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Proposition 20. Let \( \mathrm{G},{\mathrm{G}}^{\prime } \) be Lie groups, \( \mathrm{X} \) a manifold of class \( {\mathrm{C}}^{r} \) and \( \left( {g, x}\right) \mapsto {gx} \) (resp. \( \left( {x,{g}^{\prime }}\right) \mapsto x{g}^{\prime } \) ) a law of left (resp. right) operation of class \( {\mathrm{C}}^{r} \) o... | \[ \left( {t * f}\right) * {t}^{\prime } = t * \left( {f * {t}^{\prime }}\right) . \] For all \( x \in \mathrm{X} \), \[ \left\langle {{\varepsilon }_{x},\left( {t * f}\right) * {t}^{\prime }}\right\rangle = \left\langle {{\varepsilon }_{x} * {t}^{\prime \nu }, t * f}\right\rangle \] \[ = \langle {t}^{ \vee }\; * \left... | Yes |
Proposition 21. Let \( \psi \) be an automorphism of \( \mathbf{X} \) commuting with the operations of G. Then \( {\mathrm{D}}_{t} \) is invariant under \( \psi \) . | For all \( x \in \mathrm{X} \) , \n\n\[ \n\left( {\psi \left( {\mathrm{D}}_{t}\right) }\right) \left( {\psi \left( x\right) }\right) = {\psi }_{ * }\left( {{\mathrm{D}}_{t}\left( x\right) }\right) = {\psi }_{ * }\left( {t * {\varepsilon }_{x}}\right) \n\] \n\n\[ \n= t * {\psi }_{ * }\left( {\varepsilon }_{x}\right) \;\... | Yes |
Proposition 22. If \( g \in \mathrm{G} \), the transform of \( {\mathrm{D}}_{t} \) under \( g \) is \( {\mathrm{D}}_{{\varepsilon }_{g} * t * {\varepsilon }_{g} - 1} \) . | The value of this transform at \( {gx} \) is\n\n\[ \tau {\left( g\right) }_{ * }\left( {{\mathrm{D}}_{t}\left( x\right) }\right) = \tau {\left( g\right) }_{ * }\left( {t * {\varepsilon }_{x}}\right) \]\n\n\[ = {\varepsilon }_{g} * \left( {t * {\varepsilon }_{x}}\right) \;\text{(Proposition 14 (i))} \]\n\n\[ = \left( {{... | Yes |
Proposition 23. (i) The mapping \( t \mapsto {\mathrm{L}}_{t} \) (resp. \( t \mapsto {\mathrm{R}}_{t} \) ) is an isomorphism of the vector space \( \mathrm{U}\left( \mathrm{G}\right) \) onto the vector field of left (resp. right) invariant distributions on \( \mathrm{G} \) . | In G every right translation commutes with every left translation. By Proposition 21 of no. 5, \( {\mathrm{L}}_{t} \) is therefore left invariant. As \( {\left( {\mathrm{L}}_{t}\right) }_{e} = t \), the mapping \( t \mapsto {\mathrm{L}}_{t} \) is injective. Let \( \Delta \) be a field of left invariant distributions on... | Yes |
Lemma 1. Let \( \mathrm{X} \) and \( {\mathrm{X}}^{\prime } \) be complete normable spaces, \( {\mathrm{X}}_{0} \) an open neighbourhood of 0 in \( \mathrm{X} \) and \( f \) an analytic mapping of \( {\mathrm{X}}_{0} \) into \( {\mathrm{X}}^{\prime } \) such that \( f\left( 0\right) = 0 \) . Let \( f = {f}_{1} + {f}_{2... | Let \( {t}_{1}^{\prime } \) be this component. Then, for every continuous linear mapping \( u \) of \( {\mathrm{X}}^{\prime } \) into a polynormed space,\n\n\[ u\left( {t}_{1}^{\prime }\right) = \left\langle {{t}^{\prime }, u}\right\rangle \;\text{ because }u\text{ is continuous and linear }\n\]\n\[ = \langle t, u \cir... | Yes |
Proposition 24. Let \( \mathrm{G} \) be a Lie group and \( \left( {\mathrm{U},\phi ,\mathrm{E}}\right) \) a chart on \( \mathrm{G} \) such that \( \phi \left( e\right) = 0 \) . Let \( \mathrm{V} \) be an open neighbourhood of \( e \) such that \( {\mathrm{V}}^{2} \subset \mathrm{U} \) . Let \( m \) be the analytic mapp... | \n\n\( m\left( {a,0}\right) = a, m\left( {0, b}\right) = b \) for all \( a, b \) in \( \phi \left( \mathrm{V}\right) \), which proves (i) and (ii). Let \( u, v \) be in \( {\mathrm{T}}_{e}\left( \mathrm{G}\right) \) . Let \( {\mathrm{T}}_{0}\left( \mathrm{E}\right) \) be identified with \( \mathrm{E} \) and hence \( \p... | Yes |
Proposition 25. Let \( \mathrm{G} \) be a Lie group and \( \mathrm{E}\left( \mathrm{G}\right) \) the enveloping algebra of \( \mathrm{L}\left( \mathrm{G}\right) \) . The canonical injection of \( \mathrm{L}\left( \mathrm{G}\right) \) into \( \mathrm{E}\left( \mathrm{G}\right) \) defines a homomorphism \( \theta \) of t... | The bigebra \( \mathrm{U}\left( \mathrm{G}\right) \) is cocommutative (Differentiable and Analytic Manifolds, \( \mathrm{R} \) ,13.5.1) and the filtration \( \left( {{\mathrm{U}}_{s}\left( \mathrm{G}\right) }\right) \) is compatible with the bigebra structure. The set of primitive elements of \( \mathrm{U}\left( \mathr... | No |
