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Proposition 4. Let \( \mathrm{G} \) be a finite-dimensional Lie group and \( \mathrm{H} \) a subgroup of \( \mathrm{G} \) . Let \( r \in {\mathrm{N}}_{\mathrm{K}} \) . The following conditions are equivalent:\n\n(i) \( \mathrm{H} \) is an integral subgroup of \( \mathrm{G} \) ;\n\n(ii) with the Lie group structure indu... | (ii) \( \Rightarrow \) (i): obvious.\n\n(i) \( \Rightarrow \) (iii): suppose that \( \mathrm{H} \) has a Lie group structure such that \( \mathrm{H} \) is an integral subgroup of G. Using the notation of Remark 3, the set of \( y \in \mathrm{H} \) such that property \( {\mathrm{P}}_{\mathrm{H}, e, y} \) holds is an ope... | Yes |
Proposition 5. Let \( \mathrm{G} \) be a connected real Lie group. Suppose that \( \mathrm{L}\left( \mathrm{G}\right) \) has a complex normable Lie algebra structure \( {\mathrm{L}}^{\prime } \) compatible with its real normable Lie algebra structure. There exists on \( \mathrm{G} \) one and only one complex Lie group ... | By \( §4 \), no. 2, Corollary 2 to Theorem 2, it suffices to prove that the structure of \( {\mathrm{L}}^{\prime } \) is invariant under \( \mathrm{{Ad}}\mathrm{G} \) . Let \( \phi \) be an exponential mapping of \( \mathrm{G} \) . By \( §4 \) , no. 4, Corollary 3 (i) to Proposition 8, there exists a neighbourhood V of... | Yes |
Proposition 6. Let \( \\mathrm{G} \) be a connected complex Lie group. If \( \\mathrm{G} \) is compact, \( \\mathrm{G} \) is commutative. | The holomorphic mapping \( \\;g \\mapsto \\mathrm{{Ad}}\\;g\\;\\mathrm{{of}}\\;\\mathrm{G}\\;\\mathrm{{into}}\\;\\mathcal{L}\\left( {\\mathrm{L}\\left( \\mathrm{G}\\right) }\\right) \\mathrm{{is}}\\;\\mathrm{{constant}}\\left( \\text{Differen-}\\right. \) tiable and Analytic Manifolds, R,3.3.7) and hence ad \( a = 0 \)... | Yes |
Proposition 7. Let \( \mathrm{G} \) be a Lie group and a an element of \( \mathrm{L}\left( \mathrm{G}\right) \) . The mapping \( \lambda \mapsto \exp \left( {\lambda a}\right) \) of \( \mathrm{K} \) into \( \mathrm{G} \) is the unique morphism \( \phi \) of the Lie group \( \mathrm{K} \) into \( \mathrm{G} \) such that... | The mappings \( \left( {\lambda ,{\lambda }^{\prime }}\right) \mapsto \exp \left( {\lambda a}\right) \exp \left( {{\lambda }^{\prime }a}\right) \) and \( \left( {\lambda ,{\lambda }^{\prime }}\right) \mapsto \exp \left( {\lambda + {\lambda }^{\prime }}\right) a \) of \( \mathrm{K} \times \mathrm{K} \) into \( \mathrm{G... | Yes |
Proposition 8. Let \( \mathrm{G} \) be a Lie group. For all \( x, y \) in \( \mathrm{L}\left( \mathrm{G}\right) \) and \( n \) an integer,\n\n(2)\n\n\[ \exp \left( {x + y}\right) = \mathop{\lim }\limits_{{n \rightarrow + \infty }}{\left( \left( \exp \frac{1}{n}x\right) \left( \exp \frac{1}{n}y\right) \right) }^{n} \]\n... | By Proposition 7, this follows from Proposition 4 of \( §4 \), no. 3, taking \( \lambda = \frac{1}{n} \). | No |
Proposition 9. Let \( \mathrm{G} \) be a complex Lie group and \( {\mathrm{G}}^{\prime } \) the underlying real Lie group. Then \( {\exp }_{\mathrm{G}} = {\exp }_{{\mathrm{G}}^{\prime }} \) . | This follows from Proposition 5 of § 4, no. 3 and the analyticity of \( {\exp }_{G} \) and \( {\exp }_{{\mathrm{G}}^{\prime }} \) . | Yes |
Proposition 10. Let \( \mathrm{G} \) and \( \mathrm{H} \) be Lie groups and \( \mathrm{o} \) a morphism of \( \mathrm{G} \) into \( \mathrm{H} \). (i) \( \phi \circ {\exp }_{\mathrm{G}} = {\exp }_{\mathrm{H}} \circ \mathrm{L}\left( \phi \right) \. | The two sides of the equality (i) are analytic mappings of \( L\left( G\right) \) into \( H \) which coincide in a neighbourhood of \( 0\left( {§4\text{, Proposition 8, no. 4}}\right) \) and hence are equal. | Yes |
Corollary 1. Let \( \mathrm{G} \) be a Lie group, \( {\mathrm{G}}^{\prime } \) a Lie subgroup of \( \mathrm{G} \) and \( a \in \mathrm{L}\left( \mathrm{G}\right) \) . The following conditions are equivalent:\n\n(i) \( a \in \mathrm{L}\left( {\mathrm{G}}^{\prime }\right) \) ;\n\n(ii) \( \exp \left( {\lambda a}\right) \i... | The argument is as in \( §4 \), no. 4, Corollary 1 to Proposition 8 . | No |
Corollary 2. Let \( \mathrm{G} \) be a Lie group, \( \mathrm{H} \) an integral subgroup of \( \mathrm{G} \) and \( a \in \mathrm{L}\left( \mathrm{G}\right) \) . Consider the following conditions: (i) \( a \in \mathbf{L}\left( \mathbf{H}\right) \) ; (ii) \( {\exp }_{\mathrm{G}}\left( {\lambda a}\right) \in \mathrm{H} \)... | Let \( i \) be the canonical injection of \( \mathrm{H} \) into \( \mathrm{G} \) . If \( a \in \mathrm{L}\left( \mathrm{H}\right) \), then \[ {\exp }_{\mathrm{G}}\left( {\lambda a}\right) = \left( {{\exp }_{\mathrm{G}} \circ \mathrm{L}\left( i\right) }\right) \left( {\lambda a}\right) = \left( {i \circ {\exp }_{\mathbf... | Yes |
Corollary 3. Let \( \mathrm{G} \) be a Lie group, \( \mathrm{p} \) an analytic linear representation of \( \mathrm{G}, x \in \mathrm{L}\left( \mathrm{G}\right) \) and \( g \in \mathrm{G} \) . (i) \( \rho \left( {\exp x}\right) = \exp \mathrm{L}\left( \rho \right) x \) ; (ii) \( \operatorname{Ad}\left( {\exp x}\right) =... | The argument is as in \( §4 \), no. 4, Corollaries 2 and 3 to Proposition 8 . | No |
Proposition 11. Let \( \mathrm{G} \) be a connected commutative Lie group.\n\n(i) \( \exp \) is an étale morphism of the additive Lie group \( \mathrm{L}\left( \mathrm{G}\right) \) onto \( \mathrm{G} \) . | By the Hausdorff formula, \( \left( {\exp x}\right) \left( {\exp y}\right) = \exp \left( {x + y}\right) \) for \( x, y \) sufficiently close to 0 and hence for all \( x, y \) in \( \mathrm{L}\left( \mathrm{G}\right) \) by analytic continuation. Hence exp is a group homomorphism and is étale since\n\n\[ \n{\mathrm{T}}_{... | Yes |
Proposition 12. Let \( \\mathrm{G} \) be a Lie group and \( \\mathrm{L} = \\mathrm{L}\\left( \\mathrm{G}\\right) \) . For all \( x \\in \\mathrm{L} \), let \( {\\mathrm{T}}_{x}\\left( \\mathrm{\\;L}\\right) \) be identified with \( \\mathbf{L} \), so that the right differential \( \\varpi \\left( x\\right) \) of \( \\e... | \[ \\varpi \\left( x\\right) = \\mathop{\\sum }\\limits_{{n \\geq 0}}\\frac{1}{\\left( {n + 1}\\right) !}{\\left( \\operatorname{ad}x\\right) }^{n} \] The two sides are analytic functions of \( x \) and are equal for \( x \) sufficiently close to \( 0 \) (§ 4, no. 3, Proposition 6). | Yes |
Lemma 2. Let \( \mathrm{E} \) be a complex Banach space, \( u \) an element of \( \mathcal{L}\left( \mathrm{E}\right) ,\mathrm{S} \) the spectrum of \( u \) in \( \mathcal{L}\left( \mathrm{E}\right) \) (Spectral Theories, Chapter I,§ 1, no. 2) and f a holomorphic complex function on an open neighbourhood \( \Omega \) o... | There exists a holomorphic function \( g \) on \( \Omega \), everywhere non-zero, such that \( f\left( z\right) = {\left( z - {z}_{1}\right) }^{{h}_{1}}\ldots {\left( z - {z}_{n}\right) }^{{h}_{n}}g\left( z\right) \) . Then \( g\left( u\right) {g}^{-1}\left( u\right) = {g}^{-1}\left( u\right) g\left( u\right) = 1 \) an... | Yes |
