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

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2.2. Singular quadratic Lie algebras Keep the notations as in Chapter 1. To describe the set of semisimple O(n)-orbits in P 1 (o(n)), we need to add maps (λ1,...,λp) ↦→ λ(λ1,...,λp), for all λ ∈ C, λ = 0 to the group Gp. We obtain a group denoted by Hp. As a consequence of Section 1.3, we have the classification result for the diagonalizable case: Theorem 2.2.41. There is a bijection between Di (n+2) and Λp/Hp with n = 2p or n = 2p+1. Moreover, if n = 2p + 1, Di red (n + 2) = /0 and if n = 2p, then Di red (2p + 2) is in bijection with Λ + p /Hp where Λ + p = {(λ1,...,λp) | λi ∈ C,λi = 0, ∀ i }. To go further in the study of diagonalizable reduced case, we need the following Lemma: Lemma 2.2.42. Let g ′ and g ′′ be solvable singular quadratic Lie algebras, g ′ = (CX ′ 1 ⊕ CY ′ 1 ) ⊥ ⊕ q ′ a decomposition of g ′ as a double extension and C ′ = ad(Y ′ 1 )| q ′. We assume that C′ is invertible. Then g ′ and g ′′ are isomorphic if and only if they are i-isomorphic. Proof. Write g ′′ = (CX ′′ ⊥ 1 ⊕ CY ′′ 1 ) ⊕ q ′′ a decomposition of g ′′ as a double extension and C ′′ = ad(Y ′′ 1 )| q ′′. Assume that g ′ and g ′′ are isomorphic. By Corollary 2.2.31, there exist Q : q ′ → q ′′ and P ∈ L (q ′ ) such that Q P −1 is an isometry, P ∗ P C ′ = C ′ and C ′′ = λ Q C ′ Q −1 for some nonzero λ ∈ C. But C ′ is invertible, so P ∗ P = Idq ′. Therefore, P is an isometry of q ′ and then Q is an isometry from q ′ to q ′′ . The conditions of Corollary 2.2.31 (1) are satisfied, so g ′ and g ′′ are i-isomorphic. Corollary 2.2.43. One has: Dred(2p + 2) = D i red (2p + 2), ∀ p ≥ 1. Next, we describe diagonalizable reduced singular quadratic Lie algebras using the amalgamated products. First, let g4(λ) be the double extension of q = C2 λ 0 by C = , λ = 0. 0 −λ By Lemma 2.2.27, g4(λ) is i-isomorphic to g4(1), call it g4. Proposition 2.2.44. Let (g,B) be a diagonalizable reduced singular quadratic Lie algebra. Then g is an amalgamated product of singular quadratic Lie algebras all i-isomorphic to g4. Proof. We write g = (CX0 ⊕CY0) ⊥ ⊕ q, C = ad(Y0), C =C|q and B = Bq×q. Then C is a diagonalizable invertible element of o(q,B). Apply Appendix A to obtain a basis {e1,...,ep, f1,..., fp} of q and λ1,...,λp ∈ C, all non-zero, such that B(ei,ej) = B( fi, f j) = 0, B(ei, f j) = δi j and C(ei) = λiei, C( fi) = −λi fi, for all 1 ≤ i, j ≤ p. Let qi = span{ei, fi}, 1 ≤ i ≤ p. Then q = ⊥ ⊕ qi. 1≤i≤p Furthermore, hi = (CX0 ⊕ CY0) ⊥ ⊕ qi is a Lie subalgebra of g for all 1 ≤ i ≤ p and g = h1 × a h2 × a ... × a hp with hi 42 i g4(λi) i g4.
2.2. Singular quadratic Lie algebras Remark 2.2.45. For non-zero λ, µ ∈ C, consider the amalgamated product: g(λ,µ) = g4(λ) × a g4(µ). Then g(λ,µ) is the double extension of C4 by ⎛ λ ⎜ ⎜0 ⎝0 0 µ 0 0 0 −λ ⎞ 0 0 ⎟ 0 ⎠ 0 0 0 −µ . Therefore g(λ,µ) is isomorphic to g(1,1) if and only if µ = ±λ (Lemma 2.2.42 and Section 1.3). So, though g4(λ) and g4(µ) are i-isomorphic to g4, the amalgamated product g(λ,µ) is not even isomorphic to g(1,1) = g4 × a g4 if µ = ±λ. This illustrates that amalgamated products may have a rather bad behavior with respect to isomorphisms. Definition 2.2.46. A double extension is called an invertible quadratic Lie algebra if the corresponding skew-symmetric map is invertible. Remark 2.2.47. • By Remark 2.2.32, the property of being an invertible quadratic Lie algebra does not depend on the chosen decomposition. • By Appendix A, the dimension of an invertible quadratic Lie algebra is even. • By Lemma 2.2.42, two invertible quadratic Lie algebras are isomorphic if and only if they are i-isomorphic. For p ≥ 1 and λ ∈ C, let Jp(λ) = diag p (λ,...,λ) + Jp and C J Jp(λ) 0 2p(λ) = 0 −t Jp(λ) in a canonical basis of quadratic vector space C 2p . Then C J 2p(λ) ∈ o(2p). Definition 2.2.48. For λ ∈ C, let j2p(λ) be the double extension of C 2p by C J 2p(λ). We say that j2p(λ) is a Jordan-type quadratic Lie algebra. When λ = 0 and p ≥ 2, we recover the nilpotent Jordan-type Lie algebras j2p from nilpotent case. When λ = 0, j2p(λ) is an invertible singular quadratic Lie algebra and j2p(−λ) j2p(λ). Proposition 2.2.49. Let g be a solvable singular quadratic Lie algebra. Then g is an invertible quadratic Lie algebra if and only if g is an amalgamated product of Lie algebras all iisomorphic to Jordan-type Lie algebras j2p(λ), with λ = 0. 43
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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
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.
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
Page 95 and 96: 3.3. Elementary quadratic Lie super
Page 97 and 98: 3.4. Quadratic Lie superalgebras wi
Page 107 and 108: 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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