=

(Λ3f6 (Λ)

f7 (Λ)

)3 f6 (Λ)

Λ33

,

f7(Λ′)

Λ′43

=

(Λ3f6 (Λ)

f7 (Λ)

)4 f7 (Λ)

Λ43

,

f8(Λ′)

Λ′53

=

(Λ3f6 (Λ)

f7 (Λ)

)5 f8 (Λ)

Λ53

,

f9(Λ′)

Λ′63

=

(Λ3f6 (Λ)

f7 (Λ)

)6 f9 (Λ)

Λ63

,

Γ′

Λ′23

=

(Λ3f6 (Λ)

f7 (Λ)

)5 Γ

Λ23

.

New parameters are represented as λ1, λ2, λ3 and λ4 to imply that f6(Λ′)

=f46 (Λ)

f37 (Λ)= λ1, f7

(Λ′)

=f46 (Λ)

f37 (Λ)= λ1,f8

(Λ′)

=f56 (Λ)f8(Λ)

f57 (Λ)= λ2,

f9(Λ′)

=f66 (Λ)f9(Λ)

f67 (Λ)= λ3 and Γ′ =

Λ33f

56 (Λ)Γ

f57 (Λ)= λ4. Thus all algebras

from U5 are isomorphic to L (1, 0, 0, λ1, λ1, λ2, λ3, λ4) for λ1 ∈ C∗ andλ2, λ3, λ4 ∈ C.

• F7 ={L (Λ) ∈ SLba10 : Λ3 6= 0,Λ5 = 0, f6 (Λ) 6= 0, f7 (Λ) = 0

}Under the conditions of subset F7, the system of equations appear as below:

Λ′3 = 1,

f7(Λ′)

Λ′43

= 0, (B.13)

Λ′4 = 0,

f8(Λ′)

Λ′53

=1

A5

f8 (Λ)

Λ53

,

Λ′5

Λ′23

= 0,f9(Λ′)

Λ′63

=1

A6

f9 (Λ)

Λ63

,

f6(Λ′)

Λ′33

=1

A3

f6 (Λ)

Λ33

,Γ′

Λ′23

=1

A5

Γ

Λ23

.

(c) F7 = U6⋃F8 :

• U6 ={L (Λ) ∈ SLba10 :

Λ3 6= 0,Λ5 = 0, f6 (Λ) 6= 0, f7 (Λ) = 0, f8 (Λ) 6= 0}.107

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By reducingf6(Λ′)

Λ′33

= 1A3

f6(Λ)

Λ33

andf8(Λ′)

Λ′53

= 1A5

f8(Λ)

Λ53

in (B.13), then we

obtain A =f26 (Λ)

Λ3f8(Λ). The values of B and D appear as B = Λ4A

2Λ33Λ5

and

D = A2


A =Λ3f8(Λ)

f26 (Λ), the system of equalities for U6

is as the following page:

Λ′3 = 1 (B.14)

Λ′4 = 0

Λ′5

Λ′23

= 0

f6(Λ′)

Λ′33

=

(Λ3f8 (Λ)

f26 (Λ)

)3f6 (Λ)

Λ33

f7(Λ′)

Λ′43

= 0

f8(Λ′)

Λ′53

=

(Λ3f8 (Λ)

f26 (Λ)

)5f8 (Λ)

Λ53

f9(Λ′)

Λ′63

=

(Λ3f8 (Λ)

f26 (Λ)

)6f9 (Λ)

Λ63

Γ′

Λ′23

=

(Λ3f8 (Λ)

f26 (Λ)

)5Γ

Λ23


f6(Λ′)

=f38 (Λ)

f56 (Λ)= λ1, f8

(Λ′)

=f68 (Λ)

f106 (Λ)= λ2

1, f9(Λ′)

=f68 (Λ)f9(Λ)

f126 (Λ)= λ2

and Γ′ =Λ33f

58 (Λ)Γ

f106 (Λ)= λ3. Thus all algebras from U6 are isomorphic to

L(1, 0, 0, λ1, 0, λ

21, λ2, λ3

)for λ1 ∈ C∗ and λ2, λ3 ∈ C.

• F8 ={L (Λ) ∈ SLba10 :

Λ3 6= 0,Λ5 = 0, f6 (Λ) 6= 0, f7 (Λ) = 0, f8 (Λ) = 0}Subset F8 shows the following system of equations:

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Λ′3 = 1,

f7(Λ′)

Λ′43

= 0, (B.15)

Λ′4 = 0,

f8(Λ′)

Λ′53

= 0,

Λ′5

Λ′23

= 0,f9(Λ′)

Λ′63

=1

A6

f9 (Λ)

Λ63

,

f6(Λ′)

Λ′33

=1

A3

f6 (Λ)

Λ33

,Γ′

Λ′23

=1

A5

Γ

Λ23

.

(d) F8 = U7⋃U8 :

• U7 ={L (Λ) ∈ SLba10 :

Λ3 6= 0,Λ5 = 0, f6 (Λ) 6= 0, f7 (Λ) = 0, f8 (Λ) = 0,Γ 6= 0}By reducing

f6(Λ′)Λ′33

= 1A3

f6(Λ)

Λ33

and Γ′

Λ′23

= 1A5


A =f26 (Λ)

Λ43Γ

. The values of B and D appear as B = Λ4A2Λ3

3Λ5and D = A2

Λ3.