Proposition 13. Let \( \mathrm{G} \) be a connected Lie group and \( \mathrm{p} \) an analytic linear representation of \( \mathrm{G} \) on a complete normable space \( \mathrm{E} \) . Let \( {\mathrm{E}}_{1},{\mathrm{E}}_{2} \) be two closed vector subspaces of \( \mathrm{E} \) such that \( {\mathrm{E}}_{2} \subset {\... | \[ \rho \left( g\right) x \equiv x\;\left( {\text{mod }{\mathrm{E}}_{2}}\right) \;\text{for all }g \in \mathrm{G}\text{ and all }x \in {\mathrm{E}}_{1} \]\n\n\[ \Leftrightarrow p\left( {\exp a}\right) x \equiv x\;\left( {\text{mod }{\mathrm{E}}_{2}}\right) \;\text{for all }a \in \mathrm{L}\left( \mathrm{G}\right) \text... | Yes |
For \( {\mathrm{E}}_{1} \) to be stable under \( \rho \), it is necessary and sufficient that \( {\mathrm{E}}_{1} \) be stable under \( \mathrm{L}\left( \rho \right) \) . | It suffices to take \( {\mathrm{E}}_{1} = {\mathrm{E}}_{2} \) in Proposition 13. | No |
Corollary 2. Suppose that \( \rho \) is finite-dimensional. For \( \rho \) to be simple \( (\mathrm{{resp}.} \) semisimple), it is necessary and sufficient that \( \mathrm{L}\left( \rho \right) \) be simple (resp. semi-simple). | This follows from Corollary 1. | No |
Corollary 3. Let \( x \in \mathbf{E} \) . For \( x \) to be invariant under \( \rho \left( \mathbf{G}\right) \), it is necessary and sufficient that \( x \) be annihilated by \( \mathrm{L}\left( \rho \right) \left( {\mathrm{L}\left( \mathrm{G}\right) }\right) \) (that is that \( x \) be invariant under \( \mathrm{L}\le... | It suffices to take \( {\mathrm{E}}_{1} = \mathrm{K}x \) and \( {\mathrm{E}}_{2} = 0 \) in Proposition 13. | No |
Corollary 5. Suppose that \( \rho \) and \( {\rho }^{\prime } \) are finite-dimensional. For \( \rho \) and \( {\rho }^{\prime } \) to be equivalent, it is necessary and sufficient that \( \mathrm{L}\left( \rho \right) \) and \( \mathrm{L}\left( {\rho }^{\prime }\right) \) be equivalent. | This is a special case of Corollary 4. | No |
Corollary 6. Suppose that G is finite-dimensional. Let \( t \in \mathrm{U}\left( \mathrm{G}\right) \) . For \( {\mathrm{L}}_{t}\left( {\text{resp. }{\mathrm{R}}_{t}}\right) \) to be right (resp. left) invariant, it is necessary and sufficient that t belong to the centre of \( U\left( G\right) \) . | For \( {\mathrm{L}}_{t} \) (resp. \( {\mathrm{R}}_{t} \) ) to be right (resp. left) invariant, it is necessary and sufficient that \( {\varepsilon }_{g} * t = t * {\varepsilon }_{g} \) for all \( g \in \mathrm{G} \), that is that \( {\left( \operatorname{Int}g\right) }_{ * }t = t \) . There exists an integer \( n \) su... | Yes |
Lemma 3. Let \( \mathrm{G} \) be a Lie group, \( {\mathrm{H}}_{1} \) and \( {\mathrm{H}}_{2} \) integral subgroups whose topology admits a countable base and \( g \in \mathrm{G} \) . Then\n\n\[ g{\mathrm{H}}_{1}{g}^{-1} = {\mathrm{H}}_{2} \Leftrightarrow \left( {\operatorname{Ad}g}\right) \left( {\mathrm{L}\left( {\mat... | \( \operatorname{Ad}g = {\mathrm{T}}_{e}\left( {\operatorname{Int}g}\right) \) . Hence, by transport of structure, \( \left( {\operatorname{Int}g}\right) \left( {\mathrm{H}}_{1}\right) \) has Lie algebra \( \left( {\operatorname{Ad}g}\right) \left( {\mathrm{L}\left( {\mathrm{H}}_{1}\right) }\right) \) . As \( {\mathrm{... | Yes |
Proposition 14. Let \( \mathrm{G} \) be a Lie group and \( \mathrm{H} \) an integral subgroup whose topology admits a countable base. The following conditions are equivalent:\n\n(i) \( \mathrm{H} \) is normal in \( \mathrm{G} \) ;\n\n(ii) \( \mathrm{L}\left( \mathrm{H}\right) \) is invariant under \( \operatorname{Ad}\... | The equivalence (i) \( \Leftrightarrow \) (ii) follows from Lemma 3. | No |
Corollary 1. Let \( \mathrm{G} \) be a finite-dimensional simply connected Lie group. Let \( \mathfrak{m},\mathfrak{h} \) be Lie subalgebras of \( \mathrm{L}\left( \mathrm{G}\right) \) such that \( \mathrm{L}\left( \mathrm{G}\right) \) is the semi-direct product of \( \mathrm{m} \) by \( \mathfrak{h} \) . Let \( \mathr... | By Proposition 14, \( \mathrm{H} \) is a normal Lie subgroup of \( \mathrm{G} \) and the Lie group \( \mathrm{G}/\mathrm{H} \) is simply connected. Let \( \pi \) be the canonical morphism of \( \mathrm{G} \) onto \( \mathrm{G}/\mathrm{H} \) . There exists a morphism \( \theta \) of \( \mathrm{G}/\mathrm{H} \) into \( \... | No |
Corollary 2. Let \( \mathrm{G} \) be a finite-dimensional simply connected Lie group, \( \mathrm{H} \) a normal connected Lie subgroup of \( \mathrm{G} \) and \( \pi \) the canonical morphism of \( \mathrm{G} \) onto \( \mathrm{G}/\mathrm{H} \) . (i) There exists an analytic mapping \( \rho \) of \( \mathrm{G}/\mathrm{... | Let \( n = \dim \mathrm{G} - \dim \mathrm{H} \) . The corollary is obvious for \( n = 0 \) . We argue by induction on \( n \) . Suppose that there exists an ideal of L(G) containing L(H) distinct from \( \mathrm{L}\left( \mathrm{G}\right) \) and \( \mathrm{L}\left( \mathrm{H}\right) \) . Let \( {\mathrm{H}}^{\prime } \... | Yes |
Corollary 3. Let \( \mathrm{G} \) be a finite-dimensional connected Lie group and \( \mathrm{H} \) a normal connected Lie subgroup of \( \mathrm{G} \). The canonical morphism of \( {\pi }_{1}\left( \mathrm{H}\right) \) into \( {\pi }_{1}\left( \mathrm{G}\right) \) is injective. | Let \( {G}_{1} \) be the universal covering of \( G \) and \( \lambda \) the canonical mapping of \( {G}_{1} \) onto \( \mathrm{G} \). The Lie algebra of \( {\mathrm{G}}_{1} \) is identified with \( \mathrm{L}\left( \mathrm{G}\right) \). The Lie subgroup \( {\lambda }^{-1}\left( \mathrm{H}\right) \) of \( {\mathrm{G}}_... | Yes |
Proposition 15. Let \( \mathrm{G} \) be a Lie group, \( \mathrm{M} \) a manifold of class \( {\mathrm{C}}^{r}\left( {r \geq 2}\right) \) and \( \alpha \) a differential form of class \( {\mathrm{C}}^{r - 1} \) and degree 1 on \( \mathrm{M} \) with values in \( \mathrm{L}\left( \mathrm{G}\right) \), such that \( {d\alph... | The uniqueness of \( f \) follows from \( §3 \), no. 17, Corollary 2 to Proposition 59 and the fact that \( \mathrm{M} \) is connected. We prove the existence of \( f \) . There exist an open covering \( {\left( {\mathrm{U}}_{i}\right) }_{i \in \mathrm{I}} \) of \( \mathrm{M} \) and, for all \( i \in \mathrm{I} \), a m... | Yes |
Lemma 4. Let \( \mathrm{G} \) be a connected topological group, \( \mathrm{X} \) a Hausdorff topological space and \( {f}_{1},{f}_{2} \) laws of left (resp. right) operation of \( \mathrm{G} \) on \( \mathrm{X} \) such that, for all \( x \in \mathrm{X} \), the mappings \( s \mapsto {f}_{1}\left( {s, x}\right), s \mapst... | Let \( x \in \mathrm{X} \) and \( \mathrm{A} \) be the set of \( g \in \mathrm{G} \) such that \( {f}_{1}\left( {g, x}\right) = {f}_{2}\left( {g, x}\right) \) . Then \( \mathrm{A} \) is closed in G. On the other hand, let \( g \in \mathrm{A} \) ; we write \( y = {f}_{1}\left( {g, x}\right) = {f}_{2}\left( {g, x}\right)... | Yes |
