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

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N (n), 40 g, 45 P 1 (o(n)), 39 dq(g), 49 k-coboundary, 21 k-cocycle, 21 rank(g), 5 2SN-Jordan algebra, 108 2SN-Lie algebra, 54 2SN-admissible pair, 110 representation, 55, 109 2SN-central extension, 54, 108 2SN-double extension, 110 2SN-representation, 54 2SNPE-Jordan algebra, 110 2SNQ-Lie algebra, 56 Jordan-type Lie algebra, 41 adjoint action, 4 map, 4 orbit, 4 representation, 100 admissible representation, 102 amalgamated product, 33, 78 annihilator, 101 anti-commutative Novikov algebra, 119 associated 3-form, 22 invariant, 68 Jordan algebra, 121 associative, vi, 98 algebra, 120 scalar product, 98 associator, 101 canonical basis, 4 Cartan subalgebra, 12 centromorphism, 49, 85 character matrix, 124 coadjoint representation, 100 commutative Novikov algebra, 120 Index 146 cyclic, 21, 112 decomposable, 98 diagonalizable double extension, 104 quadratic Lie algebra, 41 quadratic Lie superalgebra, 82 differential super-exterior, 67 direct sum, 55, 109 double extension, 19, 33, 52, 78, 88, 103 dup number, 26, 70 elementary quadratic Lie algebra, 27 quadratic Lie superalgebra, 73 Euclidean, 98 Fitting components, 45 decomposition, 16, 45 flexible, 98 Frobenius algebra, 121 generalized double extension, 111 Gerstenhaber, 10 graded Lie algebra, 66 highest weight, 7 i-isomorphic, 18, 69 i-isomorphism, 18, 69 indecomposable, 28, 72 invariant, vi, 17, 51, 67 invertible orbit, 13 quadratic Lie algebra, 43 quadratic Lie superalgebra, 82 isometry group, 4 isotropic, 13 Jacobson representation, 99 Jacobson-Morozov theorem, 8 Jordan algebra, 97 block, 5 decomposition, 9 identity, 97
normal form, 5 Jordan-admissible, 121 Jordan-Novikov algebra, 120 Jordan-type Lie algebra, 43 Lie superalgebra, 84 Kostant, 10 left-annihilator, 101 left-symmetric algebra, 116 Lie super-bracket, 63 Lie super-derivation, 68 Lie-admissible, 121 line of skew-symmetric maps, 32 maximal vector, 6 multiplicity, 5 neutral element, 7 nilnegative element, 7 nilpositive element, 7 nilpotent double extension, 104 nilpotent Jordan-type Lie algebra, viii Lie superalgebra, 83 non-commutative Jordan algebra, 98 non-degenerate, vi, 3, 114 Novikov algebra, 116 nucleus, 102 ordinary quadratic Lie algebra, 26 quadratic Lie superalgebra, 71 partition, 5 Poisson bracket, 62 power-associative algebra, 97 pseudo-Euclidean Jordan algebra, 98 quadratic Z2-graded vector space, 62 dimension, 49 Lie algebra, 17 Lie superalgebra, 67 vector space, 3 quasi-singular, 92 Index 147 reduced, 19, 71, 125 right-annihilator, 101 semi-direct product, 55, 109 singular quadratic Lie algebra, 26 quadratic Lie superalgebra, 71 skew-supersymmetric, 93 skew-symmetric map, 3 structure constants, 124 sub-adjacent Lie algebra, 116 super Z × Z2-Poisson bracket, 62 super-antisymmetric, 62 super-derivation, 63 super-exterior algebra, 62 super-exterior product, 62 super-Poisson bracket, 22 supersymmetric, 62 symmetric Jordan-Novikov algebra, 120 Novikov algebra, 116 symplectic vector space, 3 totally isotropic, 13 trace-admissible, 97 type S1, 26, 28, 71 S3, 26, 35, 71 unital extension, 98 Jordan algebra, 98 weight, 6 weight space, 6 Weyl group, 12
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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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3.6. Quasi-singular quadratic Lie s
Page 115 and 116: 3.6. Quasi-singular quadratic Lie s
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 and 160: Appendix D In the last appendix, we
Page 161 and 162: Bibliography [AB10] I. Ayadi and S.
Page 163 and 164: [Sch55] R. D. Schafer, Noncommutati
Page 165: Index Bε, 3 E(g,B,δ), 53 Iε(V ),
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