Λ43Γ

f26 (Λ), the system of equalities for U7 is as the

following page:

Λ′3 = 1, (B.16)

Λ′4 = 0,

Λ′5

Λ′23

= 0,

f6(Λ′)

Λ′33

=

(Λ4

3Γ

f26 (Λ)

)3f6 (Λ)

Λ33

,

f7(Λ′)

Λ′43

= 0,

f8(Λ′)

Λ′53

=0,

f9(Λ′)

Λ′63

=

(Λ4

3Γ

f26 (Λ)

)6f9 (Λ)

Λ63

,

Γ′

Λ′23

=

(Λ4

3Γ

f26 (Λ)

)5Γ

Λ23

.

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New parameters are represented as λ1 and λ2 to imply that

f6(Λ′)

=Λ93Γ3

f56 (Λ)= λ1, f9

(Λ′)

=Λ183 Γ6f9(Λ)

f126 (Λ)= λ2 and Γ′ =

Λ183 Γ6

f106 (Λ)= λ2

1.

Thus all algebras from U7 are isomorphic to L(1, 0, 0, λ1, 0, 0, λ2, λ

21

)for

λ1 ∈ C∗ and λ2 ∈ C.

• U8 ={L (Λ) ∈ SLba10 :

Λ3 6= 0,Λ5 = 0, f6 (Λ) 6= 0, f7 (Λ) = 0, f8 (Λ) = 0,Γ = 0}.By reducing

f6(Λ′)Λ′33

= 1A3

f6(Λ)

Λ33

in (B.15), then we obtain A3 =f6(Λ)

Λ33

. The

values of B and D appear as B = Λ4A2Λ3

3Λ5and D = A2

Λ3. Then by using value

1A3 =

Λ33

f6(Λ), the system of equalities for U8 is as the following page:

Λ′3 = 1, (B.17)

Λ′4 = 0,

Λ′5

Λ′23

= 0,

f6(Λ′)

Λ′33

=

(Λ3

3

f6 (Λ)

)f6 (Λ)

Λ33

,

f7(Λ′)

Λ′43

= 0,

f8(Λ′)

Λ′53

=0,

f9(Λ′)

Λ′63

=

(Λ3

3

f6 (Λ)

)2f9 (Λ)

Λ63

,

Γ′

Λ′23

=0.

New parameter is represented as λ to imply that f9(Λ′)

=f9(Λ)

f26 (Λ)= λ.

Thus all algebras from U8 are isomorphic to L (1, 0, 0, 1, 0, 0, λ, 0) for λ ∈ C.

The subsets in SLba10 are represented by the following graph tree.

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Figure

B.1:GraphtreeofSLba 10

SLba 10

Λ56=

0

U1

f 6(Λ

)=

0

U2

f 8(Λ

)=

0

U3

U4

Λ5

=0

f 6(Λ

)6=

0

U5

f 7(Λ

)=

0

U6

f 8(Λ

)=

0

U7

U8

f 6(Λ

)=

0

f 7(Λ

)6=

0

U9

f 8(Λ

)=

0

U10

Γ=

0

U11

U31

f 7(Λ

)=

0

f 8(Λ

)6=

0

U12

U13

f 8(Λ

)=

0

f 9(Λ

)6=

0

U14

U32

f 9(Λ

)=

0

U33

U34

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B.2 Classi�cation of SLbb10

The isomorphism criteria for SLbb10 can be expressed as follows:

Λ′3 = 0, t7

(Λ′)

=1

A11

(D

A

)4

t7 (Λ) , (B.18)

Λ′4 =

1

A2

(D

A

)Λ4, t8

(Λ′)

=1

A14

(D

A

)5

t8 (Λ) ,

Λ′5 =

1

A3

(D

A

)Λ5, t9

(Λ′)

=1

A17

(D

A

)6

t9 (Λ) ,

Λ′6 = 0, Γ

′=

1

A7

(D

A

)2

Γ.

Under condition Λ3 = 0 and Λ4 6= 0, the system (B.18) is isomorphic to thefollowing system of equations:

Λ′3 = 0,

t7(Λ′)

Λ′44

=1

A3

t7 (Λ)

Λ44

, (B.19)

Λ′4 = 1,

t8(Λ′)

Λ′54

=1

A4

t8 (Λ)

Λ54

,

Λ′5

Λ′4

=1

A

Λ5

Λ4,

t9(Λ′)

Λ′64

=1

A5

t9 (Λ)

Λ64

,

Λ′6 = 0,

Γ′

Λ′24

=1

A3

Γ

Λ24

.

It shows that SLbb10 = T1⋃T2, where T1 =

{L (Λ) ∈ SLbb10 : Λ3 = 0,Λ4 6= 0

}and T2 =

{L (Λ) ∈ SLbb10 : β3 = 0, β4 = 0

}.

(i) T1 ={L (Λ) ∈ SLbb10 : Λ3 = 0,Λ4 6= 0

}(a) T1 = U15

⋃T3 :

• U15 ={L (Λ) ∈ SLbb10 : Λ3 = 0,Λ4 6= 0,Λ5 6= 0

}By reducing

Λ′5

Λ′4

= 1A

Λ5Λ3

in (B.19), then we obtain A = Λ5Λ4

. The values of


4and D = A3


A = Λ4Λ5

,

the system of equalities for U15 is as follows:

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Λ′3 = 0,

t7(Λ′)

Λ′44

=

(Λ4

Λ5

)3 t7 (Λ)

Λ44

, (B.20)

Λ′4 = 1,

t8(Λ′)

Λ′54

=

(Λ4

Λ5

)4 t8 (Λ)

Λ54

,

Λ′5

Λ′4

=

(Λ4

Λ5

)Λ5

Λ4,

t9(Λ′)

Λ′64

=

(Λ4

Λ5

)5 t9 (Λ)

Λ64

,

Λ′6 = 0,

Γ′

Λ′24

=

(Λ4

Λ5

)3 Γ

Λ24

.