Proposition 16. Let \( \mathrm{G} \) be a connected Lie group, \( \mathrm{X} \) a Hausdorff manifold of class \( {\mathrm{C}}^{r} \) and \( {f}_{1},{f}_{2} \) laws of left (resp. right) operation of class \( {\mathrm{C}}^{r} \) of \( \mathrm{G} \) on \( \mathrm{X} \) . If the laws of infinitesimal operation associated ... | By \( §4 \), no. 7, Corollary to Proposition 11, there exists a neighbourhood \( \mathrm{V} \) of \( \{ e\} \times \mathrm{X} \) in \( \mathrm{G} \times \mathrm{X} \) such that \( {f}_{1} \) and \( {f}_{2} \) coincide on \( \mathrm{V} \) . Hence \( {f}_{1} = {f}_{2} \) (Lemma 4). | Yes |
Lemma 5. Let \( \mathrm{G} \) be a simply connected topological group, \( \mathrm{X} \) a Hausdorff topological space, \( \mathrm{U} \) an open neighbourhood of \( e \) in \( \mathrm{G} \) and \( \psi \) a continuous mapping of \( \mathrm{U} \times \mathrm{X} \) into \( \mathrm{X} \) such that \( \psi \left( {e, x}\rig... | The uniqueness on \( \psi \) follows from Lemma 4. Let \( \mathrm{P} \) be the permutation group of \( \mathrm{X} \) . For \( u \in {\mathrm{W}}^{3} \), the mapping \( x \mapsto \psi \left( {u, x}\right) \) is an element \( f\left( u\right) \) of \( \mathrm{P} \) and\n\n\[ \left| {f\left( {{u}_{1}{u}_{2}{u}_{3}}\right)... | Yes |
Proposition 17. Let \( \Delta \) be the set of \( z \in \mathbf{C} \) such that \( - \pi < \mathcal{I}\left( z\right) < \pi \) and \( {\Delta }^{\prime } \) the set of \( z \in \mathbf{C} \) which are not real \( \leq 0 \) . Let \( \mathrm{E} \) be a complete normable space over \( \mathbf{C} \) and \( \mathrm{A} \) (r... | This follows from Spectral Theories, Chapter I,§ 4, Proposition 10 and no. 9. | No |
Theorem 6. Let \( \mathrm{E} \) be a real or complex Hilbert space and \( \mathrm{U} \) the unitary group of \( \mathrm{E} \) . (i) The set \( \mathrm{H} \) of Hermitian elements of \( \mathcal{L}\left( \mathrm{E}\right) \) is, with the real normed space structure, a closed vector subspace of \( \mathcal{L}\left( \math... | Recall that, if \( x \in \mathcal{L}\left( \mathrm{E}\right) ,{x}^{ * } \) denotes the adjoint of \( x \) . Let \( {\mathrm{H}}_{1} \) be the set of \( x \in \mathcal{L}\left( E\right) \) such that \( {x}^{ * } = - x \) . The formula \( x = \frac{1}{2}\left( {x + {x}^{ * }}\right) + \frac{1}{2}\left( {x - {x}^{ * }}\ri... | Yes |
Proposition 18. Let \( \mathrm{E} \) be a complete normable space over \( \mathbf{C}, v \in \mathcal{L}\left( \mathrm{E}\right) \) and \( g = \exp v. \) Suppose that \( \operatorname{Sp}\left( v\right) \) contains none of the points \( {2i\pi n} \) with \( n \in \mathbf{Z} - \{ 0\} \) . Then, for all \( x \in \mathrm{E... | This follows from Lemma 2 of no. 4, applied to the function \( z \mapsto {e}^{z} - 1 \) . | No |
Let \( u \in \mathcal{L}\left( \mathrm{E}\right) \) be such that every \( z \in \operatorname{Sp}u \) satisfies \( \left| {\mathcal{I}\left( z\right) }\right| < \frac{2\pi }{n + 1} \). Then, for all \( f \in \mathrm{F} \), the conditions \( \sigma \left( u\right) f = 0 \) and \( \wp \left( {\exp u}\right) f = f \) are ... | \[ L\left( \rho \right) = \sigma \] (§ 3, no. 11, Corollary 1 to Proposition 41) and hence \[ \rho \left( {\exp u}\right) = \exp \sigma \left( u\right) \] (no. 4, Corollary 3 to Proposition 10). By Proposition 18 it then suffices to prove that \( \operatorname{Sp}\sigma \left( u\right) \) does not meet \( {2i\pi }\left... | No |
Lemma 6. If \( v \in \mathcal{L}\left( \mathrm{E}\right) \), then \( \operatorname{Sp}\sigma \left( v\right) \subset \operatorname{Sp}v + \operatorname{Sp}v + \cdots + \operatorname{Sp}v \), where the sum comprises \( n + 1 \) terms. | We define elements \( {v}_{0},{v}_{1},\ldots ,{v}_{n} \) of \( \mathcal{L}\left( \mathrm{F}\right) \) by writing, for all \( f \in \mathrm{F} \) ,\n\n\[ \left( {{v}_{0}f}\right) \left( {{x}_{1},\ldots ,{x}_{n}}\right) = v\left( {f\left( {{x}_{1},\ldots ,{x}_{n}}\right) }\right) \]\n\n\[ \left( {{v}_{i}f}\right) \left( ... | Yes |
Corollary 2. Let \( \mathrm{E} \) be a complete normable algebra over \( \mathbf{C} \) and \( w \in \mathcal{L}\left( \mathrm{E}\right) \) . Suppose that every \( z \in \mathrm{{Sp}}w \) satisfies \( \left| {\mathcal{I}\left( z\right) }\right| < \frac{2\pi }{3} \) . The following conditions are equivalent :\n\n(i) \( w... | This follows from Corollary 1 with \( n = 2 \) and \( f \) the multiplication of E. | Yes |
Proposition 19. Let \( \mathrm{E} \) be a complete normable space over \( \mathbf{C}, v \in \mathcal{L}\left( \mathrm{E}\right) \) and \( g = \exp v \) . Suppose that every \( z \in \operatorname{Sp}v \) satisfies \( - \pi < \mathcal{I}\left( z\right) < \pi \) . Then, for every closed vector subspace \( {\mathrm{E}}^{\... | The condition \( v\left( {\mathrm{E}}^{\prime }\right) \subset {\mathrm{E}}^{\prime } \) implies \( g\left( {\mathrm{E}}^{\prime }\right) \subset {\mathrm{E}}^{\prime } \) and \( {g}^{-1}\left( {\mathrm{E}}^{\prime }\right) \subset {\mathrm{E}}^{\prime } \) and hence \( g\left( {\mathrm{E}}^{\prime }\right) = {\mathrm{... | Yes |
Lemma 7. Let \( \mathrm{B} \) be a group, \( \mathrm{A} \) a normal subgroup of \( \mathrm{B} \) , \( \mathrm{C} \) the group \( \mathrm{B}/\mathrm{A} \) and \( i \) : \( \mathrm{A} \rightarrow \mathrm{B} \) and \( p : \mathrm{B} \rightarrow \mathrm{C} \) the canonical morphisms. Let \( {\mathrm{A}}^{\prime } \) be a g... | (i) For \( {a}_{1},{a}_{2} \) in \( \mathrm{A} \), we have, in \( {\mathrm{B}}^{\prime \prime } \) , \[ \left( {f\left( {a}_{1}^{-1}\right), i\left( {a}_{1}\right) }\right) \left( {f\left( {a}_{2}^{-1}\right), i\left( {a}_{2}\right) }\right) = \left( {f\left( {a}_{1}^{-1}\right) \left( {\omega \left( {a}_{1}\right) f\l... | Yes |
Proposition 1. Let \( \mathrm{G} \) be a Lie group germ with identity element e. There exists a fundamental system of open neighbourhoods of \( e \) in \( \mathrm{G} \) consisting of Lie subgroups of \( \mathrm{G} \) . | Let \( \mathrm{L}\left( \mathrm{G}\right) \) be given a norm compatible with its topology and such that \( \parallel \left\lbrack {x, y}\right\rbrack \parallel \leq \parallel x\parallel \parallel y\parallel \) for all \( x, y \) in \( \mathrm{L}\left( \widetilde{\mathrm{G}}\right) \) . Let \( {\mathrm{G}}_{1} \) be the... | No |
Proposition 2. Let \( \mathrm{G} \) be a Lie group and \( \mathfrak{h} \) a Lie subalgebra of \( \mathrm{L}\left( \mathrm{G}\right) \) admitting a topological supplement. The following conditions are equivalent:\n\n(i) There exist an open subgroup \( {\mathrm{G}}^{\prime } \) of \( \mathrm{G} \) and a normal Lie subgro... | If there exist \( {\mathrm{G}}^{\prime } \) and \( \mathrm{H} \) with the properties of (i), then \( \mathrm{L}\left( {\mathrm{G}}^{\prime }\right) = \mathrm{L}\left( \mathrm{G}\right) \) and \( \mathrm{L}\left( \mathrm{H}\right) \) is an ideal of \( \mathrm{L}\left( {\mathrm{G}}^{\prime }\right) \) by \( §3 \), no. 12... | Yes |
Proposition 3. Let \( \mathrm{G} \) be a Lie group. There exists an exponential mapping \( \phi \) of \( \mathrm{G} \) with the following properties:\n\n(i) \( \phi \) is defined on an open subgroup \( \mathrm{U} \) of the additive group \( \mathrm{L}\left( \mathrm{G}\right) \) ;\n\n(ii) \( \phi \left( \mathbf{U}\right... | Let \( \mathrm{L}\left( \mathrm{G}\right) \) be given a norm compatible with its topology and such that \( \parallel \left\lbrack {x, y}\right\rbrack \parallel \leq \parallel x\parallel \parallel y\parallel \) for \( x, y \) in \( \mathrm{L}\left( \mathrm{G}\right) \) . Let \( {\mathrm{G}}_{1} \) be the Lie group defin... | Yes |
