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

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4.4. Symmetric Novikov algebras Proof. By Lemma 4.4.24, N 2 ⊂ Z(N). Moreover, N non-commutative implies that g(N) is non-Abelian and by Remark 2.2.10, dim([N,N]) ≥ 3. Therefore, dim(Z(N)) ≤ dim(N) − 3 since Z(N) = [N,N] ⊥ . Consequently, dim(N 2 ) ≤ dim(N)−3 and then dim(Ann(N)) ≥ 3. Corollary 4.4.30. Let N be a non-commutative symmetric Novikov algebra of dimension 7. If N is 2-step nilpotent then N is not reduced. Proof. Assume that N is reduced then dim(Ann(N)) = 3 and dim(N 2 ) = 4. It implies that there must have a non-zero element x ∈ N 2 such that xN = {0} and then N is not 2-step nilpotent. Now, we give a more general result for symmetric Novikov algebra of dimension 7 as follows: Proposition 4.4.31. Let N be a non-commutative symmetric Novikov algebra of dimension 7. If N is reduced then there are only two cases: (1) N is 3-step nilpotent and indecomposable. (2) N is decomposable by N = Cx ⊥ ⊕ N6 where x 2 = x and N6 is a non-commutative symmetric Novikov algebra of dimension 6. Proof. Assume that N is reduced then dim(Ann(N)) = 3, dim(N 2 ) = 4 since Ann(N) ⊂ N 2 and Ann(N) = (N 2 ) ⊥ . By [Bou59], Ann(N) is totally isotropic then there exist a totally isotropic subspace V and a non-zero x of N such that N = Ann(N) ⊕ Cx ⊕V, where Ann(N) ⊕V is non-degenerate, B(x,x) = 0 and x ⊥ = Ann(N) ⊕V . As a consequence, Ann(N) ⊕ Cx = (Ann(N)) ⊥ = N 2 . Consider the left-multiplication operator Lx : Cx ⊕V → Ann(N) ⊕ Cx, Lx(y) = xy, for all y ∈ Cx ⊕V . Denote by p the projection Ann(N) ⊕ Cx → Cx. • If p ◦ Lx = 0 then (NN)N = xN ⊂ Ann(N). Therefore, ((NN)N)N = {0}. That implies N is 3-nilpotent. If N is decomposable then N must be 2-step nilpotent. This is in contradiction to Corollary 4.4.30. • If p ◦ Lx = 0 then there is a non-zero y ∈ Cx ⊕V such that xy = ax +z with 0 = a ∈ C and z ∈ Ann(N). In this case, we can choose y such that a = 1. It implies that (x 2 )y = x(xy) = x 2 . If x 2 = 0 then 0 = B(x 2 ,y) = B(x,xy) = B(x,x). This is a contradiction. Therefore, x 2 = 0. Since x 2 ∈ Ann(N) ⊕ Cx then x 2 = z ′ + µx where z ′ ∈ Ann(N) and µ ∈ C must be nonzero. By setting x ′ := x µ and z′′ = z′ µ 2 , we get (x ′ ) 2 = z ′′ + x ′ . Let x1 := (x ′ ) 2 , one has: x 2 1 = (x ′ ) 2 (x ′ ) 2 = (z ′′ + x ′ )(z ′′ + x ′ ) = x1. Moreover, for all t = λx+v ∈ Cx⊕V , we have t(x 2 ) = (x 2 )t = x(xt) = λ µ(x 2 ). It implies that Cx 2 = Cx1 is an ideal of N. ⊥ Since B(x1,x1) = 0, by Lemma 4.4.5 one has N = Cx1 ⊕ (x1) ⊥ . Certainly, (x1) ⊥ is a non-commutative symmetric Novikov algebra of dimension 6. 126
4.4. Symmetric Novikov algebras Proposition 4.4.32. Let N be a symmetric Novikov algebra. If g(N) or J(N) is reduced then N is reduced. Proof. Assume that N is not reduced then there is a non-zero x ∈ Ann(N) such that B(x,x) = 1. Since [x,N] = [x,N]+ = 0 then g(N) and J(N) are not reduced. Corollary 4.4.33. Let N be a symmetric Novikov algebra. If g(N) is reduced then N must be 2-step nilpotent. Proof. Since g(N) is reduced then Ann(N) ⊂ N 2 . On the other hand, dim(Z(N)) = dim([N,N]) = 1 2 dim(N) so dim(Ann(N)) = dim(N 2 ). Therefore, Ann(N) = N 2 and N is 2-step nilpotent. Example 4.4.34. By Example 4.4.2, every 2-step nilpotent algebra is Novikov then we will give here an example of non-commutative symmetric Novikov algebras of dimension 7 which is 3step nilpotent. Let N = Cx ⊕ N6 be a 7-dimensional vector space where N6 is the symmetric Novikov algebra of dimension 6 in Example 4.4.26. Define the product on N by xe4 = e4x = e1,e4e4 = x,e4e5 = e3,e5e6 = e1,e6e4 = e2, and the symmetric bilinear form B defined by B(x,x) = B(e1,e4) = B(e2,e5) = B(e3,e6) = 1 B(e4,e1) = B(e5,e2) = B(e6,e3) = 1, 0 otherwise. Note that in above Example, g(N) is not reduced since x ∈ Z(N). 127
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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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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
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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
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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 ),
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TH`ESE A NEW INVARIANT OF QUADRATIC LIE ALGEBRAS AND ...

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