New parameters are represented as λ1, λ2, λ3 and λ4 to imply that

t7(Λ′)

=t7(Λ)

Λ4Λ35

= λ1,t8(Λ′)

=t8(Λ)

Λ4Λ45

= λ2, t9(Λ′)

=t9(Λ)

Λ4Λ55

= λ3 and

Γ′ = Λ4ΓΛ35

= λ4. Thus all algebras from U15 are isomorphic to

L (0, 1, 1, 0, λ1, λ2, λ3, λ4) for λ1, λ2, λ3, λ4 ∈ C.

• T3 ={L (Λ) ∈ SLbb10 : Λ3 = 0,Λ4 6= 0,Λ5 = 0

}The system of equations in subset T3 becomes:

Λ′3 = 0,

t7(Λ′)

Λ′44

=1

A3

t7 (Λ)

Λ44

, (B.21)

Λ′4 = 1,

t8(Λ′)

Λ′54

=1

A4

t8 (Λ)

Λ54

,

Λ′5

Λ′4

= 0,t9(Λ′)

Λ′64

=1

A5

t9 (Λ)

Λ64

,

Λ′6 = 0,

Γ′

Λ′24

=1

A3

Γ

Λ24

.

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(b) T3 = T4⋃T5 :

• T4 ={L (Λ) ∈ SLbb10 : Λ3 = 0,Λ4 6= 0,Λ5 = 0, t7 (Λ) 6= 0


Λ′3 = 0,

t7(Λ′)

Λ′44

=1

A3

t7 (Λ)

Λ44

, (B.22)

Λ′4 = 1,

t8(Λ′)

Λ′54

=1

A4

t8 (Λ)

Λ54

,

Λ′5

Λ′4

= 0,t9(Λ′)

Λ′64

=1

A5

t9 (Λ)

Λ64

,

Λ′6 = 0,

Γ′

Λ′24

=1

A3

Γ

Λ24

.

• T5 ={L (Λ) ∈ SLbb10 : Λ3 = 0,Λ4 6= 0,Λ5 = 0, t7 (Λ) = 0


Λ′3 = 0,

t7(Λ′)

Λ′44

= 0, (B.23)

Λ′4 = 1,

t8(Λ′)

Λ′54

=1

A4

t8 (Λ)

Λ54

,

Λ′5

Λ′4

= 0,t9(Λ′)

Λ′64

=1

A5

t9 (Λ)

Λ64

,

Λ′6 = 0,

Γ′

Λ′24

=1

A3

Γ

Λ24

.

(c) T4 = U16⋃T6 :

• U16 ={L (Λ) ∈ SLbb10 : Λ3 = 0,Λ4 6= 0,Λ5 = 0, t7 (Λ) 6= 0, t8 (Λ) 6= 0

}By reducing

t7(Λ′)Λ′44

= 1A3

t7(Λ)

Λ44

andt8(Λ′)

Λ′54

= 1A4

t8(Λ)

Λ54

in (B.22), then we

obtain A =t8(Λ)

Λ4t7(Λ). The values of B and D appear as B = Λ6A

3Λ24and

D = A3


A =Λ4t7(Λ)t8(Λ)

, the system of equalities for

U16 is as follows:

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Λ′3 = 0,

t7(Λ′)

Λ′44

=

(Λ4t7 (Λ)

t8 (Λ)

)3 t7 (Λ)

Λ44

, (B.24)

Λ′4 = 1,

t8(Λ′)

Λ′54

=

(Λ4t7 (Λ)

t8 (Λ)

)4 t8 (Λ)

Λ54

,

Λ′5

Λ′4

= 0,t9(Λ′)

Λ′64

=

(Λ4t7 (Λ)

t8 (Λ)

)5 t9 (Λ)

Λ64

,

Λ′6 = 0,

Γ′

Λ′24

=

(Λ4t7 (Λ)

t8 (Λ)

)3 Γ

Λ24

.


t7(Λ′)

=t47(Λ)

Λ4t38(Λ)

= λ1,t8(Λ′)

=t47(Λ)

Λ4t38(Λ)

= λ1, t9(Λ′)

=t57(Λ)t9(Λ)

Λ4t58(Λ)

= λ2

and Γ′ =Λ4t

37(Λ)Γ

t38(Λ)= λ3. Thus all algebras from U16 are isomorphic to

L (0, 1, 0, 0, λ1, λ1, λ2, λ3) for λ1 ∈ C∗ and λ2, λ3 ∈ C.

• T6 ={L (Λ) ∈ SLbb10 : Λ3 = 0,Λ4 6= 0,Λ5 = 0, t7 (Λ) 6= 0, t8 (Λ) = 0


Λ′3 = 0,

t7(Λ′)

Λ′44

=1

A3

t7 (Λ)

Λ44

, (B.25)

Λ′4 = 1,

t8(Λ′)

Λ′54

= 0,

Λ′5

Λ′4

= 0,t9(Λ′)

Λ′64

=1

A5

t9 (Λ)

Λ64

,

Λ′6 = 0,

Γ′

Λ′24

=1

A3

Γ

Λ24

.