Proposition 4. Let \( \mathrm{G} \) be a Lie group and \( \phi \) an injective exponential mapping of \( \mathrm{G} \) . Suppose that \( p > 0 \) . For all \( x, y \) in \( \mathrm{L}\left( \mathrm{G}\right) \) ,\n\n(1)\n\n\[ x + y = \mathop{\lim }\limits_{{n \rightarrow + \infty }}{p}^{-n}{\phi }^{-1}\left( {\phi \lef... | These are special cases of Proposition 4 of \( §4 \), no. 3 . | No |
Proposition 5. Let \( \mathrm{G} \) be a standard group.\n\n(i) If \( \mathfrak{a} \) is a non-zero ideal of \( \mathrm{A} \) contained in \( \mathfrak{m},\mathrm{G}\left( \mathfrak{a}\right) \) is an open normal subgroup of \( G \) . | If \( x \in \mathrm{G} \) and \( y \in \mathrm{G}\left( \mathfrak{a}\right) \), the coordinates of \( x \) and \( x.y \) are equal modulo \( \mathfrak{a} \) . Hence, for \( {x}^{\prime },{x}^{\prime \prime } \) in \( \mathrm{G} \) and \( {y}^{\prime },{y}^{\prime \prime } \) in \( \mathrm{G}\left( \mathrm{a}\right) \),... | No |
Proposition 6. Let \( \mathfrak{a},\mathfrak{b},\mathfrak{c},{\mathfrak{c}}^{\prime } \) be non-zero ideals of \( \mathrm{A} \) contained in \( \mathfrak{m} \) such that\n\n\[{\mathfrak{c}}^{\prime } \subset \mathfrak{c},\;\mathfrak{a}\mathfrak{b} \subset \mathfrak{c},\;\mathfrak{a}{\mathfrak{b}}^{2} \subset {\mathfrak... | By \( §5 \), no. 2, Proposition 1, there exist \( {c}_{\alpha \beta } \in {\mathrm{A}}^{r} \) such that\n\n\[{x}^{\left\lbrack -1\right\rbrack } \cdot {y}^{\left\lbrack -1\right\rbrack } \cdot x \cdot y - \left\lbrack {x, y}\right\rbrack = \mathop{\sum }\limits_{{\left| \alpha \right| + \left| \beta \right| \geq 3}}{c}... | Yes |
Proposition 7. (i) The family \( \left( {\mathrm{G}\left( {\mathfrak{a}}_{\lambda }\right) }\right) \) is a central filtration on \( \mathrm{G} \) (Chapter II, § 4, no. 4, Definition 2). | (ii) is obvious. We prove (i). Clearly \( G\left( {\mathfrak{a}}_{\lambda }\right) = \mathop{\bigcap }\limits_{{\mu < \lambda }}G\left( {\mathfrak{a}}_{\mu }\right) \) and \( G = \mathop{\bigcup }\limits_{{\lambda > 0}}G\left( {\mathfrak{a}}_{\lambda }\right) \) .\n\nOn the other hand, if \( x \in \mathrm{G}\left( {\ma... | Yes |
Proposition 8. Let \( n \in \mathbf{Z} \) and \( {h}_{n} \) be the mapping \( x \mapsto {x}^{n} \) of \( \mathrm{G} \) into \( \mathrm{G} \) . Let \( \mathfrak{a} \) be a non-zero ideal of \( \mathrm{A} \) contained in \( \mathfrak{m} \), such that \( n \notin \mathfrak{a} \) . Then \( {h}_{n} \mid \mathrm{G}\left( \ma... | By definition of standard groups, \( {h}_{n} \) is equal on the whole of \( \mathrm{G} \) to the sum of an integral series with coefficients in \( {\mathrm{A}}^{r} \) . By \( §5 \), formula (4), this series is of the form\n\n\[ \n{h}_{n}\left( x\right) = {nx} + \mathop{\sum }\limits_{{\left| \alpha \right| \geq 2}}{a}_... | Yes |
Proposition 9. Suppose that \( p \neq 0 \) .\n\n(i) Let \( \mathfrak{a} \) , \( \mathfrak{b} \) be non-zero ideals of \( \mathrm{A} \) such that \( \mathfrak{b} \subset \mathfrak{a} \subset \mathfrak{m} \) . In the group \( \mathrm{G}\left( \mathfrak{a}\right) /\mathrm{G}\left( \mathfrak{b}\right) \) , every element ha... | By \( §5 \), formula (4), for all \( x \in \mathrm{G} \) ,\n\n\[ {x}^{p} = {px} + \mathop{\sum }\limits_{{\left| \alpha \right| \geq 2}}{c}_{\alpha }{x}^{\alpha } \]\n\nwhere \( {c}_{\alpha } \in {\mathrm{A}}^{r} \) for all \( \alpha \) . Even for proving (i) it can be assumed that \( v\left( p\right) = 1 \) . Then if ... | Yes |
Lemma 1. Suppose that \( p \neq 0 \) . Let \( \mathrm{G} \) be a Lie group, \( {\mathrm{G}}_{1} \) an open subgroup of \( \mathrm{G} \) which is isomorphic to a standard group and \( x \in \mathrm{G} \) . The following conditions are equivalent:\n\n(i) there exists a power of \( x \) which belongs to \( {\mathrm{G}}_{1... | (ii) \( \Rightarrow \) (i): obvious.\n\n(i) \( \Rightarrow \) (ii): suppose that \( y = {x}^{m} \in {\mathrm{G}}_{1} \) . By Proposition 9 (i) of no. \( 5,{y}^{{p}^{n}} \) tends to \( e \) as \( n \) tends to \( + \infty \), in other words \( {x}^{m{p}^{n}} \) tends to \( e \) as \( n \) tends to \( + \infty \) . | Yes |
Proposition 10. Suppose that \( p \neq 0 \) . Let \( \mathrm{G} \) be a finite-dimensional Lie group. Let \( {\mathrm{G}}_{f} \) be the set of \( x \in \mathrm{G} \) for which there exists a strictly increasing sequence \( \left( {n}_{i}\right) \) of integers such that \( {x}^{{n}_{i}} \) tends to e as i tends to \( + ... | There exists an open subgroup of \( \mathrm{G} \) which is isomorphic to a standard group (no. 3, Theorem 4). Assertion (i) then follows from Lemma 1.\n\nLet \( \mathrm{U} \) be an open subgroup of \( \mathrm{L}\left( \mathrm{G}\right) \) and \( \phi : \mathrm{U} \rightarrow \phi \left( \mathrm{U}\right) \) an exponent... | Yes |
Proposition 11. Suppose that \( p \neq 0 \) . Let \( x, y \) be two permutable elements of \( {\mathrm{G}}_{f} \) . Then \( {xy} \in {\mathrm{G}}_{f} \) and \( \log \left( {xy}\right) = \log x + \log y \) . | The fact that \( {xy} \in {\mathrm{G}}_{f} \) follows from Lemma 1. Let \( \mathrm{U} \) be an open subgroup of the additive group \( \mathrm{L}\left( \mathrm{G}\right) \) and \( \phi : \mathrm{U} \rightarrow \phi \left( \mathrm{U}\right) \) an exponential mapping of \( \overline{\mathrm{G}} \) with the properties of P... | Yes |
Proposition 12. Suppose that \( p \neq 0 \) . Let \( x \in {\mathrm{G}}_{f} \) . The following conditions are equivalent :\n\n(i) \( \log x = 0 \) ;\n\n(ii) \( x \) is of finite order in \( \mathrm{G} \) . | If there exists an integer \( n > 0 \) such that \( {x}^{n} = e \), it follows that\n\n\[ n\log x = \log {x}^{n} = 0, \]\n\nwhence \( \log x = 0 \) . If \( \log x = 0 \), let \( \mathrm{V} \) be a neighbourhood of \( e \) in \( {\mathrm{G}}_{f} \) such that \( \log \mid \mathrm{V} \) is the inverse mapping of an inject... | Yes |
Proposition 13. Suppose that \( p \neq 0 \) . If \( \mathrm{G} \) is compact or standard, then \( {\mathrm{G}}_{f} = \mathrm{G} \) . | If \( \mathrm{G} \) is standard, it suffices to use Lemma 1. Suppose that \( \mathrm{G} \) is compact. Let \( x \in \mathrm{G} \) and \( \mathrm{V} \) be a neighbourhood of \( e \) in \( \mathrm{G} \) . Let \( y \) be a limit point of the sequence \( {\left( {x}^{n}\right) }_{n \geq 0} \) . For all \( n > 0 \), there e... | Yes |
Proposition 14. Suppose that \( p \neq 0 \) and that \( v \) is chosen such that \( v\left( p\right) = 1 \) . Let \( \mathrm{G} \) be a standard group and \( \mathrm{E}\left( \mathrm{X}\right) \) (resp. \( \mathrm{L}\left( \mathrm{X}\right) \) ) the expansion of the exponential function of \( \mathrm{G} \) (resp. the l... | Using the notation of \( §5 \), nos. 3 and 4, \( \mathrm{E} = \mathop{\sum }\limits_{{m \geq 1}}\frac{{\psi }_{m, m}}{m!}(§5 \), no. 4, Proposition 3). As the coefficients \( {c}_{\alpha \beta \gamma } \) belong to \( \mathrm{A},\begin{Vmatrix}{\psi }_{m, m}\end{Vmatrix} \leq 1 \) (Differentiable and Analytic Manifolds... | Yes |