Proposition 26. Suppose that \( \mathrm{K} \) is of characteristic \( p > 0 \) . For all \( a \in \mathbf{L}\left( \mathrm{G}\right) \) , \( {a}^{p} \in \mathrm{L}\left( \mathrm{G}\right) \) and \( \operatorname{ad}\left( {a}^{p}\right) = {\left( \operatorname{ad}a\right) }^{p} \) (the power \( {a}^{p} \) being calcula... | If \( a \in \mathrm{L}\left( \mathrm{G}\right), a \) is primitive in \( \mathrm{U}\left( \mathrm{G}\right) \), hence \( {a}^{p} \) is primitive in \( \mathrm{U}\left( \mathrm{G}\right) \) (Chapter II,§ 1, no. 2, Remark 1) and hence \( {a}^{p} \in \mathrm{L}\left( \mathrm{G}\right) \) . Let \( {\sigma }_{a} \) (resp. \(... | Yes |
Proposition 27. Let \( \mathrm{G} \) be a Lie group and \( \mathrm{X} \) a manifold of class \( {\mathrm{C}}^{r} \) . Suppose that a law of left (resp. right) operation of class \( {\mathrm{C}}^{r} \) of \( \mathrm{G} \) on \( \mathrm{X} \) is given. For all \( a \in \mathrm{L}\left( \mathrm{G}\right) \) , let \( {\mat... | Suppose that \( \mathrm{G} \) operates on \( \mathrm{X} \) on the left. Let \( \phi : \mathrm{G} \times \mathrm{X} \rightarrow \mathrm{X} \) be the law of operation. Then \( \mathrm{T}\left( \phi \right) \) is a \( \phi \) -morphism of class \( {\mathrm{C}}^{r - 1} \) of the vector bundle \( T\left( G\right) \times T\l... | Yes |
Proposition 28. Let \( \mathrm{G} \) and \( \mathrm{H} \) be Lie groups and \( \phi \) a morphism of \( \mathrm{G} \) into \( \mathrm{H} \) . Suppose that \( \mathrm{K} \) is of characteristic 0 and that \( \mathrm{H} \) is finite-dimensional.\n\n(i) The kernel \( \mathrm{N} \) of \( \phi \) is a Lie subgroup of \( \ma... | Let \( \mathrm{G} \) operate on \( \mathrm{H} \) on the left by the mapping \( \left( {g, h}\right) \mapsto \phi \left( g\right) h \) . It suffices to apply Proposition 14 of \( §1 \), no. 7, to the orbit of \( e \) . | Yes |
Proposition 29. Let \( \mathrm{G} \) and \( \mathrm{H} \) be Lie groups and \( \phi \) a morphism of \( \mathrm{G} \) into \( \mathrm{H} \) . Suppose that \( \mathrm{K} \) is of characteristic 0 and that \( \mathrm{H} \) is finite-dimensional. If \( {\mathrm{H}}^{\prime } \) is a Lie subgroup of \( \mathrm{H} \), then ... | Let \( \pi \) be the canonical mapping of \( \mathrm{H} \) into the homogeneous space \( \mathrm{X} = \mathrm{H}/{\mathrm{H}}^{\prime } \) . Let \( \mathrm{G} \) operate on \( \mathrm{X} \) on the left by the mapping \( \left( {g, x}\right) \mapsto \phi \left( g\right) x \) . The stabilizer of \( \pi \left( e\right) \)... | Yes |
Corollary 1. Let \( \mathrm{G},\mathrm{H} \) be Lie groups and \( {\phi }_{1} \) and \( {\phi }_{2} \) morphisms of \( \mathrm{G} \) into \( \mathrm{H} \) . Suppose that \( \mathrm{K} \) is of characteristic 0 and that \( \mathrm{H} \) is finite-dimensional. The set of \( g \in \mathrm{G} \) such that \( {\phi }_{1}\le... | We write \( \phi \left( g\right) = \left( {{\phi }_{1}\left( g\right) ,{\phi }_{2}\left( g\right) }\right) \) for all \( g \in \mathrm{G} \), so that \( \phi \) is a morphism of \( \mathrm{G} \) into \( \mathrm{H} \times \mathrm{H} \) . Let \( \Delta \) be the diagonal subgroup of \( \mathrm{H} \times \mathrm{H} \) . T... | Yes |
Corollary 2. Let \( \mathrm{G} \) be a finite-dimensional Lie group and \( {\mathrm{G}}_{1} \) and \( {\mathrm{G}}_{2} \) two Lie subgroups of \( \mathrm{G} \) . Suppose that \( \mathrm{K} \) is of characteristic 0 . Then \( {G}_{1} \cap {\mathrm{G}}_{2} \) is a Lie subgroup of \( \mathrm{G} \) with Lie algebra \( \mat... | We apply Proposition 29 to the canonical injection of \( {\mathrm{G}}_{1} \) into \( \mathrm{G} \) and the subgroup \( {\mathrm{G}}_{2} \) . | No |