(d) T6 = U17⋃U18 :

• U17 ={L (Λ) ∈ SLbb10 :

Λ3 = 0,Λ4 6= 0,Λ5 = 0, t7 (Λ) 6= 0, t8 (Λ) = 0, t9 (Λ) 6= 0}By reducing

t7(Λ′)Λ′44

= 1A3

t7(Λ)

Λ44

andt9(Λ′)

Λ′64

= 1A5

t9(Λ)

Λ64

in (B.25), then we

obtain A =t27(Λ)

Λ24t9(Λ)


4and

D = A3


A =Λ24t9(Λ)

t27(Λ), the system of equalities for

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U17 is as follows:

Λ′3 = 0,

t7(Λ′)

Λ′44

=

(Λ2

4t9 (Λ)

t27 (Λ)

)3t7 (Λ)

Λ44

, (B.26)

Λ′4 = 1,

t8(Λ′)

Λ′54

= 0,

Λ′5

Λ′4

= 0,t9(Λ′)

Λ′64

=

(Λ2

4t9 (Λ)

t27 (Λ)

)5t9 (Λ)

Λ64

,

Λ′6 = 0,

Γ′

Λ′24

=

(Λ2

4t9 (Λ)

t27 (Λ)

)3Γ

Λ24

.


t7(Λ′)

=Λ24t

39(Λ)

t57(Λ)= λ1, t9

(Λ′)

=Λ44t

69(Λ)

t107 (Λ)= λ2

1 and Γ′ =Λ44t

39(Λ)Γ

t67(Λ)= λ2.

Thus all algebras from U17 are isomorphic to L(0, 1, 0, 0, λ1, 0, λ

21, λ2

)for

λ1 ∈ C∗ and λ2 ∈ C.

• U18 ={L (Λ) ∈ SLbb10 : Λ3 = 0,Λ4 6= 0,Λ5 = 0, t7 (Λ) 6= 0, t8 (Λ) = 0, t9 (Λ) = 0

}By reducing

t7(Λ′)Λ′44

= 1A3

t7(Λ)

Λ44

in (B.25), then we obtain A3 =t7(Λ)

Λ44. The


4and D = A3


1A3 =

Λ44

t7(Λ), the system of equalities for U18 is as follows:

Λ′3 = 0,

t7(Λ′)

Λ′44

=

(Λ4

4

t7 (Λ)

)t7 (Λ)

Λ44

, (B.27)

Λ′4 = 1,

t8(Λ′)

Λ′54

= 0,

Λ′5

Λ′4

= 0,t9(Λ′)

Λ′64

= 0,

Λ′6 = 0,

Γ′

Λ′24

=

(Λ4

4

t7 (Λ)

)Γ

Λ24

.

New parameters are represented as λ to imply that Γ′ =Λ24Γ

t7(Λ)= λ. Thus

all algebras from U18 are isomorphic to L (0, 1, 0, 0, 1, 0, 0, λ) for λ ∈ C.

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(e) T5 = T8⋃T9 :

• T8 ={L (Λ) ∈ SLbb10 : Λ3 = 0,Λ4 6= 0,Λ5 = 0, t7 (Λ) = 0, t8 (Λ) 6= 0


Λ′3 = 0,

t7(Λ′)

Λ′44

= 0, (B.28)

Λ′4 = 1,

t8(Λ′)

Λ′54

=1

A4

t8 (Λ)

Λ54

,

Λ′5

Λ′4

= 0,t9(Λ′)

Λ′64

=1

A5

t9 (Λ)

Λ64

,

Λ′6 = 0,

Γ′

Λ′24

=1

A3

Γ

Λ24

.

• T9 ={L (Λ) ∈ SLbb10 : Λ3 = 0,Λ4 6= 0,Λ5 = 0, t7 (Λ) = 0, t8 (Λ) = 0


Λ′3 = 0,

t7(Λ′)

Λ′44

= 0, (B.29)

Λ′4 = 1,

t8(Λ′)

Λ′54

= 0,

Λ′5

Λ′4

= 0,t9(Λ′)

Λ′64

=1

A5

t9 (Λ)

Λ64

,

Λ′6 = 0,

Γ′

Λ′24

=1

A3

Γ

Λ24

.

(f) T8 = U19⋃T10 :

• U19 ={L (Λ) ∈ SLbb10 : Λ3 = 0,Λ4 6= 0,Λ5 = 0, t7 (Λ) = 0, t8 (Λ) 6= 0, t9 (Λ) 6= 0

}By reducing

t8(Λ′)Λ′54

= 1A4

t8(Λ)

Λ54

andt9(Λ′)

Λ′64

= 1A5

t9(Λ)

Λ64

in (B.28), then we ob-

tain A =t9(Λ)

Λ4t8(Λ). The values of B and D appear as B = Λ6A

3Λ24and D = A3

Λ4.


Λ4t8(Λ)t9(Λ)

, the system of equalities for U19 is as

follows:

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Λ′3 = 0,

t7(Λ′)

Λ′44

= 0, (B.30)

Λ′4 = 1,

t8(Λ′)

Λ′54

=

(Λ4t8 (Λ)

t9 (Λ)

)4 t8 (Λ)

Λ54

,

Λ′5

Λ′4

= 0,t9(Λ′)

Λ′64

=

(Λ4t8 (Λ)

t9 (Λ)

)5 t9 (Λ)

Λ64

,

Λ′6 = 0,

Γ′

Λ′24

=

(Λ4t8 (Λ)

t9 (Λ)

)3 Γ

Λ24

.


t8(Λ′)

=t58(Λ)

Λ4t49(Λ)

= λ1, t9(Λ′)

=t58(Λ)

Λ4t49(Λ)

= λ1 and Γ′ =Λ4t

38(Λ)Γ

t39(Λ)= λ2.

Thus all algebras from U19 are isomorphic to L (0, 1, 0, 0, 0, λ1, λ1, λ2) forλ1 ∈ C∗ and λ2 ∈ C.

