TH`ESE A NEW INVARIANT OF QUADRATIC LIE ALGEBRAS AND ...

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Remerciements Je remercie l’Ambassade de France au Vietnam et le CROUS de Dijon pour leurs aides de financement et d’hébergement durant mon séjour à Dijon. Mes remerciements vont aussi à l’Équipe de Mathématique-Physique, au Département de Physique et à l’Université de Pédagogie de Ho Chi Minh ville dans laquelle j’ai un poste d’enseignant, pour leur soutien durant ma période de préparation de thèse. Je terminerai en remerciant ma famille, tout particulièrement ma femme et mon petit garçon qui m’ont constamment soutenu et accompagné la préparation de ma thèse. iii
Contents Introduction vi Notations 1 1 Adjoint orbits of sp(2n) and o(m) 3 1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Nilpotent orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Semisimple orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Invertible orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5 Adjoint orbits in the general case . . . . . . . . . . . . . . . . . . . . . . . . . 16 2 Quadratic Lie algebras 17 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Singular quadratic Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.1 Super-Poisson bracket and quadratic Lie algebras . . . . . . . . . . . . 22 2.2.2 The dup number of a quadratic Lie algebra . . . . . . . . . . . . . . . 23 2.2.3 Quadratic Lie algebras of type S1 . . . . . . . . . . . . . . . . . . . . 28 2.2.4 Solvable singular quadratic Lie algebras and double extensions . . . . . 32 2.2.5 Classification singular quadratic Lie algebras . . . . . . . . . . . . . . 36 2.3 Quadratic dimension of quadratic Lie algebras . . . . . . . . . . . . . . . . . . 48 2.3.1 Centromorphisms of a quadratic Lie algebra . . . . . . . . . . . . . . . 48 2.3.2 Quadratic dimension of reduced singular quadratic Lie algebras and the invariance of dup number . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.3.3 Centromorphisms and extensions of a quadratic Lie algebra . . . . . . 52 2.4 2-step nilpotent quadratic Lie algebras . . . . . . . . . . . . . . . . . . . . . . 54 2.4.1 Some extensions of 2-step nilpotent Lie algebras . . . . . . . . . . . . 54 2.4.2 2-step nilpotent quadratic Lie algebras . . . . . . . . . . . . . . . . . . 56 3 Singular quadratic Lie superalgebras 61 3.1 Application of Z × Z2-graded Lie superalgebras to quadratic Lie superalgebras 61 3.2 The dup-number of a quadratic Lie superalgebra . . . . . . . . . . . . . . . . . 70 3.3 Elementary quadratic Lie superalgebras . . . . . . . . . . . . . . . . . . . . . 73 3.4 Quadratic Lie superalgebras with 2-dimensional even part . . . . . . . . . . . . 76 3.4.1 Double extension of a symplectic vector space . . . . . . . . . . . . . 78 iv
Page 1: Université de Bourgogne - Dijon UF
Page 6 and 7: Contents 3.4.2 Quadratic dimension
Page 8 and 9: Introduction structures can be dete
Page 10 and 11: Introduction and Di red (n + 2) th
Page 12 and 13: Introduction and obtain the followi
Page 14 and 15: y Introduction We realize that with
Page 16 and 17: Introduction by I. Bajo, S. Benayad
Page 18 and 19: Introduction (2) there exists an i-
Page 20 and 21: xix
Page 22 and 23: Notations 0 Ik where J = ∈ gl(2
Page 24 and 25: 1.1. Definitions and gε(V ) = {C
Page 26 and 27: 1.2. Nilpotent orbits Lie algebra g
Page 28 and 29: 1.2. Nilpotent orbits where µi = i
Page 30 and 31: 1.2. Nilpotent orbits The point of
Page 32 and 33: 1.3 Semisimple orbits We recall the
Page 34 and 35: 1.4. Invertible orbits Let us now c
Page 36 and 37: 1.5. Adjoint orbits in the general
Page 38 and 39: 2.1. Preliminaries (3) Set the map
Page 40 and 41: 2.1. Preliminaries Proof. Let I be
Page 42 and 43: 2.2. Singular quadratic Lie algebra
Page 44 and 45: If l = k then we stand at (k − 1)
Page 48 and 49: Consider Then dim(g) = 6 ∑ i=3 2.
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2.2. Singular quadratic Lie algebra
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(2) g g ′ if and only if g i g
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2.3. Quadratic dimension of quadrat
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2.4. 2-step nilpotent quadratic Lie
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Chapter 3 Singular quadratic Lie su
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3.1. Application of Z × Z2-graded
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3.2. The dup-number of a quadratic
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3.3. Elementary quadratic Lie super
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3.3. Elementary quadratic Lie super
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3.4. Quadratic Lie superalgebras wi
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3.5. Singular quadratic Lie superal
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3.6. Quasi-singular quadratic Lie s
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3.6. Quasi-singular quadratic Lie s
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Chapter 4 Pseudo-Euclidean Jordan a
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4.1. Preliminaries Proof. The asser
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4.1. Preliminaries T (x,y) = φ(x)(
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4.1. Preliminaries (3) π(xx ′ )y
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4.2. Jordanian double extension of
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4.2. Jordanian double extension of
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4.3. Pseudo-Euclidean 2-step nilpot
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Proof. 4.3. Pseudo-Euclidean 2-step
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4.4. Symmetric Novikov algebras (2)
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4.4. Symmetric Novikov algebras Lem
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4.4. Symmetric Novikov algebras Lem
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4.4. Symmetric Novikov algebras Pro
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Appendix A In this appendix, we rec
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Appendix A Lemma A.4. Assume that C
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Appendix B Here we prove: Lemma B.1
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Appendix C We will classify (up to
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Appendix C Lemma C.3. The element I
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Appendix D In the last appendix, we
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Bibliography [AB10] I. Ayadi and S.
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[Sch55] R. D. Schafer, Noncommutati
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Index Bε, 3 E(g,B,δ), 53 Iε(V ),
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normal form, 5 Jordan-admissible, 1
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TH`ESE A NEW INVARIANT OF QUADRATIC LIE ALGEBRAS AND ...

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