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

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Introduction and Di red (n + 2) the corresponding sets of isomorphism and i-isomorphism classes of elements in D(n + 2) and Dred(n + 2). It is clear by Theorem 2 that Di (n + 2) is in bijection with the well-known set of semisimple O(n)-orbits in P1 (o(n)). More precisely, we have the following result (Proposition 2.2.40, Corollary 2.2.43 and Proposition 2.2.44): THEOREM 4: (1) There is a bijection between D i (n+2) and the set of semisimple O(n)-orbits in P 1 (o(n)). The same result holds for D i red (n + 2) and semisimple invertible orbits in P1 (o(n)). (2) Let g and g ′ be inDred(n+2). Then n must be even and g i g ′ if and only if g g ′ . Thus, Dred(2p + 2) = Di red (2p + 2), for all p ≥ 1. (3) Let (g,B) be a diagonalizable reduced singular quadratic Lie algebra. Consider g4 the double extension of C2 1 0 by C = . Then g is an amalgamated product of singular 0 −1 quadratic Lie algebras all i-isomorphic to g4. Combined with the classification of semisimple O(n)-adjoint orbits of o(n) in Chapter 1, the sets D(2p + 2) and Dred(2p + 2) can be parametrized as in Theorem 2.2.41 where the set D(2p + 2) is in bijection with Λp/Hp and the set Dred(2p + 2) is in bijection with Λ + p /Hp (see Section 1.3 and Subsection 2.2.5 for the respective definitions). Clearly, the parametrization of i-isomorphic classes of nilpotent or diagonalizable singular quadratic Lie algebras can be regarded as a direct corollary of the classification of nilpotent or semisimple O(n)-adjoint orbits of o(n). However, we do not find any reference that shows how to parametrize O(n)-adjoint orbits of o(n) in the general case. Therefore, our next objective is to determine such a parametrization. This solution allows us to go further in the classification of singular quadratic Lie algebras. We continue with the notion of an invertible singular quadratic Lie algebra (i.e. C is invertible). Let Sinv(2p+2) be the set of such structures on C2p+2 and Sinv(2p+2) be the set of isomorphism classes of elements in Sinv(2p + 2). The isomorphic and i-isomorphic notions coincide in the invertible case as we show in Lemma 2.2.42. Moreover, a description of invertible singular quadratic Lie algebras in term of amalgamated product can be found in Proposition 2.2.49. The classification of the set Sinv(2p + 2) is deduced from Theorem 6 below. We turn our attention to the general case. Given a solvable singular quadratic Lie algebra g, realized as the double extension of Cn by C ∈ o(n), we consider the Fitting components CI and CN of C and the corresponding double extensions gI = g CI and gN = g CN that we call the Fitting components of g. We have gI invertible, gN nilpotent and g is the amalgamated product of gI and gN. We prove that (Theorem 2.2.52): THEOREM 5: Let g and g ′ be solvable singular quadratic Lie algebras and let gN, gI, g ′ N , g′ I Fitting components, respectively. Then: ix be their
(1) g i g ′ if and only if ⎧ ⎨ ⎩ i gN g ′ N i gI g ′ I . Introduction The result remains valid if we replace i by . (2) g g ′ if and only if g i g ′ . Therefore Ss(n + 2) = i Ss (n + 2). Theorem 5 is a really interesting and unexpected property of solvable singular quadratic Lie algebras. From the above facts, in order to describe more precisely the set Ss(n + 2), we turn to the problem of classification of O(n)-adjoint orbits in o(n). Since the study of the nilpotent orbits is complete, we begin with the invertible case. Let I (n) be the set of invertible elements in o(n) and I (n) be the set of O(n)-adjoint orbits of elements in I (n). Notice that I (2p + 1) = /0 (Appendix A) then we consider n = 2p. Define the set D = {(d1,...,dr) ∈ N r | d1 ≥ d2 ≥ ··· ≥ dr ≥ 1} r∈N ∗ and the map Φ : D → N defined by Φ(d1,...,dr) = ∑ r i=1 di. We introduce the set Jp of all triples (Λ,m,d) such that: (1) Λ is a subset of C \ {0} with ♯Λ ≤ 2p and λ ∈ Λ if and only if −λ ∈ Λ. (2) m : Λ → N∗ satisfies m(λ) = m(−λ), for all λ ∈ Λ and ∑ m(λ) = 2p. λ∈Λ (3) d : Λ → D satisfies d(λ) = d(−λ), for all λ ∈ Λ and Φ ◦ d = m. To every C ∈ I (2p), we can associate an element (Λ,m,d) of Jp as follows: write C = S + N as a sum of its semisimple and nilpotent parts. Then Λ is the spectrum of S, m is the multiplicity map on Λ and d gives the size of the Jordan blocks of N. Therefore, we obtain a map i : I (2p) → Jp and we prove: THEOREM 6: The map i : I (2p) → Jp induces a bijection from I (2p) onto Jp. As a corollary, we deduce a bijection from Sinv(2p + 2) onto Jp/C ∗ (Theorem 2.2.50) where the action of µ ∈ C ∗ = C \ {0} on Jp is defined by µ · (Λ,m,d) = (µΛ,m ′ ,d ′ ), with m ′ (µλ) = m(λ) and d ′ (µλ) = d(λ), ∀ λ ∈ Λ. Combined with the previous theorems, we obtain a complete classification of Ss(n + 2) as follows. Let D(n) be the set of all pairs ([d],T ) such that [d] ∈ P1(m), the set of partitions of m in which even parts occur with even multiplicity, and T ∈ Jℓ satisfying m + 2ℓ = n. We set an action of the multiplicative group C ∗ on D(n) by: µ · ([d],T ) = ([d], µ · T ) , ∀ µ ∈ C ∗ , ([d],T ) ∈ D(n). x
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 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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