• T10 ={L (Λ) ∈ SLbb10 : Λ3 = 0,Λ4 6= 0,Λ5 = 0, t7 (Λ) = 0, t8 (Λ) 6= 0, t9 (Λ) = 0


Λ′3 = 0,

t7(Λ′)

Λ′44

= 0, (B.31)

Λ′4 = 1,

t8(Λ′)

Λ′54

=1

A4

t8 (Λ)

Λ54

,

Λ′5

Λ′4

= 0,t9(Λ′)

Λ′64

= 0,

Λ′6 = 0,

Γ′

Λ′24

=1

A3

Γ

Λ24

.

(g) T10 = U20⋃U37 :

• U20 ={L (Λ) ∈ SLbb10 : Λ3 = 0,Λ4 6= 0,Λ5 = 0, t7 (Λ) = 0, t8 (Λ) 6= 0,

t9 (Λ) = 0,Γ 6= 0}By reducing

t8(Λ′)Λ′54

= 1A4

t8(Λ)

Λ54

and Γ′

Λ′24

= 1A3


A =t8(Λ)

Λ34Γ


4and D = A3

Λ4. Then

by using value 1A =

Λ34Γ


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Λ′3 = 0,

t7(Λ′)

Λ′44

= 0, (B.32)

Λ′4 = 1,

t8(Λ′)

Λ′54

=

(Λ3

4Γ

t8 (Λ)

)4t8 (Λ)

Λ54

,

Λ′5

Λ′4

= 0,t9(Λ′)

Λ′64

= 0,

Λ′6 = 0,

Γ′

Λ′24

=

(Λ3

4Γ

t8 (Λ)

)3Γ

Λ24

.

New parameter is represented as λ to imply that t8(Λ′)

=Λ74Γ4

t38(Λ)= λ and

Γ′ =Λ74Γ4

t38(Λ)= λ. Thus all algebras from U20 are isomorphic to

L (0, 1, 0, 0, 0, λ, 0, λ) for λ ∈ C∗.

• U37 ={L (Λ) ∈ SLbb10 : Λ3 = 0,Λ4 6= 0,Λ5 = 0, t7 (Λ) = 0, t8 (Λ) 6= 0,

t9 (Λ) = 0,Γ = 0}By reducing

t8(Λ′)Λ′54

= 1A4

t8(Λ)

Λ54


Λ54. The


4and D = A3


1A4 =

Λ54


Λ′3 = 0,

t7(Λ′)

Λ′44

= 0, (B.33)

Λ′4 = 1,

t8(Λ′)

Λ′54

=

(Λ5

4

t8 (Λ)

)t8 (Λ)

Λ54

,

Λ′5

Λ′4

= 0,t9(Λ′)

Λ′64

= 0,

Λ′6 = 0,

Γ′

Λ′24

= 0.

Thus all algebras from U37 are isomorphic to L (0, 1, 0, 0, 0, 1, 0, 0).

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(h) T9 = T11⋃T12 :

• T11 ={L (Λ) ∈ SLbb10 : Λ3 = 0,Λ4 6= 0,Λ5 = 0, t7 (Λ) = 0, t8 (Λ) = 0,

t9 (Λ) 6= 0}The system of equations in subset T11 becomes:

Λ′3 = 0,

t7(Λ′)

Λ′44

= 0, (B.34)

Λ′4 = 1,

t8(Λ′)

Λ′54

= 0,

Λ′5

Λ′4

= 0,t9(Λ′)

Λ′64

=1

A5

t9 (Λ)

Λ64

,

Λ′6 = 0,

Γ′

Λ′24

=1

A3

Γ

Λ24

.

• T12 ={L (Λ) ∈ SLbb10 : Λ3 = 0,Λ4 6= 0,Λ5 = 0, t7 (Λ) = 0, t8 (Λ) = 0,

t9 (Λ) = 0}The system of equations in subset T12 becomes:

Λ′3 = 0,

t7(Λ′)

Λ′44

= 0, (B.35)

Λ′4 = 1,

t8(Λ′)

Λ′54

= 0,

Λ′5

Λ′4

= 0,t9(Λ′)

Λ′64

= 0,

Λ′6 = 0,

Γ′

Λ′24

=1

A3

Γ

Λ24

.

(i) T11 = U21⋃U38 :

• U21 ={L (Λ) ∈ SLbb10 : Λ3 = 0,Λ4 6= 0,Λ5 = 0, t7 (Λ) = 0, t8 (Λ) = 0,

t9 (Λ) 6= 0,Γ 6= 0}By reducing

t9(Λ′)Λ′64

= 1A5

t9(Λ)

Λ64

and Γ′

Λ′24

= 1A3


A =Λ24Γ2

t9(Λ). The values of B and D appear as B = Λ6A

3Λ24and D = A3

Λ4. Then

by using value 1A =

t9(Λ)

Λ24Γ2 , the system of equalities for U21 is as follows:

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Λ′3 = 0,

t7(Λ′)

Λ′44

= 0, (B.36)

Λ′4 = 1,

t8(Λ′)

Λ′54

= 0,

Λ′5

Λ′4

= 0,t9(Λ′)

Λ′64

=

(t9 (Λ)

Λ24Γ2

)5t9 (Λ)

Λ64

,

Λ′6 = 0,

Γ′

Λ′24

=

(t9 (Λ)

Λ24Γ2

)3Γ

Λ24

.

