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

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Bibliography [AB10] I. Ayadi and S. Benayadi, Symmetric Novikov superalgebras, J. Math. Phys. 51 (2010), no. 2, 023501. ↑xvii, xviii, 119, 120, 123, 124 [ACL95] D. Arnal, M. Cahen, and J. Ludwig, Lie groups whose coadjoint orbits are of dimension smaller or equal to two, Lett. Math. Phys. 33 (1995), no. 2, 183-186. ↑vii [Alb49] A. A. Albert, A theory of trace-admissible algebras, Proc. Natl. Acad. Sci. USA. 35 (1949), no. 6, 317 – 322. ↑98 [BB] A. Baklouti and S. Benayadi, Pseudo-Euclidean Jordan algebras, arXiv:0811.3702v1. ↑xvi, 100, 102, 103, 108 [BB97] I. Bajo and S. Benayadi, Lie algebras admitting a unique quadratic structure, Communications in Algebra 25 (1997), no. 9, 2795 – 2805. ↑xi, 49 [BB99] H. Benamor and S. Benayadi, Double extension of quadratic Lie superalgebras, Comm. in Algebra 27 (1999), no. 1, 67 – 88. ↑ [BB07] I. Bajo and S. Benayadi, Lie algebras with quadratic dimension equal to 2, J. Pure and Appl. Alg. 209 (2007), no. 3, 725 – 737. ↑49, 53 [BBB] I. Bajo, S. Benayadi, and M. Bordemann, Generalized double extension and descriptions of quadratic Lie superalgebras, arXiv:0712.0228v1. ↑xv, xviii, 96 [BBCM02] A. Bialynicki-Birula, J. Carrell, and W. M. McGovern, Algebraic Quotients. Torus Actions and Cohomology. The Adjoint Representation and the Adjoint Action, Encyclopaedia of Mathematical Sciences, vol. 131, Springer-Verlag, Berlin, 2002. ↑3 [Ben03] S. Benayadi, Socle and some invariants of quadratic Lie superalgebras, J. of Algebra 261 (2003), no. 2, 245 – 291. ↑49 [BM01] C. Bai and D. Meng, The classification of Novikov algebras in low dimensions, J. Phys. A: Math. Gen. 34 (2001), 1581 – 1594. ↑xvii [BM02] , Bilinear forms on Novikov algebras, Int. J. Theor. Phys. 41 (2002), no. 3, 495 – 502. ↑ [BMH02] C. Bai, D. Meng, and L He, On fermionic Novikov algebras, J. Phys. A: Math. Gen. 35 (2002), no. 47, 10053 – 10063. ↑121 [BN85] A. A. Balinskii and S. P. Novikov, Poisson brackets of hydrodynamic type, Frobenius algabras and Lie algebras, Dokl. Akad. Nauk SSSR 283 (1985), no. 5, 1036 – 1039. ↑xvii [Bor97] M. Bordemann, Nondegenerate invariant bilinear forms on nonassociative algebras, Acta Math. Univ. Comenianac LXVI (1997), no. 2, 151 – 201. ↑vi, xi, 17, 19, 20, 21, 57, 100 [Bou58] N. Bourbaki, Éléments de Mathématiques. Algèbre, Algèbre Multilinéaire, Vol. Fasc. VII, Livre II, Hermann, Paris, 1958. ↑vi, 23 [Bou59] , Éléments de Mathématiques. Algèbre, Formes sesquilinéaires et formes quadratiques, Vol. Fasc. XXIV, Livre II, Hermann, Paris, 1959. ↑4, 73, 126, 130 141
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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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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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Page 113 and 114: 3.6. Quasi-singular quadratic Lie s
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Page 117 and 118: Chapter 4 Pseudo-Euclidean Jordan a
Page 119 and 120: 4.1. Preliminaries Proof. The asser
Page 121 and 122: 4.1. Preliminaries T (x,y) = φ(x)(
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)
Page 139 and 140: 4.4. Symmetric Novikov algebras Lem
Page 145 and 146: 4.4. Symmetric Novikov algebras Lem
Page 147 and 148: 4.4. Symmetric Novikov algebras Pro
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: Appendix D In the last appendix, we
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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