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

More documents

Recommendations

Info

y Introduction We realize that with a quadratic Lie superalgebra (g,B), if we define a trilinear form I on g I(X,Y,Z) = B([X,Y ],Z), ∀ X,Y,Z ∈ g then I is a super-antisymmetric trilinear form in E (3,0) (g) and therefore, it seems to be natural to ask the question: does it happen {I,I} = 0? We give an affirmative answer this in the first part of Chapter 3 (Theorem 3.1.17). Moreover, we obtain that quadratic Lie superalgebra structures with bilinear form B are in one-to-one correspondence with elements I in E (3,0) (g) satisfying {I,I} = 0 (Proposition 3.1.18) As in Chapter 2, we give the notion of dup-number of a non-Abelian quadratic Lie superalgebra g defined by dup(g) = dim({α ∈ g ∗ | α ∧ I = 0}) and suggest considering the set of quadratic Lie superalgebras where the dup-number is nonzero. Thanks to Lemma 3.2.1, we focus on a singular quadratic Lie superalgebra g with dup(g) = 1. It is called a singular quadratic Lie superalgebra of type S1. Remark that in this case, the element I may be decomposable. We detail some particular cases. When the element I is decomposable, we obtain a classification of reduced corresponding Lie superalgebras as in Proposition 3.3.3 where the even part g 0 of g is a singular quadratic Lie algebra or 2-dimensional. Actually, we prove in Proposition 3.4.1 that if g is a non-Abelian quadratic Lie superalgebra with 2-dimensional even part then g is a singular quadratic Lie superalgebra of type S1. Note that if we replace the quadratic vector space q in the definition of double extension by a symplectic vector space then we obtain a quadratic Lie superalgebra with 2-dimensional even part (Definitions 2.2.26 and 3.4.6). Thus, we have the following result (Theorem 3.4.8): THEOREM 10: A quadratic Lie superalgebra has the 2-dimensional even part if and only if it is a double extension. By a completely similar process as in Chapter 2, a classification of quadratic Lie superalgebras with 2-dimensional even part is given as follows. Let S(2 + 2n) be the set of such structures on C 2+n . We call an algebra g ∈ S(2 + 2n) diagonalizable (resp. invertible) if it is the double extension by a diagonalizable (resp. invertible) map. Denote the subsets of nilpotent elements, diagonalizable elements and invertible elements in S(2 + 2n), respectively by N(2 + 2n), D(2 + 2n) and by Sinv(2 + 2n). Denote by N(2 + 2n), D(2 + 2n), Sinv(2 + 2n) the sets of isomorphic classes in N(2 + 2n), D(2 + 2n), Sinv(2 + 2n), respectively and Dred(2 + 2n) the subset of D(2 + 2n) including reduced ones. Keeping the other notations then we have the classification result (Theorems 3.4.13 and 3.4.14): THEOREM 11: (1) Let g and g ′ be elements in S(2 + 2n). Then g and g ′ are i-isomorphic if and only if they are isomorphic. (2) There is a bijection between N(2 + 2n) and the set of nilpotent Sp(2n)-adjoint orbits of sp(2n) that induces a bijection between N(2 + 2n) and the set of partitions P−1(2n) of 2n in which odd parts occur with even multiplicity. xiii
Introduction (3) There is a bijection between D(2 + 2n) and the set of semisimple Sp(2n)- orbits of P 1 (sp(2n)) that induces a bijection between D(2 + 2n) and Λn/Hn. In the reduced case, Dred(2 + 2n) is bijective to Λ + n /Hn. (4) There is a bijection between Sinv(2 + 2n) and the set of invertible Sp(2n)- orbits of P 1 (sp(2n)) that induces a bijection between Sinv(2 + 2n) and Jn/C ∗ . (5) There is a bijection between S(2 + 2n) and the set of Sp(2n)- orbits of P 1 (sp(2n)) that induces a bijection between S(2 + 2n) and D(2n)/C ∗ . As for quadratic Lie algebras, we have the notion of quadratic dimension for quadratic Lie superalgebras. In the case g having a 2-dimensional even part, we can compute its quadratic dimension as follows: dq(g) = 2 + (dim(Z(g) − 1))(dim(Z(g) − 2) . 2 We turn now to (g,B) a singular quadratic Lie superalgebra of type S1. By Definition 3.5.3 and Lemma 3.5.5, the Lie superalgebra g can be realized as the double extension of a quadratic Z2-graded vector space q = q 0 ⊕ q 1 by a map C = C0 +C1 ∈ o(q 0) ⊕ sp(q 1). Denote by L (q 0) (resp. L (q 1)) the set of endomorphisms of q 0 (resp. q 1). We give a characterization as follows (Theorem 3.5.7). THEOREM 12: Let g and g ′ be two double extensions of q by C = C0 +C1 and C ′ = C ′ 0 +C ′ Assume that C1 is non-zero. Then 1, respectively. (1) there exists a Lie superalgebra isomorphism between g and g ′ if and only if there exist invertible maps P ∈ L (q 0), Q ∈ L (q 1) and a non-zero λ ∈ C such that (i) C ′ 0 = λPC0P −1 and P ∗ PC0 = C0. (ii) C ′ 1 = λQC1Q −1 and Q ∗ QC1 = C1. where P ∗ and Q ∗ are the adjoint maps of P and Q with respect to B|q 0 ×q 0 and B|q 1 ×q 1 . (2) there exists an i-isomorphism between g and g ′ if and only if there is a non-zero λ ∈ C such that C ′ 0 is in the O(q0)-adjoint orbit through λC0 and C ′ 1 is in the Sp(q1)-adjoint orbit through λC1. We close the problem on singular quadratic Lie superalgebras by an assertion that the dupnumber is invariant under Lie superalgebra isomorphisms (Theorem 3.5.9). In the last section of Chapter 3, we study the structure of a quadratic Lie superalgebra g such that its element I has the form: I = J ∧ p where p ∈ g ∗ 1 is non-zero and J ∈ A 1 (g 0) ⊗ S 1 (g 1) is indecomposable. We call g a quasisingular quadratic Lie superalgebra. With the notion of generalized double extension given xiv
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 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 64 and 65:
2.2. Singular quadratic Lie algebra
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 ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?