New parameter is represented as λ to imply that

t9(Λ′)

=t69(Λ)

Λ164 Γ10 = λ2 and Γ′ =

t39(Λ)

Λ84Γ5 = λ. Thus all algebras from U21 are

isomorphic to L(0, 1, 0, 0, 0, 0, λ2, λ

)for λ ∈ C∗.

• U38 ={L (Λ) ∈ SLbb10 : Λ3 = 0,Λ4 6= 0,Λ5 = 0, t7 (Λ) = 0, t8 (Λ) = 0,

t9 (Λ) 6= 0,Γ = 0}By reducing

t9(Λ′)Λ′64

= 1A5

t9(Λ)

Λ64


Λ64. The


4and D = A3


1A5 =

Λ64


Λ′3 = 0,

t7(Λ′)

Λ′44

= 0, (B.37)

Λ′4 = 1,

t8(Λ′)

Λ′54

= 0,

Λ′5

Λ′4

= 0,t9(Λ′)

Λ′64

=

(Λ6

4

t9 (Λ)

)t9 (Λ)

Λ64

,

Λ′6 = 0,

Γ′

Λ′24

= 0.


(j) T12 = U39⋃U40 :

• U39 ={L (Λ) ∈ SLbb10 : Λ3 = 0,Λ4 6= 0,Λ5 = 0, t7 (Λ) = 0, t8 (Λ) = 0,

t9 (Λ) = 0,Γ 6= 0}By reducing Γ′

Λ′24

= 1A3

ΓΛ24in (B.35), then we obtain A3 = Γ

Λ24. The values of


4and D = A3

Λ4.

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Then by using value 1A3 =

Λ24

Γ , the system of equalities for U39 is as follows:

Λ′3 = 0,

t7(Λ′)

Λ′44

= 0, (B.38)

Λ′4 = 1,

t8(Λ′)

Λ′54

= 0,

Λ′5

Λ′4

= 0,t9(Λ′)

Λ′64

= 0,

Λ′6 = 0,

Γ′

Λ′24

=

(Λ2

4

Γ

)Γ

Λ24

.


• U40 ={L (Λ) ∈ SLbb10 : Λ3 = 0,Λ4 6= 0,Λ5 = 0, t7 (Λ) = 0, t8 (Λ) = 0,

t9 (Λ) = 0,Γ = 0}All algebras from U40 are isomorphic to L (0, 1, 0, 0, 0, 0, 0, 0).

(ii) T2 ={L (β) ∈ SLbb10 : β3 = 0, β4 = 0

}The subset of T2 brings the following system of equations:

β′3 = 0, β

′7 =

1

A5

(D

A

)β7, (B.39)

β′4 = 0, β

′8 =

1

A6

(D

A

)[β8 − 4

(B

A

)β2

5

],

β′5 =

1

A3

(D

A

)β5, β

′9 =

1

A7

(D

A

)[β9 +

(B

A

)γ − 9

(B

A

)β5β6

],

β′6 =

1

A4

(D

A

)β6, γ

′=

1

A7

(D

A

)2

γ.

(a) T2 = T13⋃T14 :

• T13 ={L (β) ∈ SLbb10 : β3 = 0, β4 = 0, β5 6= 0

}The system of equations of subset T13 is as following:

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β′3 = 0,

β′7

β′5

=1

A2

β7

β5, (B.40)

β′4 = 0,

β′8

β′5

=1

A3

[β8

β5− 4

(B

A

)β5

],

β′5 = 1,

β′9

β′5

=1

A4

[β9

β5+

(B

A

)γ

β5− 9

(B

A

)β6

],

β′6

β′5

=1

A

β6

β5,

γ′

β′25

=1

A

γ

β25

.

• T14 ={L (β) ∈ SLbb10 : β3 = 0, β4 = 0, β5 = 0

}The subset of T14 brings the following system of equations:

β′3 = 0, β

′7 =

1

A5

(D

A

)β7, (B.41)

β′4 = 0, β

′8 =

1

A6

(D

A

)β8,

β′5 = 0, β

′9 =

1

A7

(D

A

)[β9 +

(B

A

)γ

],

β′6 =

1

A4

(D

A

)β6, γ

′=

1

A7

(D

A

)2

γ.

(b) T13 = U22⋃T15 :

• U22 ={L (β) ∈ SLbb10 : β3 = 0, β4 = 0, β5 6= 0, β6 6= 0

}By reducing

β′6

β′5

= 1Aβ6β5

in (B.40), then we obtain A = β6β5. The values of B

and D appear as B = β8A4β25

and D = A4

β5. Then by using value 1

A = β5β6

and

BA = β8β6

4β35, the system of equalities for U22 is as follows:

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β′3 = 0, (B.42)

β′4 = 0,

β′5 = 1,

β′6

β′5

=

(β5

β6

)β6

β5,

β′7

β′5

=

(β5

β6

)2 β7

β5,

β′8

β′5

= 0,

β′9

β′5

=

(β5

β6

)4[β9

β5+

(β8β6

4β35

)γ

β5− 9

(β8β6

4β35

)β6

],

γ′

β′25

=

(β5

β6

)γ

β25

.


β′7 = β5β7

β26= λ1, β

′9 = 1

4β46

(4β3

5β9 + β6β8γ − 9β5β26β8)

= λ2 and

γ′ = γβ5β6

= λ3. Thus all algebras from U22 are isomorphic to

L (0, 0, 1, 1, λ1, 0, λ2, λ3) for λ1, λ2, λ3 ∈ C.

• T15 ={L (β) ∈ SLbb10 : β3 = 0, β4 = 0, β5 6= 0, β6 = 0


β′3 = 0,

β′7

β′5

=1

A2

β7

β5, (B.43)

β′4 = 0,

β′8

β′5

= 0,

β′5 = 1,

β′9

β′5

=1

A4

[β9

β5+

(B

A

)γ

β5

],

β′6

β′5

= 0,γ′

β′25

=1

A

γ

β25

.

