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

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1.2. Nilpotent orbits Lie algebra g. We start with a review of the basic results give in [Hum72]. Recall that sl(2) is spanned by 1 0 0 1 0 0 H = , X = , Y = 0 −1 0 0 1 0 and these satisfy the following relations: [H,X] = 2X, [H,Y ] = −2Y and [X,Y ] = H. Let V be a finite-dimensional sl(2)-module. Since H is semisimple then H acts diagonally on V . Therefore, we can decompose V as a direct sum of eigenspaces V λ = {v ∈ V | H.v = λv}, λ ∈ C where the notation H.v denotes H acting on v by the representation. Definition 1.2.2. If V λ = {0} then we call λ a weight of H in V and we call V λ a weight space. Lemma 1.2.3. If v ∈ V λ then X.v ∈ V λ+2 and Y.v ∈ V λ−2. Proof. Since [H,X] = 2X one has H.(X.v) = X.(H.v) + 2X.v = (λ + 2)X.v. So X.v ∈ Vλ+2. And this is done similarly for Y . Since V = Vλ and dim(V ) is finite then there must exist Vλ = {0} such that Vλ+2 = {0}. λ In this case, each non-zero x ∈ V λ is called a maximal vector of weight λ (note that X.v = 0 if v is a maximal vector). Now, we assume that V is an irreducible sl(2)-module. Choose a maximal vector, say v0 ∈ V λ . Set v−1 = 0, vi = 1 i! Y i .v0 (i ≥ 0). Then one has the following lemma. Lemma 1.2.4. (1) H.vi = (λ − 2i)vi, (2) Y.vi = (i + 1)vi+1, (3) X.vi = (λ − i + 1)vi−1, (i ≥ 0). Proof. (1) One has H.vi = 1 i! HY i .v0 = 1 i! (Y H − 2Y )Y i−1 .v0. Since H.v0. = λv0, Y.vi−1 = ivi and by induction on i, we get H.vi = (λ − 2i)vi. (2) It follows from the definition of vi. (3) We prove (3) by induction on i. If i = 0, it is clear since vi−1 = 0 and X.v0 = 0. If i > 0, one has iX.vi = XY.vi−1 = [X,Y ].vi−1 +Y X.vi−1 = H.vi−1 +Y X.vi−1. By (1), (2) and induction, we obtain iX.vi = (λ − 2(i − 1))vi−1 + (λ − i + 2)Y.vi−2 = i(λ − i + 1)vi−1. 6
1.2. Nilpotent orbits By induction, it is easy to show that the non-zero vi are all linearly independent. Since dim(V ) is finite then there must exist the smallest m such that vm = 0 and vm+1 = 0, obviously vm+i = 0 for all i > 0. Therefore, the subspace of V spanned by vectors v0,...,vm is a sl(2)module. Since V is irreducible then V = span{v0,...,vm}. Moreover, the formula (3) shows λ = m by checking with i = m + 1. It means that the weight λ of a maximal vector is a nonnegative integer (equal to dim(V ) − 1) and we call it the highest weight of V . Conversely, for arbitrary m ≥ 0, formulas (1)- (3) of Lemma 1.2.4 can be used to define a representation of sl(2) on an m+1-dimensional vector space with a basis {v0,...,vm}. Moreover, it is easy to check that this representation is irreducible and then we have the following corollary: Corollary 1.2.5. (1) Let V be an irreducible sl(2)-module then V is the direct sum of its weight spaces Vµ, µ = m,m − 2,...,−(m − 2),−m where m = dim(V ) − 1 and dim(Vµ) = 1 for each µ. (2) For each integer m ≥ 0, there is (up to isomorphisms) one irreducible sl(2)-module of dimension m + 1. Next, let g be a complex semisimple Lie algebra. If there is a subalgebra of g isomorphic to sl(2) and spanned by {H,X,Y } then we called {H,X,Y } a sl(2)-triple of g. In this case, the triple {H,X,Y } satisfies the bracket relations: [H,X] = 2X, [H,Y ] = −2Y and [X,Y ] = H. We call H (resp. X, Y ) the neutral (resp. nilpositive, nilnegative) element of the triple {H,X,Y }. Since ad(H) is semisimple in the subalgebra a = span{H,X,Y } of g then it is known that H is also semisimple in g. Similarly, X,Y are nilpotent in g. Fix an integer r ≥ 0 and define a linear map ρr : sl(2) → sl(r + 1) by ⎛ r ⎜ ⎜0 ⎜ ρr(H) = ⎜ . ⎝0 0 r − 2 . 0 0 0 . 0 ... ... . .. ... 0 0 . −r + 2 ⎞ 0 0 ⎟ . ⎟, ⎟ 0 ⎠ 0 0 0 ... 0 −r ⎛ 0 ⎜ ⎜0 ⎜ ρr(X) = ⎜ . ⎝0 1 0 . 0 0 1 . 0 ... ... . .. ... 0 0 . 0 ⎞ 0 0 ⎟ . ⎟, ⎟ 1⎠ 0 0 0 ... 0 0 ⎛ 0 ⎜ ⎜µ1 ⎜ ρr(Y ) = ⎜ . ⎝ 0 0 0 . 0 0 0 . 0 ... ... . .. ... 0 0 . 0 ⎞ 0 0 ⎟ . ⎟, ⎟ 0⎠ 0 0 0 ... µr 0 7
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 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
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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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TH`ESE A NEW INVARIANT OF QUADRATIC LIE ALGEBRAS AND ...

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