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

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If l = k then we stand at (k − 1)-step and 2.2. Singular quadratic Lie algebras I = α1 ∧ ··· ∧ αk−1 ∧ ∑ ξ1,...,k−1, jk k≤ jk≤n α jk . Since αk ∧ I = 0 then ξJ = 0 if jk = k. Therefore, we obtain I = ξ α1 ∧ ··· ∧ αk. Lemma 2.2.5. WI = GI = {ιA(I) | A ∈ k−1 (V )}. Proof. Let X ∈ V such that ιX(I) = 0. If A ∈ k−1 (V ) then one has ιA(I)(X) = I(A ∧ X) = (−1) k−1 I(X ∧ A) = (−1) k−1 ιX(I)(A) = 0. So GI ⊂ WI. Let X ∈ G ⊥ I . If A ∈ k−1 (V ) then ιA(I)(X) = 0. That means ιX(I)(A) = 0, for all A ∈ k−1 (V ). Therefore ιX(I) = 0 and then X ∈ W ⊥ I . It implies that WI = GI. We turn now to the proof of the proposition. (1) Let α ∈ VI, we show that α ∈ WI. It means that if X ∈ V such that ιX(I) = 0 then α(X) = 0. Indeed, let X ∈ V such that ιX(I) = 0. Since α ∈ VI one has α ∧ I = 0. This implies that 0 = ιX(α ∧ I) = ιX(α) ∧ I − α ∧ ιX(I) = α(X)I. Therefore, α(X) = 0 since I = 0. To prove dim(VI) ≤ k, we assume to the contrary, i.e. dim(VI) ≥ k+1. Let {α1,...,αk+1} be an independent system of vectors of VI. Apply Lemma 2.2.4 one has a non-zero complex ξ such that I = ξ α1 ··· ∧ αk. Since αk+1 ∧I = 0 we get ξ α1 ···∧αk = 0, i.e. ξ = 0. This is a contradiction. Therefore, dim(VI) ≤ k. To prove dim(WI) ≥ k, let {α1,...,αr} be a basis of VI and complete it to get a basis {α1,...,αn} of V ∗ . By Lemma 2.2.4, there exists a (k − r)-form β such that We can write β as follows: I = α1 ··· ∧ αr ∧ β. β = ∑ ξ jr+1,... jk r+1≤ jr+1
2.2. Singular quadratic Lie algebras • if As = X1 ∧ ··· ∧ Xr ∧ Xjr+1 ∧ ··· ∧ Xjr+s ∧ ··· ∧ Xjk then ιAs I = ±ξJα jr+s ∈ WI(1 ≤ s ≤ k − r). Therefore, dim(WI) ≥ k. (2) The first statement of (2) follows Lemma 2.2.4. By the proof of (1), α1,...,αr,αjr+1 , ...,αjr+s ∈ WI then I ∈ k (WI). (3) Lemma 2.2.4 shows that I is decomposable if and only if dim(VI) = k. We only prove that I is decomposable if and only if dim(WI) = k. The last assertion follows. Indeed, if I is decomposable, then I = α1 ∧ ··· ∧ αk. Therefore, WI = span{α1,...,αk}, i.e. dim(WI) = k. Conversely, if dim(WI) = k, we assume that I is not decomposable. Let {α1,...,αr} be a basis of VI and complete it to get a basis {α1,...,αn} of V ∗ . By Lemma 2.2.4, there exists a (k − r)-form β such that I = α1 ··· ∧ αr ∧ β. We can write β as follows: β = ∑ ξ jr+1,... jk r+1≤ jr+1
Page 1: Université de Bourgogne - Dijon UF
Page 4 and 5: Remerciements Je remercie l’Ambas
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 48 and 49: Consider Then dim(g) = 6 ∑ i=3 2.
Page 66 and 67: (2) g g ′ if and only if g i g
Page 68 and 69: 2.3. Quadratic dimension of quadrat
Page 74 and 75: 2.4. 2-step nilpotent quadratic Lie
Page 81 and 82: Chapter 3 Singular quadratic Lie su
Page 83 and 84: 3.1. Application of Z × Z2-graded
Page 91 and 92: 3.2. The dup-number of a quadratic
Page 93 and 94: 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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