Corollary 3. Let \( \mathrm{G},{\mathrm{G}}^{\prime },\mathrm{H} \) be Lie groups and \( \phi : \mathrm{G} \rightarrow \mathrm{H} \) and \( {\phi }^{\prime } : {\mathrm{G}}^{\prime } \rightarrow \mathrm{H} \) Lie group morphisms. Suppose that \( \mathrm{K} \) is of characteristic \( 0 \) and that \( \mathrm{H} \) is fi... | We apply Corollary 1 to the morphisms \( \left( {g,{g}^{\prime }}\right) \mapsto \phi \left( g\right) \) and \( \left( {g,{g}^{\prime }}\right) \mapsto {\phi }^{\prime }\left( {g}^{\prime }\right) \) of \( \mathrm{G} \times {\mathrm{G}}^{\prime } \) into \( \mathrm{H} \) . | Yes |
Proposition 30. Let \( \mathrm{G} \) be a finite-dimensional Lie group with a countable base and \( \mathrm{H} \) and \( {\mathrm{H}}^{\prime } \) Lie subgroups of \( \mathrm{G} \) . Suppose that \( \mathrm{K} \) is of characteristic 0 and that \( {\mathrm{{HH}}}^{\prime } \) is locally closed in \( \mathrm{G} \) . (i)... | Let \( \mathrm{H} \times {\mathrm{H}}^{\prime } \) operate on \( \mathrm{G} \) on the right by the mapping \( \left( {\left( {h,{h}^{\prime }}\right), g}\right) \mapsto {hg}{h}^{\prime - 1} \) . The orbital mapping \( \rho \) of \( e \) is \( \left( {h,{h}^{\prime }}\right) \mapsto h{h}^{\prime - 1} \) . By Proposition... | Yes |
Proposition 31. Let \( \mathrm{G} \) be a finite-dimensional Lie group with countable base, \( \mathrm{H} \) a normal Lie subgroup of \( \mathrm{G} \) and \( \mathrm{A} \) a Lie subgroup of \( \overline{\mathrm{G}} \) . Suppose that \( \mathrm{K} \) is of characteristic 0 and that \( \mathrm{{AH}} \) is closed. Let \( ... | By Proposition 30, AH is a Lie subgroup of G. By Corollary 2 to Proposition \( {29},\mathrm{H} \cap \mathrm{A} \) is a Lie subgroup of \( \mathrm{G} \) . It is therefore meaningful to speak of the groups \( \mathrm{{AH}}/\mathrm{H} \) and \( \mathrm{A}/\left( {\mathrm{H} \cap \mathrm{A}}\right) \) . On the other hand, ... | Yes |
Proposition 32. Let \( \mathrm{G} \) and \( \mathrm{H} \) be Lie groups, \( k \) a non-discrete closed subfield of \( \mathrm{K} \) and \( \phi \) a morphism of \( \mathrm{G} \) into \( \mathrm{H} \) for the Lie group structures over \( k \) . Suppose that \( \mathrm{K} \) is of characteristic 0 . If \( \mathrm{L}\left... | For all \( g \in \mathrm{G} \), \[ {\mathrm{T}}_{g}\left( \phi \right) = {\mathrm{T}}_{e}\left( {\gamma \left( {\phi \left( g\right) }\right) }\right) \circ \mathrm{L}\left( \phi \right) \circ {\mathrm{T}}_{g}\left( {\gamma {\left( g\right) }^{-1}}\right) \] and hence \( {\mathrm{T}}_{g}\left( \phi \right) \) is K-line... | Yes |
Proposition 33. The mapping \( t \mapsto \langle t, f\rangle \) is a morphism of the algebra \( {\mathcal{T}}^{\left( \infty \right) }\left( \mathrm{G}\right) \) into the algebra A. | It suffices to verify that, if \( t \) and \( {t}^{\prime } \) are point distributions on \( \mathrm{G} \), then \( \left\langle {t * {t}^{\prime }, f}\right\rangle = \langle t, f\rangle \left\langle {{t}^{\prime }, f}\right\rangle \) . But\n\n\[ \left\langle {t * {t}^{\prime }, f}\right\rangle = \left\langle {t \otime... | Yes |
Proposition 34. Let \( \mathrm{H} \) be a Lie group, \( \mathrm{A} \) a unital complete normable associative algebra and \( \phi : \mathrm{H} \rightarrow {\mathrm{A}}^{ * } \) a Lie group morphism. The associated morphism \( {\phi }^{\prime } \) of \( {\mathcal{T}}^{\left( \infty \right) }\left( \mathrm{H}\right) \) in... | Let \( i \) be the identity mapping of \( {\mathrm{A}}^{ * } \) into \( \mathrm{A} \) . Then, for all \( t \in {\mathcal{T}}^{\left( \infty \right) }\left( \mathrm{H}\right) \) , \[ {\phi }^{\prime }\left( t\right) = \langle t,\phi \rangle = \langle t, i \circ \phi \rangle \] \( = \left\langle {{\phi }_{ * }\left( t\ri... | Yes |
