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

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Contents 3.4.2 Quadratic dimension of reduced quadratic Lie superalgebras with 2dimensional even part . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.5 Singular quadratic Lie superalgebras of type S1 with non-Abelian even part . . 87 3.6 Quasi-singular quadratic Lie superalgebras . . . . . . . . . . . . . . . . . . . . 92 4 Pseudo-Euclidean Jordan algebras 97 4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.2 Jordanian double extension of a quadratic vector space . . . . . . . . . . . . . 104 4.2.1 Nilpotent double extensions . . . . . . . . . . . . . . . . . . . . . . . 105 4.2.2 Diagonalizable double extensions . . . . . . . . . . . . . . . . . . . . 106 4.3 Pseudo-Euclidean 2-step nilpotent Jordan algebras . . . . . . . . . . . . . . . 108 4.3.1 2-step nilpotent Jordan algebras . . . . . . . . . . . . . . . . . . . . . 108 4.3.2 T ∗ -extensions of pseudo-Euclidean 2-step nilpotent . . . . . . . . . . . 112 4.4 Symmetric Novikov algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Appendix A 129 Appendix B 133 Appendix C 135 Appendix D 139 Bibliography 141 Index 145 v
Introduction In this thesis, we study Lie algebras, Lie superalgebras, Jordan algebras and Novikov algebras equipped with a non-degenerate associative bilinear form. Such algebras are considered over the field of complex numbers and finite-dimensional. We add the condition that the bilinear form is symmetric or even, supersymmetric in the graded case. We call them respectively quadratic Lie algebras, quadratic Lie superalgebras, pseudo-Euclidean Jordan algebras and symmetric Novikov algebras. Let g be a finite-dimensional algebra over C and (X,Y ) ↦→ XY be its product. A bilinear form B : g × g → C is called associative (or invariant) if it satisfies: B(XY,Z) = B(X,Y Z) for all X,Y,Z in g and non-degenerate if B(X,g) = 0 implies X = 0. Such a bilinear form has arisen in several areas of Mathematics and Physics. It can be seen as a generalization of the Killing form on a semisimple Lie algebra, the inner product of an Euclidean Jordan algebra or simply, as the Frobenius form of a Frobenious algebra. The associativity of a bilinear form also can be found in the conditions of an admissible trace function defined on a power-associative algebra. For details, the reader can refer to a paper by M. Bordemann [Bor97]. We begin with a quadratic Lie algebra g and its product, the bracket [ , ]. A result in the work of G. Pinczon and R. Ushirobira [PU07] leads to our first problem: define the 3-form I on g by I(X,Y,Z) = B([X,Y ],Z) for all X,Y,Z in g. Then I satisfies {I,I} = 0 where { , } is the super-Poisson bracket defined on A (g), the Grassmann algebra of skew-symmetric multilinear forms on g by: {Ω,Ω ′ n k+1 } = (−1) ∑ ιXj j=1 (Ω) ∧ ιXj (Ω′ ), ∀ Ω ∈ A k (g),Ω ′ ∈ A (g) in a fixed orthonormal basis {X1, ..., Xn} of g. In this case, the element I is called the 3-form associated to g. Conversely, given a quadratic vector space (g,B) and a non-zero 3-form I on g such that {I,I} = 0, then there is a non-Abelian quadratic Lie algebra structure on g such that I is the 3-form associated to g. By a classical result in a N. Bourbaki’s book [Bou58] that is also recalled in Proposition 2.2.3, we set the following vector space: VI = {α ∈ g ∗ | α ∧ I = 0} The dup-number dup(g) of a non-Abelian quadratic Lie algebra g is defined by dup(g) = dim(VI). It measures the decomposability of the 3-form I and its range is {0,1,3}. For instance, I is decomposable if and only if dup(g) = 3 and then the corresponding quadratic Lie algebra vi
Page 1: Université de Bourgogne - Dijon UF
Page 4 and 5: Remerciements Je remercie l’Ambas
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 44 and 45: If l = k then we stand at (k − 1)
Page 48 and 49: Consider Then dim(g) = 6 ∑ i=3 2.
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2.2. Singular quadratic Lie algebra
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(2) g g ′ if and only if g i g
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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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