Corollary 1. Let \( \mathrm{G} \) be a topological group. There exists on \( \mathrm{G} \) at most one analytic manifold structure over \( \mathbf{R} \) (resp. \( {\widetilde{\mathbf{Q}}}_{p} \) ) compatible with the topological group structure on \( \mathrm{G} \) . | This follows immediately from Theorem 1. | No |
Corollary 2. Let \( \mathrm{G} \) be a topological group and \( \mathrm{V} \) an open neighbourhood of e. Suppose that \( \mathrm{V} \) has an analytic manifold structure which makes it into a real (resp. p-adic) Lie group germ. Then \( \mathrm{G} \) is a real (resp. p-adic) Lie group. | Let \( g \in \mathrm{G} \) . There exists an open neighbourhood \( {\mathrm{V}}^{\prime } \) of \( e \) in \( \mathrm{G} \) such that \( {\mathrm{V}}^{\prime } \cup g{\mathrm{\;V}}^{\prime }{g}^{-1} \subset \mathrm{V} \) . The mapping \( v \mapsto {gv}{g}^{-1} \) of \( {\mathrm{V}}^{\prime } \) into \( \mathrm{V} \) is... | Yes |
Proposition 1. Let \( \mathrm{G},{\mathrm{G}}^{\prime } \) be topological groups and fa continuous morphism of \( \mathrm{G} \) into \( {\mathrm{G}}^{\prime } \) . Suppose that one of the following three cases holds:\n\n(a) \( \mathrm{G} \) is a real Lie group and \( {\mathrm{G}}^{\prime } \) is a p-adic Lie group;\n\n... | Case (a). Let \( {\mathrm{G}}_{0} \) be the identity component of \( \mathrm{G} \) . Then \( f\left( {\mathrm{G}}_{0}\right) \) is a connected subgroup of \( {\mathrm{G}}^{\prime } \) and hence \( f\left( {\mathrm{G}}_{0}\right) = \{ e\} \) and \( {\mathrm{G}}_{0} \) is open in \( \mathrm{G} \) .\n\nCase (b). Let \( {\... | Yes |
Corollary 1. Let \( {\mathrm{G}}^{\prime } \) be a locally compact group, \( \mathrm{G} \) a finite-dimensional Lie group over \( \mathbf{R} \) (resp. \( {\mathbf{Q}}_{p} \) ) and \( f \) a continuous morphism of \( {\mathbf{G}}^{\prime } \) into \( \mathbf{G} \) . If the kernel of \( f \) is discrete, \( {\mathrm{G}}^... | There exists a compact neighbourhood \( \mathrm{V} \) of \( e \) in \( {\mathrm{G}}^{\prime } \) such that \( f \mid \mathrm{V} \) is a homeomorphism of V onto a compact subspace of G. If U is a sufficiently small open neighbourhood of \( e \) in \( \mathrm{G} \), the hypotheses of Theorem 2 are satisfied with \( \math... | Yes |
Corollary 2. Let \( \mathrm{G} \) be a finite-dimensional Lie group over \( \mathrm{K} \) , \( \mathrm{H} \) a subgroup of \( \mathrm{G} \) , \( \mathrm{V} \) an open neighbourhood of \( e \) in \( \mathrm{G} \) and \( {\left( {\mathrm{M}}_{i}\right) }_{i \in \mathrm{I}} \) a family of analytic manifolds over \( \mathr... | (i) Suppose that \( \mathrm{K} = \mathbf{C} \) . We consider \( \mathrm{G} \) as a real Lie group. Then \( \mathrm{H} \) is a real Lie subgroup of \( \mathrm{G} \) (Theorem 2). Let \( a \in \mathrm{L}\left( \mathrm{H}\right) \) . There exists a connected open neighbourhood \( \mathrm{W} \) of 0 in \( \mathbf{C} \) such... | Yes |
Proposition 1. Let \( \mathrm{G} \) be a topological group and \( \mathrm{A} \) and \( \mathrm{B} \) subgroups of \( \mathrm{G} \). Then \( \left( \overline{\mathrm{A},\mathrm{B}}\right) = \left( \overline{\overline{\mathrm{A}},\overline{\mathrm{B}}}\right) ,\overline{{\mathrm{D}}^{i}}\overline{\mathrm{A}} = \overline{... | Let \( \phi \) be the continuous mapping \( \left( {x, y}\right) \mapsto {x}^{-1}{y}^{-1}{xy} \) of \( \mathrm{G} \times \mathrm{G} \) into \( \mathrm{G} \). Then \( \phi \left( {\mathrm{A} \times \mathrm{B}}\right) \subset \left( {\mathrm{A},\mathrm{B}}\right) \), hence \( \phi \left( {\bar{\mathrm{A}} \times \overlin... | Yes |
Corollary 2. Let \( \mathrm{G} \) be a Hausdorff topological group and \( \mathrm{A} \) a subgroup of \( \mathrm{G} \). For A to be solvable (resp. nilpotent, commutative), it is necessary and sufficient that \( \overline{\mathrm{A}} \) be so. | This follows immediately from Proposition 1. | No |
Proposition 2. Let \( \mathrm{G} \) be a topological group and \( \mathrm{A} \) and \( \mathrm{B} \) subgroups of \( \mathrm{G} \) . If \( \mathrm{A} \) is connected, \( \left( {\mathrm{A},\mathrm{B}}\right) \) is connected. | For fixed \( y \) in B, the set \( {\mathrm{M}}_{y} \) of \( \left( {x, y}\right) \) with \( x \in \mathrm{A} \) is connected (for the mapping \( x \mapsto \left( {x, y}\right) \) of A into G is continuous). \( e \in {\mathrm{M}}_{y} \) and hence the union \( \mathrm{R} \) of the \( {\mathrm{M}}_{y} \) with \( y \in \m... | No |
Proposition 3. Let \( \mathrm{G} \) be a finite-dimensional Lie group and \( {\mathrm{H}}_{1} \) and \( {\mathrm{H}}_{2} \) subgroups of \( \mathrm{G} \) . Let \( {\mathfrak{h}}_{1},{\mathfrak{h}}_{2} \) and \( \mathfrak{h} \) be the Lie subalgebras tangent at e to \( {\mathrm{H}}_{1}{\mathrm{H}}_{2} \) and \( \left( {... | Let \( a \in {\mathfrak{h}}_{1}, b \in {\mathfrak{h}}_{2} \) . There exist an open neighbourhood I of 0 in \( \mathrm{K} \) and analytic mappings \( {f}_{1},{f}_{2} \) of \( \mathrm{I} \) into \( \mathrm{G} \) such that\n\n\[ \n{f}_{1}\left( 0\right) = {f}_{2}\left( 0\right) = e,\;{f}_{1}\left( \mathrm{I}\right) \subse... | Yes |
Proposition 4. Let \( \mathrm{G} \) be a finite-dimensional real or complex Lie group. Let \( \mathrm{A},\mathrm{B},\mathrm{C} \) be integral subgroups of \( \mathrm{G} \) such that \( \left\lbrack {\mathrm{L}\left( \mathrm{A}\right) ,\mathrm{L}\left( \mathrm{C}\right) }\right\rbrack \subset \mathrm{L}\left( \mathrm{C}... | Suppose that \( \left\lbrack {\mathrm{L}\left( \mathrm{A}\right) ,\mathrm{L}\left( \mathrm{B}\right) }\right\rbrack \subset \mathrm{L}\left( \mathrm{C}\right) \). The sum \( \mathrm{L}\left( \mathrm{A}\right) + \mathrm{L}\left( \mathrm{B}\right) + \mathrm{L}\left( \mathrm{C}\right) \) is a Lie subalgebra of \( \mathrm{... | Yes |
Proposition 5. Let \( \mathrm{G} \) be a finite-dimensional real or complex Lie group and \( \mathrm{A} \) an integral subgroup of \( \mathrm{G} \) . Then \( \mathrm{D}\overline{\mathrm{A}} = \mathrm{{DA}} \) . In particular, \( \widehat{\mathrm{A}} \) is a normal subgroup of \( \overline{\mathrm{A}} \) and \( \overlin... | Let \( \mathfrak{a} = \mathrm{L}\left( \mathrm{A}\right) \) . Let \( {\mathrm{G}}_{1} \) be the set of \( g \in \mathrm{G} \) such that\n\n\[ \left( {\operatorname{Ad}g}\right) x \equiv x\left( {{\;\operatorname{mod}\;\mathcal{D}}\mathfrak{a}}\right) \;\text{ for all }x \in \mathfrak{a}. \]\n\nThen \( {\mathrm{G}}_{1} ... | Yes |
Proposition 6. Suppose that \( \mathrm{K} \) is ultrametric. Let \( \mathrm{G} \) be a finite-dimensional Lie group. Let \( \mathrm{A},\mathrm{B},\ddot{\mathrm{C}} \) be Lie subgroups of \( \mathrm{G} \) such that \( \left\lbrack {\mathrm{L}\left( \mathrm{A}\right) ,\mathrm{L}\left( \mathrm{C}\right) }\right\rbrack \su... | Suppose that \( \left\lbrack {\mathrm{L}\left( \mathrm{A}\right) ,\mathrm{L}\left( \mathrm{B}\right) }\right\rbrack \subset \mathrm{L}\left( \mathrm{C}\right) \) . As in the proof of Proposition 4, the problem reduces to the case where \( \mathrm{L}\left( \mathrm{C}\right) \) is an ideal of \( \mathrm{L}\left( \mathrm{... | Yes |
