3.1. Application of Z × Z2-graded Lie superalgebras to quadratic Lie superalgebras = − n ∑ k=1 ι X k 1 (I) ∧ ιY k(γl) 1 (X,Y ) = − ι X l 1 (I) ∧ ιY l(γl) (X,Y ) 1 = ιX l(I)(X,Y ) = B(X 1 l 1 ,[X,Y ]) = γl([X,Y ]) = −d(γl)(X,Y ). Therefore, d = −adP(I). Moreover, adP({I,I}) = [adP(I),adP(I)] = [d,d] = 2d2 = 0. Therefore, for all 1 ≤ i ≤ m, 1 ≤ j,k ≤ n one has {αi,{I,I}} = {β j,{I,I}} = {γk,{I,I}} = 0. Those imply ιX ({I,I}) = 0 for all X ∈ g and hence, we obtain {I,I} = 0. Conversely, let g be a quadratic Z2-graded vector space equipped with a bilinear form B and I be an element in E (3,0) (g). Define d = −adP(I) then d ∈ D 1 0 (E(g)). Therefore, d2 = 0 if and only if {I,I} = 0. Let F be the struture in g corresponding to d by the isomorphism D in Lemma 3.1.15, one has Proposition 3.1.18. F becomes a Lie superalgebra structure if and only if {I,I} = 0. In this case, with the notation [X,Y ] := F(X,Y ) one has: Moreover, the bilinear form B is invariant. I(X,Y,Z) = B([X,Y ],Z), ∀ X,Y,Z ∈ g. Proof. We need to prove that if F is a Lie superalgebra structure then I(X,Y,Z) = B([X,Y ],Z), for all X,Y,Z ∈ g. Indeed, let {X 1 0 ,...,X m 0 } be an orthonormal basis of V0 and {X 1 1 ,...,X n 1 , Y 1,...,Y n 1 1 } be a canonical basis of V1 then one has It implies that d = −adP(I) = − F = m ∑ j=1 Therefore, for all i we obtain m ∑ j=1 ι X j 0 ι j(I) ⊗ X X 0 j (I) ∧ ι X j 0 0 + n ∑ k=1 + n ∑ k=1 ι X k 1 (I) ∧ ι Y k 1 ιX k(I) ⊗Y 1 k 1 − n ∑ k=1 − n ∑ k=1 ι Y k 1 ιY k(I) ⊗ X 1 k 1 . B([X,Y ],X i 0 ) = ιX i(I)(X,Y ) = I(X 0 i 0 ,X,Y ) = I(X,Y,X i 0 ), B([X,Y ],X i 1 ) = −ι Xi (I)(X,Y ) = −I(X 1 i 1 ,X,Y ) = I(X,Y,X i 1 ), B([X,Y ],Y i 1 ) = −ι Y i(I)(X,Y ) = −I(Y i i 1 ,X,Y ) = I(X,Y,Y 1 ). These show that I(X,Y,Z) = B([X,Y ],Z), for all X,Y,Z ∈ g. 1 (I) ∧ ιX k . 1 Remark 3.1.19. The element I defined as above is also an invariant of g since LX(I) = 0, for all X ∈ g where LX = D(ad(X)) the Lie super-derivation of g. Therefore, I is called the associated invariant of g. Lemma 3.1.20. Let (g,B) be a quadratic Lie superalgebra and I be its associated invariant. Then ιX(I) = 0 if and only if X ∈ Z(g). 68
3.1. Application of Z × Z2-graded Lie superalgebras to quadratic Lie superalgebras Proof. Since ιX(I)(g,g) = B(X,[g,g]) and Z(g) = [g,g] ⊥ then one has ιX(I) = 0 if and only if X ∈ Z(g). Definition 3.1.21. Let (g,B) and (g ′ ,B ′ ) be two quadratic Lie superalgebras. We say that (g,B) and (g ′ ,B ′ ) are isometrically isomorphic (or i-isomorphic) if there exists a Lie superalgebra isomorphism A from g onto g ′ satisfying B ′ (A(X),A(Y )) = B(X,Y ), ∀ X,Y ∈ g. In other words, A is an i-isomorphism if it is a (necessarily even) Lie superalgebra isomorphism and an isometry. We write g i g ′ . Note that two isomorphic quadratic Lie superalgebras (g,B) and (g ′ ,B ′ ) are not necessarily i-isomorphic by the example below: Example 3.1.22. Let g = osp(1,2) and B its Killing form. Recall that g 0 = o(3). Consider another bilinear form B ′ = λB, λ ∈ C, λ = 0. In this case, (g,B) and (g,λB) cannot be iisomorphic if λ = 1 since (g 0 ,B) and (g 0 ,λB) are not i-isomorphic (see Example 2.1.4). 69
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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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4.4. Symmetric Novikov algebras Lem
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4.4. Symmetric Novikov algebras (3)
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4.4. Symmetric Novikov algebras Lem
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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