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

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1.5. Adjoint orbits in the general case 1.5 Adjoint orbits in the general case Let us now classify Iε-adjoint orbits of gε in the general case as follows. Let C be an element in gε and consider the Fitting decomposition of C C m = VN ⊕VI, where VN and VI are stable by C, CN = C|VN is nilpotent and CI = C|VI is invertible. Since C is skew-symmetric, Bε(C k (VN),VI) = (−1) k Bε(VN,C k (VI)) for any k then one has VI = (VN) ⊥ . Also, the restrictions B N ε = Bε|VN×VN and BI ε = Bε|VI×VI are non-degenerate. Clearly, CN ∈ gε(VN) and CI ∈ gε(VI). By Section 1.2 and Section 1.4, CN is attached with a partition [d] ∈ Pε(n) and CI corresponds to a triple T ∈ Jℓ where n = dim(VN), 2ℓ = dim(VI). Let D(m) be the set of all pairs ([d],T ) such that [d] ∈ Pε(n) and T ∈ Jℓ satisfying n + 2ℓ = m. By the preceding remarks, there exists a map p : gε → D(m) . Denote by O(gε) the set of Iε-adjoint orbits of gε then we obtain the classification of O(gε) as follows: Proposition 1.5.1. The map p : gε → D(m) induces a bijection p : O(gε) → D(m). Proof. Let C and C ′ be two elements in gε. Assume that C and C ′ lie in the same Iε-adjoint orbit. It means that there exists an isometry P such that C ′ = PCP−1 . So C ′k P = P Ck for any k in N. As a consequence, P(VN) ⊂ V ′ N and P(VI) ⊂ V ′ . However, P is an isometry then V ′ N = P(VN) and V ′ I = P(VI). Therefore, one has C ′ N = PN CNP −1 N and C′ I = PI CIP −1 I , where PN = P : VN → V ′ N and PI = P : VI → V ′ I are isometries. It implies that CN, C ′ N have the same partition and CI, C ′ I have the same triple. Hence, the map p is well defined. For a pair ([d],T ) ∈ D(m) with [d] ∈ Pε(n) and T ∈ Jℓ, we set a nilpotent map CN ∈ gε(VN) corresponding to [d] as in Section 1.2 and an invertible map CI ∈ gε(VI) as in Proposition 1.4.4 where dim(VN) = n and dim(VI) = 2ℓ. Define C ∈ gε by C(XN + XI) = CN(XN) +CI(XI), for all XN ∈ VN, XI ∈ VI. By construction, p(C) = ([d],T ) and p is onto. To prove p is one-to-one, let C,C ′ ∈ gε such that p(C) = p(C ′ ) = ([d],T ). Keep the above notations, since CN and C ′ N have the same partition then there exists an isometry PN : VN → V ′ N such that C ′ N = PN CN P −1 N . Similarly CI and C ′ I have the same triple and then there exists an isometry PI : VI → V ′ I such that C′ I = PI CI P −1 I . Define P : V → V by P(XN + XI) = PN(XN) + PI(XI), for all XN ∈ VN,XI ∈ VI then P is an isometry and C ′ = P C P−1 . Therefore, p is one-toone. 16 I
Chapter 2 Quadratic Lie algebras In the first part of this chapter, we recall some preliminary definitions and results of quadratic Lie algebras. Next, we study a new invariant under isomorphisms of quadratic Lie algebras that we call dup-number. Moreover, we give a classification of singular quadratic Lie algebras, i.e. those for which the invariant does not vanish. The classification is closely related to O(m)-adjoint orbits of o(m) mentioned in the Chapter 1. To prove these results, we need to fully describe the structure of singular quadratic Lie algebras by properties of super-Poisson bracket defined on the (Z-graded) Grassmann algebra of alternating multilinear forms of an ndimensional quadratic vector space [PU07] and in terms of double extensions ([Kac85], [FS87] and [MR85]). The invariance of dup-number is a consequence of calculating the quadratic dimension of singular quadratic Lie algebras in the reduced case. Another effective method called T ∗ -extension is introduced by M. Bordemann to descibe solvable quadratic Lie algebras [Bor97] and we use it to study the 2-step nilpotent case. Finally, we also obtain a familiar result: the classification of 2-step nilpotent quadratic Lie algebras up to isometrical isomorphisms is equivalent to the classification all of associated 3-forms. 2.1 Preliminaries Definition 2.1.1. A quadratic Lie algebra (g,B) is a vector space g equipped with a nondegenerate symmetric bilinear form B and a Lie algebra structure on g such that B is invariant (that means, B([X,Y ],Z) = B(X,[Y,Z]), for all X, Y , Z ∈ g). Let (g,B) be a quadratic Lie algebra. Since B is non-degenerate and invariant, we have some simple properties of g as follows: Proposition 2.1.2. (1) If I is an ideal of g then I⊥ is also an ideal of g. Moreover, if I is non-degenerate then so is I⊥ and g = I ⊕ I⊥ . Conveniently, in this case we use the notation g = I ⊥ ⊕ I⊥ . (2) Z(g) = [g,g] ⊥ where Z(g) is the center of g. And then dim(Z(g)) + dim([g,g]) = dim(g). 17
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-
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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 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
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3.1. Application of Z × Z2-graded
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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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