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

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Introduction (2) there exists an i-isomorphism between J1 and J2 if and only if there exists an isomorphism A1 of a satisfying θ1(x,y) = θ2(A1(x),A1(y)) ◦ A1, ∀ x,y ∈ a. As a consequence, the classification of i-isomorphic T ∗ -extensions of a is equivalent to the classification of symmetric 3-forms on a. We will detail it in the cases of dim(a) = 1 and 2 (Examples 4.3.18 and 4.3.19). In the last of Chapter 4, we study Novikov algebras. These objects appeared in the study of the Hamiltonian condition of an operator in the formal calculus of variations (see the paper by I. M. Gel’fand and I. Y. Dorfman [GD79]) as well as in the classification of Poisson brackets of hydrodynamic type (done by A. A. Balinskii and S. P. Novikov in [BN85]). A detailed classification of Novikov algebras up to dimension 3 is given by C. Bai and D. Meng in [BM01]. It is known that an associative algebra is both Lie-admissible and Jordan-admissible. This is not true for Novikov algebras although they are Lie-admissible. Therefore, it is natural to search a condition for a Novikov algebra to become Jordan-admissible. The condition we give here (weaker than associativity) is the following (Theorem 4.4.17): THEOREM 17: A Novikov algebra N is Jordan-admissible if it satisfies the condition (x,x,x) = 0, ∀ x ∈ N. A corollary of Theorem 17 is that Novikov algebras are not power-associative since there exist Novikov algebras not Jordan-admissible. Next, we consider symmetric Novikov algebras. In this case, N will be associative, its subadjacent Lie algebra g(N) is a 2-step nilpotent quadratic Lie algebra (shown in the paper [AB10] of I. Ayadi and S. Benayadi) and the associated Jordan algebra J(N) is pseudo-Euclidean. Therefore, the study of 2-step nilpotent quadratic Lie algebras and pseudo-Euclidean Jordan algebras is closely related to symmetric Novikov algebras. It is known that every symmetric Novikov algebra up to dimension 5 is commutative [AB10] and a non-commutative example of dimension 6 is given by F. Zhu and Z. Chen in [ZC07]. This algebra is 2-step nilpotent. We will show in Proposition 4.4.28 that every non-commutative symmetric Novikov algebra of dimension 6 is 2-step nilpotent. As for quadratic Lie algebras and pseudo-Euclidean Jordan algebras, we define the notion of a reduced symmetric Novikov algebra. Using this notion, we obtain the result (Proposition 4.4.29): if N is a non-commutative symmetric Novikov algebra such that it is reduced then 3 ≤ dim(Ann(N)) ≤ dim(N 2 ) ≤ dim(N) − 3. In other words, we do not have N 2 = N in the non-commutative case . Note that this may happen in the commutative case (see Example 4.4.13). As a consequence, we have a characterization for the non-commutative case of dimension 7 (Proposition 4.4.31). Finally, we give an indecomposable non-commutative example for 3-step nilpotent symmetric Novikov algebras of dimension 7. xvii
Introduction The thesis has been divided into four parts. The classifications of O(m)-adjoint orbits of o(m) and Sp(2n)-adjoint orbits of sp(2n) are presented in Chapter 1. Singular quadratic Lie algebras and 2-step nilpotent quadratic Lie algebras are studied in Chapter 2. We will prove the equality {I,I} = 0 and introduce the class of singular quadratic Lie superalgebras in Chapter 3. However, since the classifying method is not new, we only focus on two cases: elementary and 2-dimensional even part. The classification of singular quadratic Lie algebras and singular quadratic Lie superalgebras having 2-dimensional even part can be regarded as an application of the problem of orbits classification in Chapter 1. We present quasi-singular quadratic Lie superalgebras without classification in Chapter 3. Such algebras can be found in [BBB] where the generalized double extension notion is reduced into the one-dimensional extension of an Abelian superalgebra. Pseudo-Euclidean Jordan algebras that are the one-dimensional double extension of an Abelian algebra and 2-step nilpotent pseudo-Euclidean Jordan algebras are given in Chapter 4 with a classifying characterization. The structure of symmetric Novikov algebras is studied in the last section of Chapter 4 with a more detail than in [AB10]. There are four appendices containing rather obvious and lengthy results but yet useful for our problems. Appendix A supplies a source about skew-symmetric maps for Chapter 1 and Chapter 2. Appendix B gives a non trivial proof of a fact that every non-Abelian 5-dimensional quadratic Lie algebra is singular. Another proof can be found in Appendix C where we classify (up to isomorphisms) 3-forms on a vector space V with 1 ≤ dim(V ) ≤ 5. Appendix D is a small result used in Chapter 4 for pseudo-Euclidean Jordan algebras. xviii
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
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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.
Page 66 and 67: (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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