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

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2.4. 2-step nilpotent quadratic Lie algebras (1) Let A : g → g ′ be a Lie algebra isomorphism. Since h ∗ = Z(g) = Z(g ′ ) is stable by A then there exist linear maps A1 : h → h, A ′ 1 : h → h∗ and A2 : h ∗ → h ∗ such that A(x + f ) = A1(x) + A ′ 1(x) + A2( f ), ∀ x ∈ h, f ∈ h ∗ . It is obvious that A2 is an isomorphism of h ∗ . We show that A1 is also an isomorphism of h. Indeed, if there exists x0 ∈ h such that A1(x0) = 0 then 0 = [A(x0),g ′ ] ′ = A([x0,A −1 (g ′ )]) = A([x0,g]). It means that [x0,g] = 0. Since h ∗ = Z(g) then x0 = 0. Therefore A1 is an isomorphism of h. For all x,y ∈ h, f ,g ∈ h ∗ , one has A([x + f ,y + g]) = A(θ1(x,y)) = A2(θ1(x,y)). and [A(x + f ),A(y + g)] ′ = θ2(A1(x),A1(y)). Therefore, A2(θ1(x,y)) = θ2(A1(x),A1(y)) for all x,y ∈ h. Conversely, if there exist an isomorphism A1 of h and an isomorphism A2 of h ∗ such that A2(θ1(x,y)) = θ2(A1(x),A1(y)) for all x,y ∈ h, we define A : g → g ′ by A(x+ f ) = A1(x)+ A2( f ) for all x + f ∈ g. Then it is easy to check that A is a Lie algebra isomorphism. (2) Asumme A : g → g ′ is an i-isomorphism then there exist A1 and A2 defined as in (1). Let x ∈ h, f ∈ h ∗ , one has B ′ (A(x),A( f )) = B(x, f ) ⇒ A2( f )(A1(x)) = f (x). Therefore, A2( f ) = f ◦ A −1 1 , for all f ∈ h∗ . On the other hand, since A2(θ1(x,y)) = θ2(A1(x),A1(y)) we obtain θ1(x,y) = θ2(A1(x),A1(y)) ◦ A1, ∀ x,y ∈ h. Conversely, define A(x + f ) = A1(x) + f ◦ A −1 1 , for all x ∈ h, f ∈ h∗ then A is an iisomorphism. Example 2.4.17. We keep the notations as above. Let g ′ be the T ∗ -extension of h by θ ′ = λθ where λ = 0. Then g and g ′ are i-isomorphic by A : g → g ′ defined by A(x + f ) = 1 3√ λ x + 3√ λ f , for all x + f ∈ g. For a non-degenerate cyclic skew-symmetric bilinear map θ of h, define the 3-form I(x,y,z) = θ(x,y)z, ∀ x,y,z ∈ h. Then I ∈ A 3 (h). The non-degenerate condition of θ is equivalent to ιx(I) = 0 for all x ∈ h\{0}. Conversely, let h be a vector space and I ∈ A 3 (h) such that ιx(I) = 0 for every non-zero vector x ∈ h. Define θ : h × h → h∗ by θ(x,y) = I(x,y,.), for all x,y ∈ h then θ is skew-symmetric and non-degenerate. Moreover, since I is alternating one has θ is cyclic and then we obtain a reduced 2SNQ-Lie algebra T ∗ θ (h) defined by θ. Therefore, there is a one-to-one map from the set of all T ∗-extension of h onto the subset {I ∈ A 3 (h) | ιx(I) = 0, ∀ x ∈ h \ {0}}. We has a corollary of the above theorem as follows: 58
2.4. 2-step nilpotent quadratic Lie algebras Corollary 2.4.18. Let g and g ′ be T ∗ -extensions of h with respect to I1 and I2. Then g and g ′ are i-isomorphic if and only if there exists an isomorphism A of h such that I1(x,y,z) = I2(A(x),A(y),A(z)), for all x,y,z ∈ h. Remark 2.4.19. The element I in this case is exactly the 3-form associated to g in Definition 2.2.2 and the above corollary is only a particular case of Lemma 2.2.8. Lemma 2.4.20. Let h be a vector space and I ∈ A 3 (h) satisfying ιx(I) = 0, ∀ x ∈ h \ {0}}. If there are nontrivial subspaces h1, h2 of h such that h = h1 ⊕ h2 and I is decomposed by I = I1 + I2 where I1 ∈ A 3 (h1), I2 ∈ A 3 (h2). Then the T ∗ -extension g of h with respect to I is decomposable. Proof. Let a = h1 ⊕ h ∗ 1 be the T ∗ -extension of h1 with respect to I1 then a is non-degenerate. We will show that a is an ideal of g. Indeed, one has: [h1,h1 ⊕ h2] = I(h1,h1 ⊕ h2,.) = I(h1,h1,.) + I(h1,h2,.). Since I(h1,h2,.) = 0 then [h1,h1 ⊕ h2] = I(h1,h1,.) = I1(h1,h1,.) ⊂ h∗ 1 . Therefore a is an ideal of g and then g is decomposable. Remark 2.4.21. Denote by N(2n) the set of i-isomorphism classes of 2n-dimensional reduced 2SNQ-Lie algebras. It is obvious that N(2) = N(4) = /0 and N(6) has only an element (see Appendix C and Example 2.4.17, also in [PU07] or [Ova07]). By Appendix C and Lemma 2.4.20, N(8) = /0, N(10) contains only an element and N(2n) = /0 if n ≥ 6. 59
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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
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Introduction and obtain the followi
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y Introduction We realize that with
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Introduction by I. Bajo, S. Benayad
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Introduction (2) there exists an i-
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xix
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Notations 0 Ik where J = ∈ gl(2
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1.1. Definitions and gε(V ) = {C
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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 76 and 77: 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
Page 117 and 118: Chapter 4 Pseudo-Euclidean Jordan a
Page 119 and 120: 4.1. Preliminaries Proof. The asser
Page 121 and 122: 4.1. Preliminaries T (x,y) = φ(x)(
Page 123 and 124: 4.1. Preliminaries (3) π(xx ′ )y
Page 125 and 126: 4.2. Jordanian double extension of
Page 127 and 128: 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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