Proposition 7. Let \( \mathrm{G} \) be a finite-dimensional Lie group, \( \mathfrak{g} \) its Lie algebra and \( \mathfrak{a} \) a subset of \( \mathfrak{g} \) . Then \( {Z}_{G}\left( \mathfrak{a}\right) \) is a Lie subgroup of \( G \) with Lie algebra \( {\mathfrak{z}}_{g}\left( \mathfrak{a}\right) \) . | This follows from \( §3 \), Proposition 44 and Corollary 2 to Proposition 39. | No |
Proposition 8. Let \( \mathrm{G} \) be a finite-dimensional real or complex Lie group, \( \mathrm{g} \) its Lie algebra and \( \mathrm{A} \) a subset of \( \mathrm{G} \) . Then \( {\mathrm{Z}}_{\mathrm{G}}\left( \mathrm{A}\right) \) is a Lie subgroup of \( \mathrm{G} \) with Lie algebra \( {\delta }_{g}\left( A\right) ... | Suppose that A consists of a single point \( a \) . Then \( {Z}_{G}\left( A\right) \) is the set of fixed points of Int \( a \) ; hence \( {Z}_{G}\left( A\right) \) is a Lie subgroup of \( G \) and \( L\left( {{Z}_{G}\left( A\right) }\right) \) is the set of fixed points of Ad \( a \), that is \( {\mathfrak{z}}_{\mathf... | Yes |
Proposition 9. Let \( \mathrm{G} \) be a finite-dimensional real or complex Lie group, \( \mathrm{g} \) its Lie algebra, A an integral subgroup of \( \mathrm{G} \) and \( \mathfrak{a} = \mathrm{L}\left( \mathrm{A}\right) \) . Then \( {\mathrm{Z}}_{\mathrm{G}}\left( \mathrm{A}\right) = {\mathrm{Z}}_{\mathrm{G}}\left( \m... | \n\[ x \in {\mathrm{Z}}_{\mathrm{G}}\left( \mathrm{A}\right) \Leftrightarrow \mathrm{A} \subset {\mathrm{Z}}_{\mathrm{G}}\left( {\{ x\} }\right) \]\n\[ \Leftrightarrow \mathfrak{a} \subset \mathrm{L}\left( {{\mathrm{Z}}_{\mathrm{G}}\left( {\{ x\} }\right) }\right) \;\left( {§6\text{, Corollary 2 to Proposition 3}}\righ... | Yes |
Proposition 10. Let \( \mathrm{G} \) be a finite-dimensional Lie group, \( \mathfrak{g} \) its Lie algebra and \( \mathfrak{a} \) a vector subspace of \( \mathfrak{g} \) . Then \( {\mathrm{N}}_{\mathfrak{G}}\left( \mathfrak{a}\right) \) is a Lie subgroup of \( \mathrm{G} \) with Lie algebra \( {\mathfrak{n}}_{\mathfrak... | This follows from § 3, Proposition 44 and Corollary 1 to Proposition 39. | No |
Proposition 11. Let \( \mathrm{G} \) be a finite-dimensional real or complex Lie group, \( \mathrm{g} \) its Lie algebra, A an integral subgroup of \( \mathrm{G} \) and \( \mathfrak{a} = \mathrm{L}\left( \mathrm{A}\right) \) . Then \( {\mathrm{N}}_{\mathrm{G}}\left( \mathrm{A}\right) = {\mathrm{N}}_{\mathrm{G}}\left( \... | The equality \( {\mathrm{N}}_{\mathrm{G}}\left( \mathrm{A}\right) = {\mathrm{N}}_{\mathrm{G}}\left( \mathrm{a}\right) \) follows from \( §6 \), no. 2, Corollary 2 to Proposition 3. By Proposition \( {10},{\mathrm{\;N}}_{\mathrm{G}}\left( \mathrm{A}\right) \) is then a Lie subgroup of \( \mathrm{G} \) with Lie algebra \... | Yes |
Proposition 12. Let \( \mathrm{G} \) be a finite-dimensional Lie group. For \( \mathrm{L}\left( \mathrm{G}\right) \) to be nilpotent, it is necessary and sufficient that \( \mathrm{G} \) have a nilpotent open subgroup. | Suppose that \( \mathrm{G} \) has a nilpotent open subgroup \( {\mathrm{G}}_{0} \) . By the Corollaries to Propositions 4 and 6, no. 2, \( {\mathcal{C}}^{i}{\mathrm{\;L}}^{i}\left( {\mathrm{G}}_{0}\right) = \{ 0\} \) for sufficiently large \( i \) . Hence \( \mathrm{L}\left( {\mathrm{G}}_{0}\right) = \mathrm{L}\left( \... | Yes |
Proposition 13. Let \( \mathrm{G} \) be a finite-dimensional simply connected nilpotent Lie group over \( \mathbf{R} \) or \( \mathbf{C} \). (i) \( {\exp }_{\mathrm{G}} \) is an isomorphism of the Lie group associated with \( \mathrm{L}\left( \mathrm{G}\right) \) onto \( \mathrm{G} \). (ii) Every integral subgroup of \... | Let \( \mathfrak{g} = \mathrm{L}\left( \mathrm{G}\right) \), which is nilpotent (Proposition 12). As two simply connected Lie groups over \( \mathbf{R} \) or \( \mathbf{C} \) which have the same Lie algebra are isomorphic (§ 6, no. 3, Theorem 3 (ii)), it suffices to prove the proposition when \( \mathrm{G} \) is the gr... | Yes |
Proposition 14. Let \( \mathbf{G} \) be a finite-dimensional connected Lie group over \( \mathbf{R} \) or \( \mathbf{C} \). (i) If \( \mathrm{G} \) is nilpotent, \( {\exp }_{\mathrm{G}} \) is étale and surjective. (ii) If \( \mathbf{K} = \widehat{\mathbf{C}} \) and \( {\exp }_{\mathrm{G}} \) is étale, then \( \mathrm{G... | Let \( {\mathrm{G}}^{\prime } \) be the universal covering space of \( \mathrm{G} \). Let \( \phi \) be the canonical morphism of \( {\mathrm{G}}^{\prime } \) onto \( \mathrm{G} \). Then \( {\exp }_{\mathrm{G}} = \phi \circ {\exp }_{{\mathrm{G}}^{\prime }}\left( {§6\text{, no. 4, Proposition 10}}\right) \) and hence (i... | Yes |
Proposition 15. Let \( \mathrm{G} \) be a finite-dimensional connected nilpotent Lie group over \( \mathbf{R} \) or \( \mathbf{C} \) and \( \mathrm{A} \) an integral subgroup of \( \mathrm{G} \) . Then \( {\mathrm{Z}}_{\mathrm{G}}\left( \mathrm{A}\right) \) is the connected Lie subgroup of \( \mathrm{G} \) with Lie alg... | By Proposition 9 of no. 3 it suffices to prove that \( {Z}_{G}\left( A\right) \) is connected. Let \( g \in {\mathrm{Z}}_{\mathrm{G}}\left( \mathrm{A}\right) \) . There exists \( x \in \mathrm{L}\left( \mathrm{G}\right) \) such that \( g = \exp x \) (Proposition 14). Then \( \operatorname{Ad}g \mid \mathrm{L}\left( \ma... | Yes |
Proposition 16. Let \( \mathbf{G} \) be a finite-dimensional nilpotent Lie group over \( \mathbf{R} \) or \( \mathbf{C} \) and \( \mathrm{A} \) an integral subgroup of \( \mathrm{G} \) distinct from \( \mathrm{G} \) . Then \( {\mathrm{N}}_{\mathrm{G}}\left( \mathrm{A}\right) \) is a connected Lie subgroup of \( \mathrm... | \( {\mathrm{N}}_{\mathrm{G}}\left( \mathrm{A}\right) \neq \mathrm{A} \) (Algebra, Chapter I,§ 6, Corollary 1 to Proposition 8). By Proposition 11 of no. 4, we need only prove that \( {\mathrm{N}}_{\mathrm{G}}\left( \mathrm{A}\right) \) is connected. Let \( g \in {\mathrm{N}}_{\mathrm{G}}\left( \mathrm{A}\right) \) . Th... | Yes |
Proposition 17. Let \( \mathfrak{g} \) be a finite-dimensional nilpotent Lie algebra over \( \mathrm{K} \) and \( \left( {{\mathfrak{g}}_{0},{\mathfrak{g}}_{1},\ldots ,{\mathfrak{g}}_{n}}\right) \) a decreasing sequence of ideals of \( \mathfrak{g} \) such that \( {\mathfrak{g}}_{0} = \mathfrak{g},{\mathfrak{g}}_{n} = ... | The proposition is obvious for \( \dim \mathfrak{g} = 0 \) . Suppose that \( \dim \mathfrak{g} > 0 \) and that the proposition has been established for dimensions \( < \dim g \) . It can be assumed that \( {\mathfrak{g}}_{n - 1} \neq \{ 0\} \) and \( {\mathfrak{g}}_{n - 1} \) is then a non-zero central ideal of \( \mat... | Yes |
Proposition 18. Let \( k \) be a commutative field. V a vector space of finite dimension \( > 0 \) over \( k \) and \( \mathbf{G} \) a subgroup of \( \mathbf{{GL}}\left( \mathbf{V}\right) \) whose elements are unipotent.\n\n(i) There exists a non-zero element \( v \) of \( \mathrm{V} \) such that \( {gv} = v \) for all... | (a) Suppose first that \( k \) is algebraically closed and that the identity representation of \( \mathrm{G} \) is simple. Let \( a, b \) be in \( \mathrm{G} \) . Then\n\n\[\n\operatorname{Tr}\left( {a\left( {b - 1}\right) }\right) = \operatorname{Tr}\left( {{ab} - 1}\right) - \operatorname{Tr}\left( {a - 1}\right) = 0... | Yes |
