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

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Abstract In this thesis, we defind a new invariant of quadratic Lie algebras and quadratic Lie superalgebras and give a complete study and classification of singular quadratic Lie algebras and singular quadratic Lie superalgebras, i.e. those for which the invariant does not vanish. The classification is related to adjoint orbits of Lie algebras o(m) and sp(2n). Also, we give an isomorphic characterization of 2-step nilpotent quadratic Lie algebras and quasi-singular quadratic Lie superalgebras for the purpose of completeness. We study pseudo-Euclidean Jordan algebras obtained as double extensions of a quadratic vector space by a one-dimensional algebra and 2-step nilpotent pseudo-Euclidean Jordan algebras, in the same manner as it was done for singular quadratic Lie algebras and 2-step nilpotent quadratic Lie algebras. Finally, we focus on the case of a symmetric Novikov algebra and study it up to dimension 7. Key-words: quadratic Lie algebras, quadratic Lie superalgebras, pseudo-Eucliean Jordan algebras, symmetric Novikov algebras, invariant, adjoint orbits, Lie algebra o(m), Lie algebra sp(2n), solvable Lie algebras, 2-step nilpotent, double extensions, T ∗ -extension, generalized double extension, Jordan-admissible.
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Université de Bourgogne - Dijon UF
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Remerciements Je remercie l’Ambas
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Contents 3.4.2 Quadratic dimension
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Introduction structures can be dete
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Introduction and Di red (n + 2) th
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Introduction and obtain the followi
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y Introduction We realize that with
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Introduction by I. Bajo, S. Benayad
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Introduction (2) there exists an i-
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xix
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Notations 0 Ik where J = ∈ gl(2
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1.1. Definitions and gε(V ) = {C
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1.2. Nilpotent orbits Lie algebra g
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1.2. Nilpotent orbits where µi = i
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1.2. Nilpotent orbits The point of
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1.3 Semisimple orbits We recall the
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1.4. Invertible orbits Let us now c
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1.5. Adjoint orbits in the general
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2.1. Preliminaries (3) Set the map
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2.1. Preliminaries Proof. Let I be
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2.2. Singular quadratic Lie algebra
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If l = k then we stand at (k − 1)
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Consider Then dim(g) = 6 ∑ i=3 2.
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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
Page 117 and 118: Chapter 4 Pseudo-Euclidean Jordan a
Page 119 and 120: 4.1. Preliminaries Proof. The asser
Page 121 and 122: 4.1. Preliminaries T (x,y) = φ(x)(
Page 123 and 124: 4.1. Preliminaries (3) π(xx ′ )y
Page 125 and 126: 4.2. Jordanian double extension of
Page 127 and 128: 4.2. Jordanian double extension of
Page 129 and 130: 4.3. Pseudo-Euclidean 2-step nilpot
Page 133 and 134: Proof. 4.3. Pseudo-Euclidean 2-step
Page 137 and 138: 4.4. Symmetric Novikov algebras (2)
Page 139 and 140: 4.4. Symmetric Novikov algebras Lem
Page 145 and 146: 4.4. Symmetric Novikov algebras Lem
Page 147 and 148: 4.4. Symmetric Novikov algebras Pro
Page 149 and 150: Appendix A In this appendix, we rec
Page 151 and 152: Appendix A Lemma A.4. Assume that C
Page 153 and 154: Appendix B Here we prove: Lemma B.1
Page 155 and 156: Appendix C We will classify (up to
Page 157 and 158: Appendix C Lemma C.3. The element I
Page 159 and 160: Appendix D In the last appendix, we
Page 161 and 162: Bibliography [AB10] I. Ayadi and S.
Page 163 and 164: [Sch55] R. D. Schafer, Noncommutati
Page 165 and 166: Index Bε, 3 E(g,B,δ), 53 Iε(V ),
Page 167: normal form, 5 Jordan-admissible, 1
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