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

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2.4. 2-step nilpotent quadratic Lie algebras 2.4 2-step nilpotent quadratic Lie algebras Conveniently, we redefine a 2-step nilpotent Lie algebra in another way as follows: Definition 2.4.1. An algebra g over C with a bilinear product g × g → g,(x,y) ↦→ [x,y] is called a 2-step nilpotent Lie algebra if it satisfies [x,y] = −[y,x] and [[x,y],z] = 0 for all x,y,z ∈ g. Sometimes, we use the notion 2SN-Lie algebra as an abbreviation. According to this definition, a commutative Lie algebra is a trivial case of 2SN-Lie algebras. 2.4.1 Some extensions of 2-step nilpotent Lie algebras Definition 2.4.2. Let g be a 2SN-Lie algebra, V be a vector space and ϕ : g × g → V be a bilinear map. On the space g = g ⊕V we define the following product: [x + u,y + v] = [x,y] + ϕ(x,y), ∀ x,y ∈ g,u,v ∈ V. Then it is easy to see that g is a 2SN-Lie algebra if and only if ϕ is skew-symmetric and ϕ([x,y],z) = 0, for all x,y,z ∈ g. In this case V is contained in the center Z(g) of g so the Lie algebra g is called the 2SN-central extension of g by V by means of ϕ. Proposition 2.4.3. Let g be a 2SN-Lie algebra then g is the 2SN-central extension of an Abelian algebra h by some vector space V . Proof. Denote by V = [g,g] and let h = g/[g,g]. Then h is Abelian. Set the map ϕ : h × h → V by ϕ(p(x), p(y)) = [x,y], ∀ x,y ∈ g, where p : g → h is the canonical projection. This map is well defined since g is 2-step nilpotent. So g is the 2SN-central extension of h by V by means of ϕ. Let g be a 2SN-Lie algebra, V be a vector space and π : g → End(V ) be a linear map. On the space g = g ⊕V we define the following product: [x + u,y + v] = [x,y] + π(x)v − π(y)u, ∀ x,y ∈ g,u,v ∈ V. Proposition 2.4.4. The vector space g is a 2SN-Lie algebra if and only if π satisfies the condition: π([x,y]) = π(x)π(y) = 0, ∀ x,y ∈ g. In this case, π is called a 2SN-representation of g in V . Proof. For all x,y,z ∈ g, u,v,w ∈ V the condition [[x + u,y + v],z + w] = 0 is equivalent to π([x,y])w−π(z)π(x)v+π(z)π(y)u = 0 for all u,v,w ∈ V . This happens if and only if π([x,y]) = π(x)π(y) = 0, for all x,y ∈ g. Remark 2.4.5. The adjoint representation and the coadjoint representation are 2SN-representations of a 2SN-Lie algebra g. Therefore, the extensions of g by itself or its dual space with respect to these representations are 2-step nilpotent. 54
2.4. 2-step nilpotent quadratic Lie algebras Definition 2.4.6. Let g be a 2SN-Lie algebra, V and W be vector spaces. If π : g → End(V ) and ρ : g → End(W) are two 2SN-representations of g then the map π ⊕ ρ : g → End(V ⊕W) defined by (π ⊕ ρ)(x)(v + w) = π(x)v + ρ(x)w, ∀ x ∈ g,v ∈ V,w ∈ W is also a 2SN-representation of g and it is called the direct sum of the two representations π and ρ. Proposition 2.4.7. Let g1, g2 be 2SN-Lie algebras and π : g1 → End(g2) be a linear map. We define on the vector space g = g1 ⊕ g2 the following product: [x + y,x ′ + y ′ ] = [x,x ′ ]g1 + π(x)y′ − π(x ′ )y + [y,y ′ ]g2 , ∀ x,x′ ∈ g1,y,y ′ ∈ g2. Then g with this product is a 2SN-Lie algebra if and only if π satisfies the following conditions: (1) π([x,x ′ ]g1 ) = π(x)π(x′ ) = 0. (2) π(x)([y,y ′ ]g2 ) = [π(x)y,y′ ]g2 = 0. for all x,x ′ ∈ g1,y,y ′ ∈ g2. Proof. Assume that g is a 2SN-Lie algebra. For all x,x ′ ,x ′′ ∈ g1, y,y ′ ,y ′′ ∈ g2, one has: 0 = [[x + y,x ′ + y ′ ],x ′′ + y ′′ ] = π([x,x ′ ]g1 )y′′ − π(x ′′ )π(x)y ′ + +π(x ′′ )π(x ′ )y + [π(x)y ′ ,y ′′ ]g2 − [π(x′ )y,y ′′ ]g2 − π(x′′ )([y,y ′ ]g2 ). Let y = y ′ = x ′′ = 0 we obtain π([x,x ′ ]g1 )y′′ = 0, for all y ′′ ∈ g. It means that π([x,x ′ ]g1 ) = 0. Similarly, π(x)π(x ′ ), π(x)([y,y ′ ]g2 ) and [π(x)y,y′ ]g2 are zero for all x,x′ ∈ g1, y,y ′ ∈ g2. Conversely, if π satisfies (1) and (2) then it is easy to check g is a 2SN-Lie algebra by Definition 2.4.1. Clearly, the map π in Proposition 2.4.7 is a 2SN-representation of g1 in g2. Hence,, we obtain the following definition: Definition 2.4.8. Let g1, g2 be 2SN-Lie algebras and π : g1 → End(g2) be a 2SN-representation of g1 in g2. Then the vector g = g1 ⊕ g2 with the product: [x + y,x ′ + y ′ ] = [x,x ′ ]g1 + π(x)y′ − π(x ′ )y + [y,y ′ ]g2 , ∀ x,x′ ∈ g1,y,y ′ ∈ g2. become a 2SN-Lie algebra if and only if π(x)([y,y ′ ]g2 ) = [π(x)y,y′ ]g2 = 0, ∀ x ∈ g1,y,y ′ ∈ g2. In this case, π is called a 2SN-admissible representation of g1 in g2 and we say that g is the semi-direct product of g2 by g1 by means of π. Remark 2.4.9. (1) The condition in the above definition ensures π(x) ∈ Der(g2), for all x ∈ g1. (2) The adjoint representation of a 2SN-Lie algebra is a 2SN-admissible representation. 55
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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
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 66 and 67: (2) g g ′ if and only if g i g
Page 68 and 69: 2.3. Quadratic dimension of quadrat
Page 76 and 77: 2.4. 2-step nilpotent quadratic Lie
Page 78 and 79: 2.4. 2-step nilpotent quadratic Lie
Page 81 and 82: Chapter 3 Singular quadratic Lie su
Page 83 and 84: 3.1. Application of Z × Z2-graded
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 107 and 108: 3.5. Singular quadratic Lie superal
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
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4.2. Jordanian double extension of
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4.2. Jordanian double extension of
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4.3. Pseudo-Euclidean 2-step nilpot
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Proof. 4.3. Pseudo-Euclidean 2-step
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4.4. Symmetric Novikov algebras (2)
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4.4. Symmetric Novikov algebras Lem
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4.4. Symmetric Novikov algebras Lem
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4.4. Symmetric Novikov algebras Pro
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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
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