Corollary 1. Let \( \mathrm{G} \) be a finite-dimensional connected real or complex Lie group. For \( \mathrm{G} \) to be nilpotent, it is necessary and sufficient that every element of \( \mathrm{{Ad}}\mathrm{G} \) be nilpotent. | If every element of Ad G is unipotent, Ad G is nilpotent (Proposition 18) and hence \( \mathrm{G} \), which is a central extension of \( \mathrm{{Ad}}\mathrm{G} \), is nilpotent. If \( \mathrm{G} \) is nilpotent, \( \mathrm{L}\left( \mathrm{G}\right) \) is nilpotent, hence ad \( x \) is nilpotent for all \( x \in \math... | Yes |
Proposition 19. Let \( \mathrm{G} \) be a finite-dimensional Lie group. For \( \mathrm{L}\left( \mathrm{G}\right) \) to be solvable, it is necessary and sufficient that \( \mathrm{G} \) possess a solvable open subgroup. | The proof is analogous to that of Proposition 12 of no. 5. | No |
Proposition 20. Let \( \mathrm{G} \) be a simply connected solvable Lie group of finite-dimension \( n \) over \( \mathbf{R} \) or \( \mathbf{C} \) and \( \mathfrak{g} = \mathrm{L}\left( \mathrm{G}\right) \) . Let \( \left( {{\mathfrak{g}}_{n},{\mathfrak{g}}_{n - 1},\ldots ,{\mathfrak{g}}_{0}}\right) \) be a sequence o... | For \( n = 0 \) the proposition is obvious. We argue by induction on \( n \) . Let \( \mathrm{H} \) be the integral subgroup of \( \mathrm{G} \) such that \( \mathrm{L}\left( \mathrm{H}\right) = \mathrm{K}{x}_{n} \) . By \( §6 \), no. 6, Corollary 1 to Proposition 14, \( \mathrm{H} \) and \( {\mathrm{G}}_{n - 1} \) are... | Yes |
Proposition 21. Let \( \mathrm{G} \) be a simply connected finite-dimensional solvable Lie group over \( \mathbf{R} \) or \( \mathbf{C} \) and \( \mathrm{M} \) an integral subgroup of \( \mathrm{G} \) . Then \( \mathrm{M} \) is a Lie subgroup of \( \mathrm{G} \) and is simply connected. | We continue to use the notation \( n, g,{g}_{i},{x}_{i} \) , \( \phi \) of Proposition 20 but impose on the \( {x}_{i} \) the following supplementary condition: let \( {i}_{p} > {i}_{p - 1} > \cdots > {i}_{1} \) be the integers \( i \) such that \( \mathrm{L}\left( \mathrm{M}\right) \cap {\mathfrak{g}}_{i} \neq \mathrm... | Yes |
Proposition 22. Suppose that \( \mathrm{K} = \mathbf{R} \) or \( \mathbf{C} \) . Let \( \mathrm{V} \) be a finite-dimensional vector space and \( \mathbf{G} \) a connected solvable subgroup of \( \mathbf{{GL}}\left( \mathrm{V}\right) \) . Suppose that the identity representation of \( \mathrm{G} \) is simple.\n\n(i) If... | (i) Suppose that \( \mathrm{K} = \mathbf{R} \) . Then the closure \( \mathrm{H} \) of \( \mathrm{G} \) in \( \mathbf{{GL}}\left( \mathrm{V}\right) \) is a solvable connected Lie subgroup of \( \mathbf{{GL}}\left( V\right) \) (no. 1, Corollary 2 to Proposition 1). Hence L(H) is solvable (Proposition 19). The identity re... | Yes |
Proposition 23. Let \( \mathrm{G} \) be a finite-dimensional rear or complex Lie group, \( \mathrm{r} \) the radical of \( \mathrm{L}\left( \mathrm{G}\right) \) (Chapter I,§ 5, Definition 2) and \( \mathrm{n} \) the largest nilpotent ideal of \( \mathrm{L}\left( \mathrm{G}\right) \) (Chapter I,§ 4, no. 4). Let \( \math... | The group \( \mathbf{R} \) is solvable (§ 6, Proposition 19). Suppose that \( \mathrm{K} = \mathbf{R} \) . Let \( {\mathrm{G}}^{\prime } \) be a connected solvable normal subgroup of \( \mathrm{G} \) . Then \( \overline{{\mathrm{G}}^{\prime }} \) is a connected solvable (no. 1, Corollary 2 to Proposition 1) normal Lie ... | Yes |
Proposition 24. Suppose that \( \mathrm{K} = \mathbf{R} \) or \( \mathbf{C} \) . Let \( {\mathrm{G}}_{1},{\mathrm{G}}_{2} \) be two finite-dimensional connected Lie groups, \( {\mathrm{R}}_{1} \) and \( {\mathrm{R}}_{2} \) their radicals and \( \phi \) a surjective morphism of \( {\mathrm{G}}_{1} \) into \( {\mathrm{G}... | By \( §3 \), no. 8, Proposition 28, \( \mathrm{L}\left( \phi \right) \) is surjective. Hence \( \mathrm{L}\left( \phi \right) \left( {\mathrm{L}\left( {\mathrm{R}}_{1}\right) }\right) = \mathrm{L}\left( {\mathrm{R}}_{2}\right) \) (Chapter I,§ 6, Corollary 3 to Proposition 2). Let \( i \) be the canonical injection of \... | Yes |
Proposition 25. Suppose that \( \mathrm{K} = \mathbf{R} \) or \( \mathbf{C} \) . Let \( {\mathrm{G}}_{1},{\mathrm{G}}_{2} \) be finite-dimensional connected Lie groups and \( {\mathrm{R}}_{1} \) and \( {\mathrm{R}}_{2} \) their radicals. The radical of \( {\mathrm{G}}_{1} \times {\mathrm{G}}_{2} \) is \( {\mathrm{R}}_{... | This follows from Chapter I, § 5, Proposition 4. | No |
Proposition 26. Let \( \mathrm{G} \) be a finite-dimensional connected real or complex Lie group. The following conditions are equivalent:\n\n(i) \( \mathrm{L}\left( \mathrm{G}\right) \) is semi-simple;\n\n(ii) the radical of \( \mathrm{G} \) is \( \{ e\} \) ;\n\n(iii) every normal commutative integral subgroup of \( \... | Condition (ii) means that the radical of \( \mathrm{L}\left( \mathrm{G}\right) \) is \( \{ 0\} \) and hence (i) \( \Leftrightarrow \) (ii) (Chapter I,§ 6, Theorem 1). The equivalence of (i) and (iii) follows from \( §6 \) , no. 6, Proposition 14. | No |
Proposition 27. Let \( \mathrm{G} \) be a finite-dimensional connected real or complex Lie group. The following conditions are equivalent:\n\n(i) \( \mathrm{L}\left( \mathrm{G}\right) \) is simple;\n\n(ii) the only normal integral subgroups of \( \mathrm{G} \) are \( \{ e\} \) and \( \mathrm{G} \) and further \( \mathr... | This follows from \( §6 \), no. 6, Proposition 14. | No |
Proposition 28. Let \( \mathrm{G} \) be a simply connected real or complex Lie group. The following conditions are equivalent:\n\n(i) \( \mathrm{G} \) is semi-simple;\n\n(ii) \( \mathrm{G} \) is isomorphic to the product of a finite number of almost simple groups. | If \( \mathrm{G} \) is a finite product of almost simple Lie groups, \( \mathrm{L}\left( \mathrm{G}\right) \) is a finite product of simple Lie algebras and is hence semi-simple. If \( \mathrm{G} \) is semi-simple, \( \mathrm{L}\left( \mathrm{G}\right) \) is isomorphic to a product of simple Lie algebras \( {\mathrm{L}... | Yes |
Lemma 1. Let \( \mathrm{G} \) be a connected topological group, \( \mathrm{Z} \) its centre and \( {\mathrm{Z}}^{\prime } \) a discrete subgroup of \( Z \) . Then the centre of \( G/{Z}^{\prime } \) is \( Z/{Z}^{\prime } \) . | Let \( y \) be an element of \( \mathrm{G} \) whose class modulo \( {\mathrm{Z}}^{\prime } \) is a central element of \( \mathrm{G}/{\mathrm{Z}}^{\prime } \) . Let \( \phi \) be the mapping \( g \mapsto {gy}{g}^{-1}{y}^{-1} \) of \( \mathrm{G} \) into \( \mathrm{G} \) . Then \( \phi \left( \mathrm{G}\right) \) is conne... | Yes |
Proposition 29. Let \( \mathrm{G} \) be a semi-simple connected real or complex Lie group.\n\n(i) \( G = \left( {G, G}\right) \).\n\n(ii) The centre \( \dot{\mathrm{Z}} \) of \( \mathrm{G} \) is discrete.\n\n(iii) The centre of \( \mathrm{G}/\mathrm{Z} \) is \( \{ e\} \) . | Assertion (i) follows from the Corollary to Proposition 4, no. 2 and Chapter \( I,§6 \), Theorem 1.\n\nAssertion (ii) follows from \( §6 \), no. 4, Corollary 4 to Proposition 10 and Chapter I, § 6, no. 1, Remark 2.\n\nAssertion (iii) follows from (ii) and Lemma 1. | Yes |