Proposition 35. Let \( \mathrm{E} \) be a finite-dimensional vector space. Let \( \phi \) be the morphism \( g \mapsto \det g \) of the Lie group \( \mathbf{{GL}}\left( \mathrm{E}\right) \) into the Lie group \( {\mathrm{K}}^{ * } \) . The mapping \( \mathrm{L}\left( \phi \right) \) of \( \mathcal{L}\left( \mathrm{E}\r... | We choose a norm and a basis of E. The expansion of the determinant proves that\n\n\[ \det \left( {1 + u}\right) \in 1 + \operatorname{Tr}u + o\left( {\parallel u\parallel }\right) \]\n\nwhen \( u \) tends to 0 in \( \mathcal{L}\left( \mathrm{E}\right) \) . Hence, using Proposition 34, no. 9, for \( x \in \mathcal{L}\l... | Yes |
Proposition 36. Let \( \mathbf{I} \) be a subset of \( \{ 1,2,\ldots, n\} \) and \( \mathbf{G} \) the subgroup of \( \mathbf{{GL}}\left( \mathbf{E}\right) \) consisting of the \( g = {\left( {g}_{ij}\right) }_{1 \leq i, j \leq n} \in \mathbf{{GL}}\left( \mathrm{E}\right) \) such that \( {g}_{ij} = 0 \) for \( i < j \) ... | Let \( \mathrm{S} \) be the set of \( \left( {x}_{ij}\right) \in \mathcal{L}\left( \mathrm{E}\right) \) such that \( {x}_{ij} = 0 \) for \( i < j \) and \( {x}_{ii} = 0 \) for \( i \in \mathrm{I} \) . Then \( \mathrm{G} \) is the intersection of \( \mathbf{{GL}}\left( \mathrm{E}\right) \) and the affine subspace \( 1 +... | Yes |
Proposition 37. Let \( \mathrm{A} \) be a complete normable unital associative algebra and \( x \mapsto {x}^{\iota } \) a continuous linear mapping of \( \mathrm{A} \) into \( \mathrm{A} \) such that \( {\left( {x}^{\mathrm{t}}\right) }^{\mathrm{t}} = x \) and \( {\left( xy\right) }^{\mathrm{t}} = {y}^{\mathrm{t}}{x}^{... | Let \( \mathrm{S} \) (resp. \( {\mathrm{S}}^{\prime } \) ) be the set of \( y \in \mathrm{A} \) such that \( y = {y}^{\iota } \) (resp. \( y = - {y}^{\iota } \) ). Then \( \mathrm{S},{\mathrm{S}}^{\prime } \) are closed vector subspaces of \( \mathrm{A} \) . The formula\n\n\[ y = \frac{1}{2}\left( {y + {y}^{\iota }}\ri... | Yes |
Corollary 1. Suppose that \( \mathrm{K} \) is of characteristic \( \neq 2 \) . Let \( \mathrm{E} \) be a finite-dimensional vector space over \( {\mathrm{K}}^{ - } \) and \( \phi \) a non-degenerate symmetric (resp. alternating) bilinear form on \( \mathrm{E} \) . For all \( u \in \mathcal{L}\left( \mathrm{E}\right) \)... | We apply Proposition 37 with \( \mathrm{A} = \mathcal{L}\left( \mathrm{E}\right) \) and \( {x}^{\iota } = {x}^{ * } \) . | Yes |
Corollary 2. Let \( \mathrm{E} \) be a complex (resp. real) Hilbert space and \( \mathrm{U} \) the unitary group of \( \mathbf{E} \) . Then \( \mathbf{U} \) is a real subgroup of \( \mathbf{{GL}}\left( \widetilde{\mathbf{E}}\right) \) and \( \mathbf{L}\left( \mathbf{U}\right) \) is the set of \( x \in \mathcal{L}\left(... | We apply Proposition 37 with \( \mathcal{A} = \mathcal{L}\left( E\right) \) considered as an algebra over \( \mathbf{R} \) and \( {x}^{\iota } = {x}^{ * } \) . | No |
Consider \( \mathrm{G} \) as operating on \( \mathrm{E} \) on the left by the mapping \( \left( {g, x}\right) \mapsto \pi \left( g\right) x \) . Let \( b \in \mathbf{E} \) and \( \rho \left( b\right) \) be its orbital mapping. Let \( {\mathrm{T}}_{b}\left( \mathrm{E}\right) \) be canonically identified with \( \mathrm{... | \n\n\( \mathrm{L}\left( \pi \right) t = \langle t,\pi \rangle \) (no. 9, Proposition 34). As the mapping \( A \mapsto {Ab} \) of \( \mathcal{L}\left( \mathrm{E}\right) \) into \( \mathrm{E} \) is continuous and linear, it follows that \n\n\[ \n\left( {L\left( \pi \right) t}\right) \left( b\right) = \langle t, g \mapsto... | Yes |
Proposition 39. Suppose that \( \mathrm{K} \) is of characteristic 0. Let \( \mathrm{G} \) be a Lie group, \( \mathrm{E} \) a finite-dimensional vector space and \( \pi \) an analytic linear representation of \( \mathrm{G} \) on \( \mathrm{E} \). Let \( {\mathrm{E}}_{1},{\mathrm{E}}_{2} \) be vector subspaces of \( \ma... | This follows from Propositions 29 (no. 8) and 36 (no. 10). | Yes |
Corollary 1. In the notation of Proposition 39, the set of \( g \in \mathrm{G} \) such that \( \pi \left( g\right) \left( {\mathrm{E}}_{1}\right) \subset {\mathrm{E}}_{1} \) is a Lie subgroup of \( \mathrm{G} \) and its Lie algebra is the set of \( a \in \mathrm{L}\left( \mathrm{G}\right) \) such that \( \mathrm{L}\lef... | We apply Proposition 39 with \( {\mathrm{E}}_{1} = {\mathrm{E}}_{2} \) . | No |
