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

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Notations 0 Ik where J = ∈ gl(2k), Ik denotes the unit matrix of size k. From here, we use the −Ik 0 notations sl(n), o(n) and sp(2k) instead of sl(n,C), o(n,C), sp(2k,C), respectively. We also denote by P1 (o(n)) and P1 (sp(2k)) the projective spaces associated to o(n) and sp(2k). For complex numbers λ1,...,λn, the notation diagn (λ1,...,λn) indicates the diagonal matrix whose diagonal entries are λ1,...,λn, respectively. Finally, if g is a finite dimensional algebra over K then we denote the algebra of derivations of g by Der(g). Recall that an endomorphism D : g → g is called a derivation of g if it satisfies D(XY ) = D(X)Y + XD(Y ) for all X,Y in g. 2
Chapter 1 Adjoint orbits of sp(2n) and o(m) In the first chapter, we will turn to a fundamental and really interesting problem in Lie theory: the classification of orbits of classical Lie algebras sl(m), sp(2n) and o(m) where m,n ∈ N ∗ . However, we only emphasize two special cases Sp(2n)-adjoint orbits of sp(2n) and O(m)adjoint orbits of o(m) that are necessary for next chapters. The Jordan decomposition allows us to consider two kinds of orbits: nilpotent and semisimple (represented respectively by diagonal matrices and strictly upper triangular matrices). The classfication of nilpotent orbits that we present here follows the work of Gerstenhaber and an important point is the Jacobson-Morozov theorem. In the semisimple case, a well-known result is that the orbits are parametrized by a Cartan subalgebra under an action of the associated Weyl group. A brief overview can be found in [Hum95] with interesting discussions. Many results with detailed proofs can be found in [CM93] and [BBCM02]. A different point here is to use the Fitting decomposition to review this problem. In particular, we parametrize the invertible component in the Fitting decomposition of a skew-symmetric map and from this, we give an explicit classification for Sp(2n)-adjoint orbits of sp(2n) and O(m)-adjoint orbits of o(m) in the general case. In other words, we establish a one-to-one correspondence between the set of orbits and some set of indices. This is an rather obvious and classical result but in our knowledge there is not a reference for that mentioned before. 1.1 Definitions Let V be a m-dimensional complex vector space endowed with a non-degenerate bilinear form Bε where ε = ±1 such that Bε(X,Y ) = εBε(Y,X), for all X,Y ∈ V . If ε = 1 then the form B1 is symmetric and we say V a quadratic vector space. If ε = −1 then m must be even and we say V a symplectic vector space with symplectic form B−1. We denote by L (V ) the algebra of linear operators of V and GL(V ) the group of invertible operators in L (V ). A map C ∈ L (V ) is called skew-symmetric (with respect to Bε) if it satisfies the following condition: We define Bε(C(X),Y ) = −Bε(X,C(Y )), ∀ X,Y ∈ V. Iε(V ) = {A ∈ GL(V ) | Bε(A(X),A(Y )) = Bε(X,Y ), ∀ X,Y ∈ V } 3
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 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 70 and 71: 2.3. Quadratic dimension of quadrat
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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
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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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