Proposition 30. (i) Let \( \mathfrak{g} \) be a semi-simple real or complex Lie algebra. Then Int \( \mathfrak{g} \) is the identity component of Aut \( \mathfrak{g} \) . | Every derivation of \( \mathfrak{g} \) is inner (Chapter I,§ 6, Corollary 3 to Proposition 1) and hence \( L\left( {\operatorname{Int}g}\right) = L\left( {\operatorname{Aut}g}\right) \), which proves (i). | Yes |
Proposition 31. Let \( \mathrm{G} \) be a simply connected finite-dimensional real or complex Lie group and \( \mathrm{R} \) its radical. There exists a semi-simple simply connected Lie subgroup \( \mathrm{S} \) of \( \mathrm{G} \) such that \( \mathrm{G} \), as a Lie group, is the semi-direct product of \( \mathrm{S} ... | This follows from \( §6 \), no. 6, Corollary 1 to Proposition 14 and Chapter I, \( §6 \), Theorem 5 and Corollary 1. | No |
(i) If \( \rho \) is semi-simple, \( {\rho }^{\prime } \) is semi-simple. | Suppose that \( \rho \) is semi-simple; we prove that \( {\rho }^{\prime } \) is semi-simple. It suffices to consider the case where \( \rho \) is simple. Let \( {\mathrm{V}}^{\prime } \) be a minimal non-zero sub- \( {\mathrm{G}}^{\prime } \) - module of \( \mathrm{V} \) . For all \( g \in \mathrm{G},\rho \left( {\mat... | Yes |
Corollary 2. Let \( \rho \) be a finite-dimensional semi-simple analytic linear representation of \( \mathrm{G} \) on a vector space \( \mathrm{V},\mathrm{S} \) the symmetric algebra of \( \mathrm{V} \) and \( {\mathrm{S}}^{\mathrm{G}} \) the subalgebra of \( \mathrm{S} \) consisting of the elements invariant under \( ... | This follows from Theorem 1, Chapter I, § 6, Theorem 6 (a) and Commutative Algebra, Chapter V, \( §1 \), Theorem 2. | No |
Proposition 32. Let \( \mathrm{G} \) be a finite-dimensional connected real Lie group. Suppose that \( \mathrm{L}\left( \mathrm{G}\right) \) is reductive. The following conditions are equivalent:\n\n(i) \( \mathrm{G}/\overline{{\mathrm{D}}^{1}}\mathrm{G} \) is compact;\n\n(ii) (resp. \( \left( {\mathrm{{ii}}}^{\prime }... | (i) \( \Rightarrow \) (ii’): Suppose that \( \mathrm{G}/\overline{\mathrm{D}}{}^{1}\mathrm{G} \) is compact. Then every continuous linear representation of \( \mathrm{G}/\overline{{\mathrm{D}}^{1}}\mathrm{G} \) on a finite-dimensional real vector space is semi-simple (Integration, Chapter VII,§ 3, Proposition 1). Let \... | Yes |
Proposition 33. Let \( \mathrm{G} \) be a finite-dimensional complex Lie group whose number of connected components is finite, \( \rho \) a finite-dimensional analytic linear representation of \( \breve{\mathrm{G}} \) and \( {\mathrm{G}}^{\prime } \) an integral subgroup of the real Lie group \( \mathrm{G} \) such that... | Let \( {\rho }^{\prime } = \rho \mid {\mathrm{G}}^{\prime } \) . For \( \rho \) (resp. \( {\rho }^{\prime } \) ) to be semi-simple, it is necessary and sufficient that \( \mathrm{L}\left( \rho \right) \) (resp. \( \mathrm{L}\left( {\rho }^{\prime }\right) \) ) be semi-simple (Theorem 1). Let \( \mathrm{V} \) be the spa... | Yes |
Lemma 1. Let \( \mathrm{G} \) be a Lie group and \( \alpha \) a vector field on \( \mathrm{G} \) . For all \( g \in \mathrm{G} \), let\n\n\[ \n\beta \left( g\right) = \alpha \left( g\right) {g}^{-1} \in \mathrm{L}\left( \mathrm{G}\right) .\n\]\n\nThe following conditions are equivalent:\n\n(i) \( \propto \) is a homomo... | But the product of \( \beta \left( g\right) \) and \( \left( {\operatorname{Ad}g}\right) \beta \left( {g}^{\prime }\right) \) in \( \mathrm{T}\left( \mathrm{G}\right) \) is just the sum of \( \beta \left( g\right) \) and \( \left( {\operatorname{Ad}g}\right) \beta \left( {g}^{\prime }\right) \) in \( \mathrm{L}\left( \... | Yes |
Proposition 1. Let \( {\mathrm{K}}^{\prime } \) be a non-discrete closed subfield of \( \mathrm{K},\mathrm{G} \) a Lie group over \( \mathrm{K},\mathrm{V} \) a manifold over \( {\mathrm{K}}^{\prime } \) and \( \left( {v, g}\right) \mapsto {vg} \) a \( {\mathrm{K}}^{\prime } \) -analytic mapping of \( \mathrm{V} \times ... | For \( v \in \mathrm{V},{g}_{1} \in \mathrm{G},{g}_{2} \in \mathrm{G}, v\left( {{g}_{1}{g}_{2}}\right) = \left( {v{g}_{1}}\right) \left( {v{g}_{2}}\right) \) . Hence, for \( {u}_{1} \in \mathrm{{TG}} \) , \( {u}_{2} \in \mathrm{{TG}}, a\left( {{u}_{1}{u}_{2}}\right) = \left( {a{u}_{1}}\right) \left( {a{u}_{2}}\right) \... | Yes |
Proposition 3. Suppose that \( \mathrm{K} \) is ultrametric. Let \( \mathrm{G} \) be a compact Lie group and \( \alpha \) an infinitesimal automorphism of \( \mathrm{G} \) . There exist an open subgroup \( \mathrm{I} \) of \( \mathrm{K} \) and a law of analytic operation \( \left( {\lambda, g}\right) \mapsto {\phi }_{\... | As \( \ddot{\mathrm{G}} \) is compact, there exist an open subgroup \( {\mathrm{I}}^{\prime } \) of \( \mathrm{K} \) and a law of analytic operation \( \left( {\lambda, g}\right) \mapsto {\phi }_{\lambda }\left( g\right) \) of \( {\mathrm{I}}^{\prime } \) on \( \mathrm{G} \) with property (1) of the proposition\n\n(§ 4... | Yes |
Lemma 3. Let \( \mathrm{G} \) and \( {\mathrm{G}}^{\prime } \) be Lie groups and \( \mathrm{o} \) a homomorphism of \( \mathrm{G} \) into \( \operatorname{Aut}\left( {\mathrm{G}}^{\prime }\right) \) . Let \( f\left( {g,{g}^{\prime }}\right) = \left( {\phi \left( g\right) }\right) \left( {g}^{\prime }\right) \) for \( g... | Clearly (i) implies (ii) and (iii). Let \( {g}_{0} \in \mathbf{G},{g}_{0}^{\prime } \in {\mathbf{G}}^{\prime } \) . For all \( g \in \mathbf{G},{g}^{\prime } \in {\mathbf{G}}^{\prime } \),\n\n\[ f\left( {g{g}_{0},{g}^{\prime }{g}_{0}^{\prime }}\right) = \left( {\phi \left( g\right) \phi \left( {g}_{0}\right) }\right) \... | Yes |
Lemma 4. Let \( \mathrm{H} \) be a finite-dimensional simply connected Lie group.\n\n(i) For all \( u \in \operatorname{Aut}\mathrm{L}\left( \mathrm{H}\right) \), let \( \theta \left( u\right) \) be the unique automorphism of \( \mathrm{H} \) such that \( \mathrm{L}\left( {\dot{\theta }\left( u\right) }\right) = u \) .... | To prove (i), it suffices, by Lemma 3 of no. 1, to verify that the mapping \( \left( {u, g}\right) \mapsto \theta \left( u\right) g \) is analytic in a neighbourhood of \( \left( {{\operatorname{Id}}_{\mathrm{L}\left( \mathrm{H}\right) }, e}\right) \) . There exists an\n\nopen neighbourhood B of 0 in L(H) such that \( ... | Yes |
Corollary 2. Let \( \mathrm{G} \) be a semi-simple connected real or complex Lie group. The group Int \( \mathrm{G} \) is the identity component of \( \mathrm{{Aut}}\mathrm{G} \) . | The mapping \( u \mapsto \mathrm{L}\left( u\right) \) is an isomorphism of Aut \( \mathrm{G} \) onto a Lie subgroup of Aut L(G) (Theorem 1). The image of Int G under this isomorphism is \( \mathrm{{Ad}}\mathrm{G} \) . But \( \mathrm{{Ad}}\mathrm{G} \) is the identity component of \( \mathrm{{Aut}}\mathrm{L}\left( \math... | Yes |
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