Corollary 2. Let \( \mathrm{G},\mathrm{E},\pi \) be as in Proposition 39. Let \( \mathrm{F} \) be a subset of \( \mathrm{E} \). The set of \( g \in \mathbf{G} \) such that \( \pi \left( g\right) x = x \) for all \( x \in \widetilde{\mathbf{F}} \) is a Lie subgroup of \( \mathbf{G} \) and its Lie algebra is the set of \... | We apply Proposition 39 with \( {\mathrm{E}}_{2} = \{ 0\} \) and \( {\mathrm{E}}_{1} \) the vector subspace of \( \mathrm{E} \) generated by \( \mathrm{F} \). | Yes |
Proposition 40. Let \( \mathrm{G} \) be a Lie group, \( \mathrm{E} \) a complete normable space, \( \pi \) an analytic linear representation of \( \mathrm{G} \) on \( \mathrm{E} \) and \( \mathrm{F} \) a closed vector subspace of \( \mathrm{E} \) stable under \( \pi \left( \mathrm{G}\right) \) . Suppose that \( \mathrm... | Let \( \mathrm{A} \) be the set of \( u \in \mathcal{L}\left( \mathrm{E}\right) \) such that \( u\left( \mathrm{\;F}\right) \subset \mathrm{F} \) . Then \( \mathrm{A} \) is a closed vector subspace of \( \mathcal{L}\left( \mathrm{E}\right) \) and \( \pi \) takes its values in A. By virtue of the hypotheses on \( \mathr... | Yes |
Proposition 41. Let \( \mathrm{G} \) be a Lie group and \( {\pi }_{1},{\pi }_{2},\ldots ,{\pi }_{n},\pi \) analytic linear representation of \( \mathrm{G} \) on complete normable spaces \( {\mathrm{E}}_{1},{\mathrm{E}}_{2},\ldots ,{\mathrm{E}}_{n} \), E. Let\n\n\[ \left( {{x}_{1},{x}_{2},\ldots ,{x}_{n}}\right) \mapsto... | As an example we perform the calculation for \( n = 2 \) .\n\n\[ \left( {L\left( \pi \right) a}\right) \left( {{x}_{1}{x}_{2}}\right) = \langle a, g \mapsto \pi \left( g\right) \left( {{x}_{1}{x}_{2}}\right) \rangle \]\n(Proposition 38)\n\n\[ = \langle a,\left( {g \mapsto {\pi }_{1}\left( g\right) {x}_{1}}\right) \left... | Yes |
Let \( \mathrm{G} \) be a Lie group, \( {\mathrm{E}}_{1},\ldots ,{\mathrm{E}}_{n + 1} \) complete normable spaces and \( {\pi }_{1},\ldots ,{\pi }_{n + 1} \) analytic linear representations of \( \mathrm{G} \) on \( {\mathrm{E}}_{1},\ldots ,{\mathrm{E}}_{n + 1} \). Let\n\n\[ \mathrm{E} = \mathcal{L}\left( {{\mathrm{E}}... | Every element \( \left( {{A}_{1},\ldots ,{A}_{n + 1}}\right) \) of \( \mathcal{L}\left( {\mathrm{E}}_{1}\right) \times \cdots \times \mathcal{L}\left( {\mathrm{E}}_{n + 1}\right) \) defines a continuous endomorphism \( \theta \left( {{A}_{1},\ldots ,{A}_{n + 1}}\right) \) of \( \mathrm{E} \) by the formula\n\n\[ \left(... | Yes |
Corollary 2. Let \( \mathrm{G} \) be a Lie group and \( \pi \) an analytic linear representation of \( \mathrm{G} \) on a complete normable space \( \mathrm{E} \). Then \( g \mapsto {}^{t}\pi {\left( g\right) }^{-1} \) is an analytic linear representation \( ▱ \) of \( \mathrm{G} \) on the complete normable space \( \m... | This is a special case of Corollary 1. | No |
Corollary 3. Let \( \mathrm{G} \) be a Lie group and \( {\pi }_{1},\ldots ,{\pi }_{n} \) analytic linear representations of \( \mathrm{G} \) on finite-dimensional vector spaces \( {\mathrm{E}}_{1},\ldots ,{\mathrm{E}}_{n} \). Then the representation \( {\pi }_{1} \otimes \cdots \otimes {\pi }_{n} \) of \( \mathrm{G} \)... | The mapping \( \left( {{A}_{1},\ldots ,{A}_{n}}\right) \mapsto {A}_{1} \otimes \cdots \otimes {A}_{n} \) of \( \mathcal{L}\left( {\mathrm{E}}_{1}\right) \times \cdots \times \mathcal{L}\left( {\mathrm{E}}_{n}\right) \) into \( \mathcal{L}\left( {{\mathrm{E}}_{1} \otimes \cdots \otimes {\mathrm{E}}_{n}}\right) \) is mul... | Yes |
Corollary 4. Let \( \mathrm{G} \) be a Lie group and \( \pi \) an analytic linear representation of \( \mathrm{G} \) on a finite-dimensional vector space \( \mathbf{E} \). Then the representations \( {\mathbf{T}}^{n}\left( \pi \right) ,{\mathbf{S}}^{n}\left( \pi \right) \) and \( { \land }^{n}\left( \pi \right) \) of G... | This follows from Corollary 3 and Proposition 40. | No |