(c) T15 = T16⋃T17 :

• T16 ={L (β) ∈ SLbb10 : β3 = 0, β4 = 0, β5 6= 0, β6 = 0, β7 6= 0

}

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The system of equations of subset T16 is as following:

β′3 = 0,

β′7

β′5

=1

A2

β7

β5, (B.44)

β′4 = 0,

β′8

β′5

= 0,

β′5 = 1,

β′9

β′5

=1

A4

[β9

β5+

(B

A

)γ

β5

],

β′6

β′5

= 0,γ′

β′25

=1

A

γ

β25

.

• T17 ={L (β) ∈ SLbb10 : β3 = 0, β4 = 0, β5 6= 0, β6 = 0, β7 = 0


β′3 = 0,

β′7

β′5

= 0, (B.45)

β′4 = 0,

β′8

β′5

= 0,

β′5 = 1,

β′9

β′5

=1

A4

[β9

β5+

(B

A

)γ

β5

],

β′6

β′5

= 0,γ′

β′25

=1

A

γ

β25

.

(d) T16 = U23⋃T18 :

• U23 ={L (β) ∈ SLbb10 : β3 = 0, β4 = 0, β5 6= 0, β6 = 0, β7 6= 0, γ 6= 0

}By reducing

β′6

β′5

= 1Aβ6β5

in (B.44), then we obtain A = β5β7γ . The values of

B and D appear as B = β8A4β25

and D = A4


A = γβ5β7

and BA = β7β8

4β5γ, the system of equalities for U23 is as follows:

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β′3 = 0, (B.46)

β′4 = 0,

β′5 = 1,

β′6

β′5

= 0,

β′7

β′5

=

(γ

β5β7

)2 β7

β5,

β′8

β′5

= 0,

β′9

β′5

=

(γ

β5β7

)4 [β9

β5+

(β7β8

4β5γ

)γ

β5

],

γ′

β′25

=

(γ

β5β7

)γ

β25

.

New parameters are represented as λ1 and λ2 to imply that β′7 = γ2

β35β7= λ1,

β′9 = γ4

4β65β47

(4β5β9 + β7β8) = λ2 and γ′ = γ2

β35β7= λ1. Thus all algebras

from U23 are isomorphic to L (0, 0, 1, 0, λ1, 0, λ2, λ1) for λ1 ∈ C∗ and λ2 ∈ C.

• T18 ={L (β) ∈ SLbb10 : β3 = 0, β4 = 0, β5 6= 0, β6 = 0, β7 6= 0, γ = 0


β′3 = 0,

β′7

β′5

=1

A2

β7

β5, (B.47)

β′4 = 0,

β′8

β′5

= 0,

β′5 = 1,

β′9

β′5

=1

A4

β9

β5,

β′6

β′5

= 0,γ′

β′25

= 0.

(e) T18 = U24⋃U41 :

• U24 ={L (β) ∈ SLbb10 : β3 = 0, β4 = 0, β5 6= 0, β6 = 0, β7 6= 0, γ = 0, β9 6= 0

}By reducing

β′7

β′5

= 1A2

β7β5

andβ′9

β′5

= 1A4

β9β5

in (B.47), then we obtain A2 = β9β7.

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The values of B and D appear as B = β8A4β25

and D = A4

β5. Then by using

value 1A2 = β7

β9, the system of equalities for U24 is as follows:

β′3 = 0,

β′7

β′5

=

(β7

β9

)β7

β5, (B.48)

β′4 = 0,

β′8

β′5

= 0,

β′5 = 1,

β′9

β′5

=

(β7

β9

)2 β9

β5,

β′6

β′5

= 0,γ′

β′25

= 0.

New parameter is represented as λ to imply that β′7 =

β27β5β9

= λ and

β′9 =

β27β5β9

= λ. Thus all algebras from U24 are isomorphic to

L (0, 0, 1, 0, λ, 0, λ, 0) for λ ∈ C∗.

• U41 ={L (β) ∈ SLbb10 : β3 = 0, β4 = 0, β5 6= 0, β6 = 0, β7 6= 0, γ = 0, β9 = 0

}By reducing

β′7

β′5

= 1A2

β7β5

in (B.47), then we obtain A2 = β7β5. The values of

B and D appear as B = β8A4β25

and D = A4


A2 = β5β7,

the system of equalities for U41 is as follows:

β′3 = 0,

β′7

β′5

=

(β5

β7

)β7

β5, (B.49)

β′4 = 0,

β′8

β′5

= 0,

β′5 = 1,

β′9

β′5

= 0,

β′6

β′5

= 0,γ′

β′25

= 0.


The subsets in SLbb10

are illustrated by the following graph trees.

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Figure

B.2:GraphtreeofSLbb 10

SLbb 10

Λ46=

0

U15

Λ5

=0

t 7(Λ

)6=

0

U16

t 8(Λ

)=

0

U17

U18

t 7(Λ

)=

0

t 7(Λ

)6=

0

U19

t 9(Λ

)=

0

U20

U36

t 8(Λ

)=

0

t 9(Λ

)6=

0

U21

U37

t 9(Λ

)=

0

U38

U39

β4

=0

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Figure

B.3:GraphtreeofSLbb 10

(continuedfrom

Figure

B.2)

β4

=0

β56=

0

U22

β66=

0

β76=

0

U23

γ=

0

U24

U40

β7

=0

U25

γ=

0

U41

U42

β5

=0

γ6=

0

U26

β6

=0

β76=

0

U27

U43

β7

=0

U44

U45

γ=

0

β66=

0

U28

β7

=0

β86=

0

U29

U46

β8

=0

U35

U47

β6

=0

β76=

0

U30

β8

=0

U48

U49

β7

=0

β86=

0

U50

U51

β8

=0

U52

U53

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APPENDIX C

This part includes two examples of computer program to represent the list ofalgebras of SLb10 into table of multiplications. Here orbits U30 and U52 are usedto ilustrate the examples.

