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

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4.1. Preliminaries (2) SxSySz + SzSySx + S (xz)y = SxSyz + SySzx + SzSxy. Remark 4.1.9. An equivalent definition of a representation S of J can be found for instance in [BB], as a necessary and sufficient condition for the vector space J1 = J ⊕V equipped with the product: (x + u)(y + v) = xy + Sx(v) + Sy(u), ∀x,y ∈ J,u,v ∈ V to be a Jordan algebra. That is: (1) S x 2Sx − SxS x 2 = 0, (2) 2SxySx + S x 2Sy − 2SxSySx − S x 2 y = 0, for all x,y ∈ J. In this case, Jacobson’s definition is different from the usual definition of representation, that is, as a homomorphism from J into the Jordan algebra of linear maps. For x ∈ J, let Rx ∈ End(J) be the endomorphism of J defined by: Rx(y) = xy = yx, ∀ y ∈ J. Then the Jordan identity is equivalent to [Rx,R x 2] = 0, for all x ∈ J where [·,·] denotes the Lie bracket on End(J). The linear maps R : J → End(J) with R(x) = Rx and R ∗ : J → End(J ∗ ) with R ∗ (x)( f ) = f ◦ Rx, ∀ x ∈ J, f ∈ J ∗ , are called respectively the adjoint representation and the coadjoint representation of J. It is easy to check that they are indeed representations of J. Recall that there exists a natural nondegenerate bilinear from 〈·,·〉 on J ⊕ J ∗ defined by 〈x, f 〉 = f (x), for all x ∈ J, f ∈ J ∗ . For all x,y ∈ J, f ∈ J ∗ , one has: f (xy) = 〈xy, f 〉 = 〈Rx(y), f 〉 = 〈y,R ∗ x( f )〉. That means that R ∗ x is the adjoint map of Rx with respect to the bilinear form 〈·,·〉. The following proposition gives a characterization of pseudo-Euclidean Jordan algebras ([BB], Proposition 2.1 or [Bor97], Proposition 2.4.) Proposition 4.1.10. Let J be a Jordan algebra. Then J is pseudo-Euclidean if and only if its adjoint representation and coadjoint representation are equivalent. Proof. Assume that (J,B) is a pseudo-Euclidean Jordan algebra. We define the map φ : J → J ∗ by φ(x) = B(x,.), for all x ∈ J. Since B is non-degenerate, φ is an isomorphism from J onto J ∗ . Moreover, it satisfies φ (Rx(y))(z) = B(xy,z) = B(y,xz) = (R ∗ (x)φ(y))(z). That means φ ◦ Rx = R ∗ (x) ◦ φ, for all x ∈ J. Therefore, the representations R and R ∗ are equivalent. Conversely, assume that R and R ∗ are equivalent then there exists an isomorphism φ : J → J ∗ such that φ ◦ Rx = R ∗ (x) ◦ φ, for all x ∈ J. Set the bilinear form T : J × J → C defined by 100
4.1. Preliminaries T (x,y) = φ(x)(y), for all x,y ∈ J then T is non-degenerate. Moreover, it is easy to check that T is also associative since R and R ∗ are equivalent by φ. However, T is not necessarily symmetric. Then we need construct a symmetric bilinear form that is still non-degenerate and associative from T as follows. Define the symmetric (resp. skew-symmetric) part Ts (resp. Ta) by Ts(x,y) = 1 2 (T (x,y) + T (y,x)) (resp. Ta(x,y) = 1 (T (x,y) − T (y,x))), ∀ x,y ∈ J. 2 By straightforward checking, Ts and Ta are also associative. Consider subspaces Js = {x ∈ J | Ts(x,J) = 0} and Ja = {x ∈ J | Ta(x,J) = 0}. If x ∈ Js ∩ Ja then T (x,J) = Ts(x,J) + Ta(x,J) = 0. So x = 0 since T is non-degenerate. It means Js ∩ Ja = {0}. Moreover, Js and Ja are also ideals of J since Ts and Ta are associative. Now, for all x,y,z ∈ J, Ta(xy,z) = Ta(x,yz) = −Ta(xy,z). Therefore, J 2 ⊂ Ja and then J 2 s ⊂ Js ∩ Ja = {0}. Let J = W ⊕ Js where W is a complemenary subspace of Js in J. It is obvious that Ja ⊂ W. Therefore, WW ⊂ J 2 ⊂ W and WJs = {0}. Consider F : Js × Js → C be a nondegenerate symmetric bilinear form on Js. Since J 2 s = {0} then F is associative. Finally, we define the bilinear form B : J × J → C by: B|W×W = Ts|W×W , B|Js×Js = F and B(W,Js) = B(Js,W) = 0. Let x = xw +xs,y = yw +ys,z = zw +zs ∈ W ⊕Js. Remark that Ts(Js,J) = 0 so if Ts(xw,W) = 0 then Ts(xw,J) = 0. It implies xw ∈ Js so xw = 0. One has B(xw + xs,J) = 0 if and only if Ts(xw,W) = 0 and F(xs,Js) = 0. By preceding remark and F non-degenerate on Js then xw = xs = 0. It means B non-degenerate. It is easy to see that Hence, B is associative. B((xw + xs)(yw + ys),zw + zs) = Ts(xwyw,zw) = Ts(xw,ywzw) = B(xw + xs,(yw + ys)(zw + zs)) We will need some special subspaces of an arbitrary algebra J: Definition 4.1.11. Let J be an algebra. (1) The subspace is the associator of J. (2) The subspaces (J,J,J) = span{(x,y,z) | x,y,z ∈ J} LAnn(J) = {x ∈ J | xJ = 0}, RAnn(J) = {x ∈ J | Jx = 0} and Ann(J) = {x ∈ J | xJ = Jx = 0} are respectively the left-annihilator, the right-annihilator and the annihilator of J. Certainly, if J is commutative then these subspaces coincide. 101
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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
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1.2. Nilpotent orbits where µi = i
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1.2. Nilpotent orbits The point of
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1.3 Semisimple orbits We recall the
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1.4. Invertible orbits Let us now c
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1.5. Adjoint orbits in the general
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2.1. Preliminaries (3) Set the map
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2.1. Preliminaries Proof. Let I be
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2.2. Singular quadratic Lie algebra
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If l = k then we stand at (k − 1)
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Consider Then dim(g) = 6 ∑ i=3 2.
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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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Page 81 and 82: Chapter 3 Singular quadratic Lie su
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Page 117 and 118: Chapter 4 Pseudo-Euclidean Jordan a
Page 119: 4.1. Preliminaries Proof. The asser
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
Page 129 and 130: 4.3. Pseudo-Euclidean 2-step nilpot
Page 133 and 134: Proof. 4.3. Pseudo-Euclidean 2-step
Page 137 and 138: 4.4. Symmetric Novikov algebras (2)
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Page 149 and 150: Appendix A In this appendix, we rec
Page 151 and 152: Appendix A Lemma A.4. Assume that C
Page 153 and 154: Appendix B Here we prove: Lemma B.1
Page 155 and 156: Appendix C We will classify (up to
Page 157 and 158: Appendix C Lemma C.3. The element I
Page 159 and 160: Appendix D In the last appendix, we
Page 161 and 162: Bibliography [AB10] I. Ayadi and S.
Page 163 and 164: [Sch55] R. D. Schafer, Noncommutati
Page 165 and 166: Index Bε, 3 E(g,B,δ), 53 Iε(V ),
Page 167 and 168: normal form, 5 Jordan-admissible, 1
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