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

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1.2. Nilpotent orbits where µi = i(r + 1 − i) for 1 ≤ i ≤ r. By Corollary 1.2.5, ρr defines an irreducible representation of sl(2) of highest weight r and dimension r + 1. Moreover, every irreducible finite-dimensional representation of sl(2) arises in this way. Conveniently, for a partition [d] = [d1,...,dk] of n, denote by O [d] the orbit through X [d]. Now, let O be a non-zero nilpotent orbit in sl(n) then there exists a partition [d1,...,dk] of n such that O = O [d1,...,dk]. We define the homomorphism φO : sl(2) → sl(n) by φO = ⊕ ρdi−1. 1≤i≤k 0 1 Then φO(X) = X [d1,...,dk] where X = ∈ sl(2). Note that if O is a non-zero nilpotent 0 0 orbit then the partition [d1,...,dk] does not coincide with [1n ], the image φO is not trivial and it is isomorphic to sl(2). Therefore, to each non-zero nilpotent orbit O = OX in sl(n), we can attach a sl(2)-triple {H,X,Y } such that the nilpositive element is X. More precisely, choose X exactly having the Jordan form and set 1 0 0 1 0 0 H := φO , X := φO and Y := φO . 0 −1 0 0 1 0 This can be done for an arbitrary complex semisimple algebra g, not necessarily sl(n), by the theorem of Jacobson-Morozov as follows. Proposition 1.2.6 (Jacobson-Morozov). Let g be a complex semisimple Lie algebra. If X is a non-zero nilpotent element of g then it is the nilpositive element of a sl(2)-triple. Equivalently, for any nilpotent element X, there 0 1 exists a homomorphism φ : sl(2) → g such that: φ = X. 0 0 Proof. We follow the proof given in [CM93]. First, we prove the following lemma. Lemma 1.2.7. Let g be a complex semisimple Lie algebra and X be a nilpotent element in g. Then one has: (1) κ(X,g X ) = 0 where κ is the Killing form and g X is the centralizer of X in g defined by g X = {Y ∈ g | [X,Y ] = 0}. (2) [g,X] = (g X ) ⊥ where the notation (g X ) ⊥ denotes the orthogonal subspace of g X by κ. Proof. (1) Let Z be an element of g X then by the Jacobi identity, one has ad(X) ◦ ad(Z) = ad(Z) ◦ ad(X). As a consequence, (ad(X) ◦ ad(Z)) k = ad k (X) ◦ ad k (Z) for any k in N. Since X is nilpotent then ad(X) is nilpotent. It implies that ad k (X) = 0 for some k and therefore ad(X) ◦ ad(Z) is nilpotent. That means trace(ad(X) ◦ ad(Z)) = κ(X,Z) = 0. (2) By the invariance of κ, one has κ([g,X],g X ) = κ(g,[X,g X ]) = 0. Hence, [g,X] ⊂ (g X ) ⊥ . Since dim(g) = dim(ker(ad(X)))+dim(Im(ad(X))) then dim(g) = dim(g X )+dim([g,X]). By the non-degeneracy of κ, we obtain [g,X] = (g X ) ⊥ . 8
1.2. Nilpotent orbits We prove the Jacobson-Morozov theorem by induction on the dimension of g. If dim(g) = 3 then g is isomorphic to sl(2) and the result follows. Assume that dim(g) > 3. If X is in a proper semisimple subalgebra of g then by induction, there is a sl(2)-triple with X its nilpositive element. Hence we may assume that X is not in any proper semisimple subalgebra of g. By Lemma 1.2.7 and [X,X] = 0, one has X ∈ (g X ) ⊥ = [g,X]. Therefore, there exists H ′ ∈ g such that [H ′ ,X] = 2X. Now, let H ′ = H ′ s + H ′ n be the Jordan decomposition of H ′ in g where H ′ s is semisimple and H ′ n is nilpotent. Remark that any subspace which is stable by ad(H ′ ) is also stable by ad(H ′ s) and ad(H ′ n). The nilpotency of the ad(H ′ n) action on the stable subspace CX gives [H ′ n,X] = 0 and therefore [H ′ s,X] = 2X. Set H = H ′ s. If H ∈ [g,X] then there exists Y ∈ g such that H = [X,Y ]. Since ad(H) acts semisimply on g then g is decomposed by g = g λ1 ⊕ ··· ⊕ g λk , where g λ1 ,..., g λk are ad(H)-eigenspaces. Write Y = Y λ1 + ··· +Y λk with Yi ∈ g λi . Let Z ∈ g λi then [H,Z] = λiZ. Hence, [X,[H,Z]] = λi[X,Z]. By the Jacobi identity, one has [Z,[H,X]] + [H,[X,Z]] = λi[X,Z]. Then we get ad(H)([X,Z]) = (λi + 2)[X,Z]. It shows that [X,g λi ] ⊂ gλi+2,1 ≤ i ≤ k. Since ad(H)(H) = 0, one has H ∈ g0. Moreover, H = [X,Y ] = ∑ i=1 k [X,Yi]. Therefore, there is some Y ′ ∈ g−2 such that H = [X,Y ′ ]. If we replace Y by Y ′ then [H,X] = 2X, [H,Y ] = −2Y and [X,Y ] = H. That means {H,X,Y } is a sl(2)-triple with X its nilpositive element. From the above reason, it remains to prove H ∈ [g,X]. By contradiction, assume that H /∈ [g,X] then κ(H,g X ) = 0. According to the Jacobi identity [X,[H,g X ]] + [g X ,[X,H]] + [H,[g X ,X]] = 0, one get [X,[H,g X ]] = 0. It implies that ad(H)(g X ) ⊂ g X , i.e. g X is ad(H)-invariant. By acting semisimply of ad(H) on g X , g X is decomposed into ad(H)-eigenspaces: g X = g X τ1 ⊕ ··· ⊕ gXτi = gX0 ⊕ ∑ g τi=0 X τi . From the invariance of the Killing form, one has κ(H,[H,g X ]) = κ([H,H],g X ) = 0. Therefore if Z is a non-zero element of g X τi with τi = 0 then 0 = κ(H,[H,Z]) = κ(H,τiZ) = τiκ(H,Z). This shows that H ∈ (g X τi )⊥ . Since κ(H,g X ) = 0 there must exist Z ∈ g X 0 = {Y ∈ gX | [H,Y ] = 0} = (g X ) H such that κ(H,Z) = 0. If Z is nilpotent then κ(H,Z) = 0 as in the proof of Lemma 1.2.7. Therefore, Z is non-nilpotent. That means its semisimple component Zs = 0 and hence g Zs is reductive. As a consequence, [g Zs ,g Zs ] is a semisimple subalgebra of g and it is a proper subalgebra since g Zs = g only if Zs = 0. On the other hand, since [X,Z] = [H,Z] = 0, apply the property of Jordan decomposition one has [X,Zs] = [H,Zs] = 0. That means Zs ∈ (g X ) H and then X ∈ g Zs . Also, H ∈ g Zs so we get 2X = [H,X] ∈ [g Zs ,g Zs ]. This is a contradiction then we obtain the result. 9
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 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 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
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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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