Let \( \mathrm{A} \) be a finite-dimensional algebra. Suppose that \( \mathrm{K} \) is of characteris- \( \textit{tic}\;0\textit{. The automorphism group}\;\textit{Aut}\left( A\right) \textit{ of }A\textit{ is a Lie subgroup of }\textbf{GL}\left( A\right) \textit{ and }\textbf{L}\left( {\textit{Aut}\left( A\right) }\ri... | This follows from Corollary 1 (applied to \( \mathrm{E} = \mathcal{L}\left( {\mathrm{A},\mathrm{A};\mathrm{A}}\right) \) ) and Corollary 2 of Proposition 39 (applied to the subset of \( \mathrm{E} \) consisting only of multiplication on A). | Yes |
Proposition 42. Let \( \mathrm{G} \) be a Lie group, \( \mathrm{X} \) an analytic manifold, \( \left( {g, x}\right) \mapsto {gx} \) (resp. xg) a law of analytic left (resp. right) operation of \( \mathrm{G} \) on \( \mathrm{X} \) and \( {x}_{0} \) a point of \( \mathrm{X} \) which is invariant under \( \mathrm{G} \) . ... | \( \tau \left( {g{g}^{\prime }}\right) = \tau \left( g\right) \tau \left( {g}^{\prime }\right) \) (resp. \( \tau \left( {g}^{\prime }\right) \tau \left( g\right) \) ) and hence \( \pi \left( {g{g}^{\prime }}\right) = \pi \left( g\right) \pi \left( {g}^{\prime }\right) \) (resp. \( \pi \left( {g}^{\prime }\right) \pi \l... | Yes |
Proposition 43. Let \( t \) , \( u \) be in \( {\mathcal{T}}^{\left( \infty \right) }\left( \mathrm{G}\right) \) . Let \( \mathop{\sum }\limits_{{i = 1}}^{n}{t}_{i} \otimes {t}_{i}^{\prime } \) be the image of \( t \) under the coproduct. Then\n\n\[ t + u = \mathop{\sum }\limits_{{i = 1}}^{n}{t}_{i} * u * {t}_{i}^{\pri... | By definition, \( t \intercal u \) is the image of \( t \otimes u \) under the mapping \( \left( {g,{g}^{\prime }}\right) \mapsto g{g}^{\prime }{g}^{-1} \) of \( \mathrm{G} \times \mathrm{G} \) into \( \mathrm{G} \) . Now this mapping is obtained by composing the following mappings:\n\n\[ \alpha : \left( {g,{g}^{\prime... | Yes |
Corollary 1. Let \( u \in \mathrm{L}\left( \mathrm{G}\right) \) and \( {u}^{\prime } \in {\mathcal{T}}^{\left( \infty \right) }\left( \mathrm{G}\right) \) . Then \( u \intercal {u}^{\prime } = u * {u}^{\prime } - {u}^{\prime } * u \) . | The image of \( u \) under the coproduct is \( u \otimes {\varepsilon }_{e} + {\varepsilon }_{e} \otimes u \), whence\n\n\[ u \intercal {u}^{\prime } = u * {u}^{\prime } * {\varepsilon }_{e} + {\varepsilon }_{e} * {u}^{\prime } * {u}^{ \vee } = u * {u}^{\prime } - {u}^{\prime } * u. \] | Yes |
Corollary 3. Let \( a \in \mathrm{L}\left( \mathrm{G}\right) \) . The vector field defined by a and the left operation \( g \mapsto \) Int \( g \) of \( \mathrm{G} \) on \( \mathrm{G} \) is the field \( {\mathrm{R}}_{a} - {\mathrm{L}}_{a} \) . | The value of this field at \( g \) is\n\n\[ \n a \intercal {\varepsilon }_{g} = a * {\varepsilon }_{g} - {\varepsilon }_{g} * a\;\text{ (Corollary 1) } \n\]\n\n\[ \n= {\left( {\mathrm{R}}_{a}\right) }_{g} - {\left( {\mathrm{L}}_{a}\right) }_{g}\;\left( {\mathrm{{Definition}}\;5}\right) . \n\] | Yes |
Proposition 44. For all \( a \in \mathrm{L}\left( \mathrm{G}\right) \) , \n\n\[ \n\left( {\mathrm{L}\left( \mathrm{{Ad}}\right) }\right) \left( a\right) = {\mathrm{{ad}}}_{\mathrm{L}\left( \mathrm{G}\right) }a. \n\] | Let \( b \in \mathrm{L}\left( \mathrm{G}\right) \) . By Proposition 42 (ii) of no. 11 and Corollary 3 to Proposition \n\n\[ \n\left( {\mathrm{L}\left( \mathrm{{Ad}}\right) }\right) \left( a\right) .b = - \left\lbrack {{\mathrm{R}}_{a} - {\mathrm{L}}_{a},{\mathrm{\;L}}_{b}}\right\rbrack \left( e\right) . \n\] \n\nNow \(... | Yes |
Proposition 45. Suppose that \( \mathrm{G} \) is finite-dimensional and that \( \mathrm{K} \) is of characteristic 0 . Let \( s \) be an integer \( \geq 0 \) . Then the mapping \( \pi : g \mapsto {\operatorname{Ad}}_{{\mathrm{U}}_{\mathrm{s}}\left( \mathrm{G}\right) }\left( g\right) \) is an analytic linear representat... | The linear representation \( \;\pi \; \) is a quotient of \( \mathop{\bigoplus }\limits_{{r = 0}}^{s}{\mathrm{T}}^{r}\left( \mathrm{{Ad}}\right) \; \) and is hence analytic. For \( a \in \mathrm{L}\left( \mathrm{G}\right) \) and \( {x}_{1},{x}_{2},\ldots ,{x}_{s} \) in \( \mathrm{L}\left( \mathrm{G}\right) \) ,\n\n\[ \... | Yes |
