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

More documents

Recommendations

Info

Introduction structures can be determined completely up to isometrical isomorphisms (or i-isomorphisms, for short) [PU07]. Such Lie algebras appear also in the classification of Lie algebras whose coadjoint orbits of dimension at most 2 (done by D. Arnal, M. Cahen and J. Ludwig [ACL95]). A remarkable point is that the dup-number is invariant by i-isomorphism, that is, two i-isomorphic quadratic Lie algebras have the same dup-number. The first goal of our study is to determine quadratic Lie algebra structures in the case dup(g) = 1. The classification of such structures is one of the aims of this thesis. Here, we want to emphasize that our classification is considered in two senses: up to i-isomorphisms and more strongly, up to isomorphisms. This study is interesting by itself. It allows us to regard two distinguished kinds of classes: quadratic Lie algebras whose dup-number is non-zero and those whose dup-number is zero. We say that a non-Abelian quadratic Lie algebra g is ordinary if dup(g) = 0. Otherwise, g is called singular. By a technical requirement, we separate singular quadratic Lie algebras into two classes: those of type S1 if their dup-number is 1 and of type S3 if their dup-number is 3. For n ≥ 1, let O(n) be the set of ordinary, S(n) be the set of singular and Q(n) be the set of non-Abelian quadratic Lie algebra structures on C n . The distinction of two sets O(n) and S(n) is shown in Theorem 2.2.13 as follows: THEOREM 1: (1) O(n) is Zariski-open and S(n) is Zariski-closed in Q(n). (2) Q(n) = /0 if and only if n ≥ 3. (3) O(n) = /0 if and only if n ≥ 6. Next, we shall give a complete classification of singular quadratic Lie algebras, up to iisomorphisms and up to isomorphisms. It is done mainly on a solvable framework by the reason below. There are four main steps to reach this goal: (1) Using the identity {I,I} = 0, we determine the Lie bracket on a solvable singular quadratic Lie algebra (Proposition 2.2.22). (2) We describe a solvable singular quadratic Lie algebra as a double extension of a quadratic vector space by a skew-symmetric map (or double extension, for short) (this notion is initiated by V. Kac [Kac85] and generally developed by A. Medina and P. Revoy [MR85]). As a consequence of (1), a quadratic Lie algebra is singular and solvable if and only if it is a double extension (Proposition 2.2.28 and Proposition 2.2.29). (3) We find the i-isomorphic and isomorphic conditions for two solvable singular quadratic Lie algebras (Theorem 2.2.30 and Corollary 2.2.31). These conditions allow us to establish a one-to-one correspondence between the set of i-isomorphic class of solvable singular quadratic Lie algebras and the set P1 (o(n)) of O(n)-orbits in P1 (o(n)), where P1 (o(n)) denotes the projective space of the Lie algebra o(n). (4) Finally, we prove that the i-isomorphic and isomorphic notions coincide for solvable singular quadratic Lie algebras. vii
Introduction What about non-solvable singular quadratic Lie algebras? Such a Lie algebra g can be written as g = s ⊥ ⊕ z where z is a central ideal of g and s o(3) equipped with a bilinear form λκ for some nonzero λ ∈ C where κ is the Killing form of o(3). Note that different from the solvable case, the notions of i-isomophism and isomorphism are not equivalent in this case. We denote by Ss(n+2) the set of solvable singular quadratic Lie algebra structures on Cn+2 , by Ss(n+2) the set of isomorphism classes of elements in Ss(n+2) and by i Ss (n+2) the set of i-isomorphism classes. Given C ∈ o(n), there is an associated double extension gC ∈ Ss(n + 2) (Definition 2.2.26) and then (Theorem 2.2.35): THEOREM 2: The map C → gC induces a bijection from P1 (o(n)) onto i Ss (n + 2). A weak form of Theorem 2 was stated in the paper by G. Favre and L. J. Santharoubane [FS87], in the case of i-isomorphisms satisfying some (dispensable) conditions. A strong improvement to Theorem 2 will be given in Theorem 5 where the i-isomorphic notion is replaced by the isomorphic notion. We detail Theorem 2 in some particular cases. Let N(n+2) be the set of nilpotent singular structures on C n+2 , N i (n + 2) be the set of i-isomorphism classes and N(n + 2) be the set of isomorphism classes of elements in N(n+2). We denote g and g ′ i-isomorphic by g i g ′ . Using the Jacobson-Morozov theorem, we prove that (Theorem 2.2.37): THEOREM 3: (1) Let g and g ′ be in N(n+2). Then g i g ′ if and only if g g ′ . Thus N i (n+2) = N(n+2). (2) Let N (n) be the set of nilpotent O(n)-adjoint orbits in o(n). Then the map C ↦→ gC induces a bijection from N (n) onto N(n + 2). In Chapter 1, we recall the well-known classification of nilpotent O(n)-adjoint orbits in o(n). An important ingredient is the Jacobson-Morosov and Kostant theorems on sl(2)-triples in semisimple Lie algebras (see the book by D. H. Collingwood and W. M. McGovern [CM93] for more details). Using this classification, we obtain a classification of N(n + 2) in term of special partitions of n, that is, there is a one-to-one correspondence between N(n + 2) and the set P1(n) of partitions of n in which even parts occur with even multiplicity (Theorem 2.2.38). In other words, we can parametrize the set N(n + 2) by the set of indices P1(n). This parametrization is detailed by means of amalgamated products of nilpotent Jordan-type Lie algebras. Let D(n+2) be the set of diagonalizable singular structures on C n+2 (i.e. C is a semisimple element of o(n)) and Dred(n + 2) be the set of reduced ones (see Definition 2.1.7 for the definition of a reduced quadratic Lie algebra). Denote by D(n + 2), D i (n + 2), Dred(n + 2) viii
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 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 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 58 and 59:
2.2. Singular quadratic Lie algebra
Page 60 and 61:
Page 62 and 63:
Page 64 and 65:
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 70 and 71:
Page 72 and 73:
Page 74 and 75:
2.4. 2-step nilpotent quadratic Lie
Page 76 and 77:
Page 78 and 79:
Page 81 and 82:
Chapter 3 Singular quadratic Lie su
Page 83 and 84:
3.1. Application of Z × Z2-graded
Page 85 and 86:
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
3.2. The dup-number of a quadratic
Page 93 and 94:
3.3. Elementary quadratic Lie super
Page 95 and 96:
3.3. Elementary quadratic Lie super
Page 97 and 98:
3.4. Quadratic Lie superalgebras wi
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
3.5. Singular quadratic Lie superal
Page 109 and 110:
Page 111 and 112:
Page 113 and 114:
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 131 and 132:
Page 133 and 134:
Proof. 4.3. Pseudo-Euclidean 2-step
Page 135 and 136:
Page 137 and 138:
4.4. Symmetric Novikov algebras (2)
Page 139 and 140:
4.4. Symmetric Novikov algebras Lem
Page 141 and 142:
Page 143 and 144:
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 and 166:
Index Bε, 3 E(g,B,δ), 53 Iε(V ),
Page 167 and 168:
normal form, 5 Jordan-admissible, 1
show all

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

Create successful ePaper yourself

Delete template?

Save as template?