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

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(2) g g ′ if and only if g i g ′ . 2.2. Singular quadratic Lie algebras Proof. We assume that g g ′ . Then by Corollary 2.2.31, there exists an invertible P : q → q ′ and a non-zero λ ∈ C such that C ′ = λ P C P −1 , so q ′ N = P(qN) and q ′ I = P(qI), then dim(q ′ N ) = dim(qN) and dim(q ′ I ) = dim(qI). Thus, there exist isometries FN : q ′ N → qN and FI : q ′ I → qI and we can define an isometry F : q ′ → q by F(X ′ N + X ′ I ) = FN(X ′ N ) + FI(X ′ I ), for all X ′ N ∈ q′ N and X ′ I ∈ q′ I . We now define F : g′ → g by F(X ′ 1 ) = X1, F(Y ′ 1 ) = Y1, F| q ′ = F and a new Lie bracket on g : [X,Y ] ′′ = F [F −1 (X),F −1 (Y )] ′ , ∀ X,Y ∈ g. Call g ′′ this new quadratic Lie algebra. We have g ′′ = (CX1 ⊕ CY1) ⊥ ⊕ q, i.e., q ′′ = q and C ′′ = F C ′ F −1 . So q ′′ N = F(q′ N ) = qN and q ′′ I = F(q′ I ) = qI. But g g ′′ , so there exists an invertible Q : q → q such that C ′′ = λ Q C Q −1 for some non-zero λ ∈ C (Corollary 2.2.31). It follows that q ′′ N = Q(qN) and q ′′ I = Q(qI), so Q(qN) = qN and Q(qI) = qI. Moreover, we have Q ∗ Q C = C (Corollary 2.2.31), so Q ∗ Q C k = C k for all k. There exists k such that qI = Im(C k ) and (Q ∗ Q C k )(X) = C k (X), for all X ∈ g. So Q ∗ Q|qI = IdqI and QI = Q|qI is an isometry. Since C ′′ I = λ Q I CI Q −1 I , then gI i g ′′ I (Corollary 2.2.31). Let QN = Q|qN . Then C′′ N = λ QN CN Q −1 N and Q∗ N QN CN = CN, so by Corollary 2.2.31, gN g ′′ N . Since gN and g ′′ N are nilpotent, then g′′ i N gN by Theorem 2.2.37. Conversely, assume that gN g ′ N and gI g ′ I . Then gN i g ′ N and gI i g ′ I by Theorem 2.2.37 and Lemma 2.2.42. So, there exist isometries PN : gN → g ′ N , PI : gI → g ′ I and non-zero λN and λI ∈ C such that C ′ N = λN PN CN P −1 N and C ′ I = λI PI CI P −1 I . By Lemma 2.2.36, since gN and g ′ N are nilpotent, we can assume that λN = λI = λ. Now we define P : q → q ′ by P(XN + XI) = PN(XN) + PI(XI), for all XN ∈ qN, XI ∈ qI, so P is an isometry. Moreover, since C(XN + XI) = CN(XN) +CI(XI), for all XN ∈ qN, XI ∈ qI and C ′ (X ′ N + X ′ I ) = C′ N(X ′ N ) + C′ I(X ′ I ), for all X ′ N ∈ qN, X ′ I ∈ qI, we conclude C ′ = λ P CP−1 and finally, g i g ′ , by Corollary 2.2.31. Remark 2.2.53. The class of solvable singular quadratic Lie algebras has the remarkable property that two Lie algebras in this class are isomorphic if and only if they are i-isomorphic. In addition, the Fitting components do not depend on the realizations of the Lie algebra as a double extension and they completely characterize the Lie algebra (up to isomorphisms). It results that the classfification of Ss(n + 2) can be reduced from the classification of O(n)orbits of o(n) in Chapter 1. We set an action of the group C ∗ on D(n) by: µ · ([d],T ) = ([d], µ · T ) , ∀ µ ∈ C ∗ , ([d],T ) ∈ D(n). Then, we have the classification result of Ss(n + 2) as follows: Theorem 2.2.54. The set Ss(n + 2) is in bijection with D(n)/C ∗ . Proof. By Theorems 2.2.35 and 2.2.52, there is a bijection between Ss(n + 2) and P 1 (o(n)). It needs only to show that there is a bijection between P 1 (o(n)) and D(n)/C ∗ . Let C ∈ o(n) then 46
2.2. Singular quadratic Lie algebras there is the bijection p mapping C onto a pair ([d],T ) in D(n) by Proposition 1.5.1. Moreover, CN and µCN have the same partition so p(µC) = ([d], µ · T ) = µ · p(C). Therefore, the map p induces a bijection p : P1 (o(n)) → D(n)/C∗ given by p([C]) = [([d],T )], if [C] is the class of C ∈ o(n) and [([d],T )] is the class of ([d],T ) ∈ D(n). The result follows. 47
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Université de Bourgogne - Dijon UF
Page 4 and 5:
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
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 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
Page 113 and 114: 3.6. Quasi-singular quadratic Lie s
Page 115 and 116: 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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