Proposition 46. Let \( h \in \mathrm{G}, x \in {\mathrm{T}}_{h}\left( \mathrm{G}\right) \) and \( a \in \mathrm{L}\left( \mathrm{G}\right) \) . Let \( \phi \) be the mapping \( \left( {g,{g}^{\prime }}\right) \mapsto g{g}^{\prime }{g}^{-1} \) of \( \mathrm{G} \times \mathrm{G} \) into \( \mathrm{G} \) . The image \( y ... | \[ y = \left( {{\mathrm{T}}_{\left( e, h\right) }\phi }\right) \left( {a \otimes {\varepsilon }_{h} + {\varepsilon }_{e} \otimes x}\right) \] \[ = a \intercal {\varepsilon }_{h} + {\varepsilon }_{e} \intercal x \] \[ = a * {\varepsilon }_{h} - {\varepsilon }_{h} * a + x \] \[ = h\left( {\left( {\operatorname{Ad}{h}^{-1... | Yes |
Proposition 47. Let \( \mathrm{G} \) be a Lie group, \( \mathrm{H} \) and \( \mathrm{E} \) Lie subgroups of \( \mathrm{G} \) and suppose that \( h\mathrm{E}{h}^{-1} = \mathrm{E} \) for all \( h \in \mathrm{H} \) . Then \( {\mathcal{T}}^{\left( \infty \right) }\left( \mathrm{H}\right) \intercal {\mathcal{T}}^{\left( \in... | If \( t \in {\mathcal{T}}^{\left( \infty \right) }\left( \mathrm{H}\right) \) and \( {t}^{\prime } \in {\mathcal{T}}^{\left( \infty \right) }\left( \mathrm{E}\right) \), then \( t \otimes {t}^{\prime } \in {\mathcal{T}}^{\left( \infty \right) }\left( {\mathrm{H} \times \mathrm{E}}\right) \) and the image of \( \mathrm{... | No |
Proposition 48. Let \( \mathrm{G} \) be a Lie group and \( \mathrm{H} \) and \( \mathrm{E} \) Lie subgroups of \( \mathrm{G} \) . Suppose that \( \mathrm{G} \) is, as a Lie group, the semi-direct product of \( \mathrm{H} \) by \( \mathrm{E} \) . Let \( \circ \) be the linear representation \( g \mapsto \left( {\operato... | (i) is obvious and (ii) follows from Proposition 47. \( L\left( \sigma \right) = L\left( \rho \right) \mid L\left( H\right) \) . Now by Propositions 40 (no. 11) and 44 (no. 12), \( \mathrm{L}\left( \mathcal{N}\right) \left( t\right) \) is, for all \( t \in \mathrm{L}\left( \mathrm{G}\right) \) , the restriction of \( {... | Yes |
Proposition 49. Let A be a complete normable unital associative algebra. We identify A with \( \mathrm{L}\left( {\mathrm{A}}^{ * }\right) \) . Then, if \( g \in {\mathrm{A}}^{ * } \) and \( y \in \mathrm{A} \) , \( \left( {\operatorname{Ad}g}\right) y = {gy}{g}^{-1} \) . | Recall that \( \operatorname{Ad}g = {\mathrm{T}}_{1}\left( {\operatorname{Int}g}\right) \) . Let \( {u}_{g} \) be the mapping \( x \mapsto {gx}{g}^{-1} \) of \( \mathrm{A} \) into \( \mathrm{A} \) . The identity chart of \( {\mathrm{A}}^{ * } \) into \( \mathrm{A} \) transforms Int \( g \) into \( {u}_{g} \mid {\mathrm... | Yes |
Proposition 50. Let \( \mathrm{G} \) be a Lie Group (assumed to be finite-dimensional if \( \mathrm{K} \) is of characteristic \( > 0 \) ). Let \( \mathrm{E} \) be the vector space of continuous alternating multilinear forms of degree \( k \) on \( {\mathrm{T}}_{e}\left( \mathrm{G}\right) \) . For all \( u \in \mathrm{... | This is a special case of what we have said above. | No |
Proposition 51. Let \( \mathrm{G} \) be a Lie group, of finite dimension if \( \mathrm{K} \) is of characteristic \( > 0 \) , and let \( {a}_{1},\ldots ,{a}_{p} \) be elements of \( \mathrm{L}\left( \mathrm{G}\right) ,\mathrm{F} \) a complete normable space and \( \alpha \) a differential form of degree \( p - 1 \) on ... | Suppose that \( \alpha \) is left invariant. By Differentiable and Analytic Manifolds, \( \mathrm{R} \) , 8.5.7, then\n\n\[ \n\left( {d\alpha }\right) \left( {{L}_{{a}_{1}},\ldots ,{\mathrm{L}}_{{a}_{p}}}\right) = \mathop{\sum }\limits_{i}{\left( -1\right) }^{i - 1}{\mathrm{\;L}}_{{a}_{i}}\alpha \left( {{\mathrm{L}}_{{... | Yes |