Calling Sequence:SC represents the list of algebras of SLb10 either in structural constantsβ3 , β4 , ..., β9 , γ or Λ3 ,Λ4 , f5 (Λ) , ..., f9 (Λ) ,Γ or Λ3 ,Λ4 ,Λ5 , t6 (Λ) , ..., t9 (Λ) ,Γ.

Parameters:isom1 − e0e1isom2 − e1e1isom3 − e2e1isom4 − e3e1isom5 − e4e1isom6 − e5e1isom7 − e6e1isom8 − e7e1

Example of U52:Program

SC:=[β3 = 0, β4 = 0, β5 = 0, β6 = 0, β7 = 0, β8 = 0, β9 = 1, γ = 0:

isom1:=eval(β3e3 + β4e4 + β5e5 + β6e6 + β7e7 + β8e8 + β9e9 , SC);

isom2:=eval(γe9 , SC);

isom3:=eval(β3e4 + β4e5 + β5e6 + β6e7 + β7e8 + β8e9 , SC);

isom4:=eval(β3e5 + β4e6 + β5e7 + β6e8 + β7e9 , SC);

isom5:=eval(β3e6 + β4e7 + β5e8 + β6e9 , SC);

isom6:=eval(β3e7 + β4e8 + β5e9 , SC);

isom7:=eval(β3e8 + β4e9 , SC);

isom8:=eval(β3e9 , SC);

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Result

isom1 := e9isom2 := 0

isom3 := 0

isom4 := 0

isom5 := 0

isom6 := 0

isom7 := 0

isom8 := 0

Example of U30:Program

SC:=[β3 = 0, β4 = 0, β5 = 0, β6 = 0, β7 = 1, β8 = 1, β9 = λ, γ = 0:

isom1:=eval(β3e3 + β4e4 + β5e5 + β6e6 + β7e7 + β8e8 + β9e9 , SC);

isom2:=eval(γe9 , SC);

isom3:=eval(β3e4 + β4e5 + β5e6 + β6e7 + β7e8 + β8e9 , SC);

isom4:=eval(β3e5 + β4e6 + β5e7 + β6e8 + β7e9 , SC);

isom5:=eval(β3e6 + β4e7 + β5e8 + β6e9 , SC);

isom6:=eval(β3e7 + β4e8 + β5e9 , SC);

isom7:=eval(β3e8 + β4e9 , SC);

isom8:=eval(β3e9 , SC);

Result

isom1 := e7 + e8 + λe9isom2 := 0

isom3 := e8 + e9isom4 := 0

isom5 := 0

isom6 := 0

isom7 := 0

isom8 := 0

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BIODATA OF STUDENT

The student, Suzila binti Mohd Kasim, was born in 1989. The author startedher tertiary education at matriculation level in 2007 at Kolej Matrikulasi Perak,Perak. She obtained her Bachelor of Science (Honours) Mathematics in 2011from Universiti Putra Malaysia, Serdang, Selangor. In the same university, shepursued her Masters study in 2011.

The student can be contacted via her supervisor,Professor Dr. Isamiddin S. Rakhimov, by address:

Institute for Mathematical Research,Universiti Putra Malaysia,43400 Serdang,Selangor,Malaysia.

Email: [email protected]

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LIST OF PUBLICATIONS

Publications that arise from the study are:

1. S. Mohd Kasim, N. S. Mohamed, I. S. Rakhimov and S. K. Said Husain.Isomorphism Criteria of Leibniz Algebras Arising from the Naturally GradedNon-Lie Filiform Leibniz Algebra, Proceedings of 2nd Regional Conferenceon Applied and Engineering Mathematics 2012, p225-228, 2012. (Penang,Malaysia)

2. S. Mohd Kasim, N. S. Mohamed, I. S. Rakhimov and S. K. Said Husain.Algorithm for Constructing the Isomorphism Criteria for 10-DimensionalComplex Filiform Leibniz Algebras, Proceedings of the Fundamental ScienceCongress 2012, p51-52, 2012. (UPM Serdang, Selangor, Malaysia)

3. S. Mohd Kasim, I. S. Rakhimov and S. K. Said Husain. Appearence of OddParts of Catalan Numbers in the Classi�cation Problem of Complex Fili-form Leibniz Algebras, Proceedings in the International Science Postgradu-ate Conference 2012, p465-477, 2012. (UTM Skudai, Johor, Malaysia)

4. S. Mohd Kasim, I. S. Rakhimov and S. K. Said Husain. On IsomorphismClasses and Invariants of A Subclass Filiform Leibniz Algebras Arising fromNaturally Graded Non-Lie Filiform Leibniz Algebras in Dimension 10, Pro-ceedings of the Fundamental Science Congress 2013, p512-516, 2013. (UPMSerdang, Selangor, Malaysia)

5. S. Mohd Kasim, I. S. Rakhimov and S. K. Said Husain. IsomorphismClasses of 10-Dimensional Filiform Leibniz Algebras, AIP Conference Pro-ceedings, 1602, p708-715, 2014.

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