Corollary 1. Let \( \mathrm{G} \) be a Lie group, of finite dimension if \( \mathrm{K} \) is of characteristic \( > 0 \) , and \( \omega \) and \( {\omega }^{\prime } \) the left and right canonical differential forms of \( \mathrm{G} \) . Then\n\n\[ \n{d\omega } + {\left\lbrack \omega \right\rbrack }^{2} = 0,\;d{\omeg... | By Proposition 51,\n\n\[ \n{\left( d\omega \right) }_{e}\left( {{a}_{1},{a}_{2}}\right) = - {\omega }_{e}\left( \left\lbrack {{a}_{1},{a}_{2}}\right\rbrack \right) = - \left\lbrack {{a}_{1},{a}_{2}}\right\rbrack = - \left\lbrack {{\omega }_{e}\left( {a}_{1}\right) ,{\omega }_{e}\left( {a}_{2}\right) }\right\rbrack \n\]... | Yes |
Corollary 2. Suppose that \( \mathrm{G} \) is finite-dimensional. Let \( \left( {{e}_{1},\ldots ,{e}_{n}}\right) \) be a basis of \( \mathrm{L}\left( \mathrm{G}\right) ,\left( {{e}_{1}^{ * },\ldots ,{e}_{n}^{ * }}\right) \) the dual basis, \( \left( {c}_{ijk}\right) \) the constants of structure of \( \mathrm{L}\left( ... | If \( r < s \) ,\n\n\[ {\left( d{\omega }_{k}\right) }_{e}\left( {{e}_{r},{e}_{s}}\right) = - {\left( {\omega }_{k}\right) }_{e}\left( \left\lbrack {{e}_{r},{e}_{s}}\right\rbrack \right) \]\n\n\[ = - \mathop{\sum }\limits_{i}{c}_{rsl}{\left( {\omega }_{k}\right) }_{e}\left( {e}_{l}\right) \]\n\n\[ = - {c}_{rsk} \]\n\n\... | Yes |
Lemma 2. Let \( \mathrm{G} \) be a Lie group, \( \mathrm{U} \) a symmetric open neighbourhood of \( \mathrm{e} \) in \( \mathrm{G} \) , \( \mathrm{E} \) a complete normable space and \( \phi : {\mathbf{U}}^{2} \rightarrow \mathbf{E} \) an analytic mapping. For all \( g \in \mathbf{U} \), let \( {\omega }_{g} \) be the ... | Clearly \( {\omega }_{e} = {d}_{e}\phi \) . For all \( g \in \mathrm{U} \) and all \( t \in {\mathrm{T}}_{e}\left( \mathrm{G}\right) \) ,\n\n\[ \left\langle {{\omega }_{g},{\mathrm{\;T}}_{e}\left( {\gamma \left( g\right) }\right) t}\right\rangle = \left\langle {{d}_{g}\left( {\phi \circ \gamma {\left( g\right) }^{-1}}\... | Yes |
Proposition 52. Let \( n \) be an integer \( > 0 \), G an \( n \) -dimensional Lie group, U a symmetric open neighbourhood of \( e \) in \( \mathrm{G} \) and \( \psi : {\mathrm{U}}^{2} \rightarrow {\mathrm{K}}^{n} \) a chart of \( \mathrm{G} \) such that \( \psi \left( e\right) = 0 \) . If \( \left( {{x}_{1},\ldots ,{x... | We apply Lemma 2 with \( E = K, \) taking \( \phi \left( g\right) \) to be the coordinate of \( \psi \left( g\right) \) of index \( k \) . We obtain a differential form \( {\omega }_{k} \) ; let \( {\varpi }_{k} \) be its transform under \( \psi \) . The value of \( {\varpi }_{k} \) at \( \left( {{x}_{1},\ldots ,{x}_{n... | Yes |
Proposition 53. Let \( \mathrm{G} \) be a Lie group, \( \mathrm{A} \) a complete normable algebra and \( \mathrm{\phi } \) a Lie group morphism of \( \mathrm{G} \) into \( {\mathrm{A}}^{ * } \) . For all \( g \in \mathrm{G} \), let \( {\omega }_{g} = \phi {\left( g\right) }^{-1} \cdot {d}_{g}\phi \) . Then \( \omega \)... | We apply Lemma 2 with \( \mathrm{E} = \mathrm{A} \) and \( \mathrm{U} = \mathrm{G} \) . The differential at \( g \) of the mapping \( h \mapsto \phi \left( {{g}^{-1}h}\right) = \phi {\left( g\right) }^{-1}\phi \left( h\right) \) is \( \phi {\left( g\right) }^{-1} \cdot {d}_{g}\phi \) . | Yes |
Proposition 54. Let \( \mathrm{G} \) be a Lie group of finite dimension \( n \) , \( \omega \) a left invariant differential form of degree \( n \) on \( \mathrm{G} \) and \( \phi \) an endomorphism of \( \mathrm{G} \) . Then\n\n\[{\phi }^{ * }\left( \omega \right) = \left( {\det \mathrm{L}\left( \phi \right) }\right) ... | We write \( L\left( \phi \right) = u,\;{\omega }_{e} = f\; \) and \( {\phi }^{ * }{\left( \omega \right) }_{e} = g. \) For all \( {x}_{1},\ldots ,{x}_{n} \) in \( L\left( G\right) , \)\n\n\[g\left( {{x}_{1},\ldots ,{x}_{n}}\right) = f\left( {u{x}_{1},\ldots, u{x}_{n}}\right) = \left( {\det u}\right) f\left( {{x}_{1},\l... | Yes |
Proposition 55. Suppose that \( \mathrm{K} \) is locally compact. Let \( \mu \) be a Haar measure on the additive group of \( \mathrm{K} \) . Let \( \mathrm{G} \) be a Lie group of finite dimension \( n \) . (i) Let \( \omega \) be a non-zero left invariant differential form of degree \( n \) on \( \mathrm{G} \) . Then... | (i) is obvious. Let \( \mathrm{V},{\mathrm{V}}^{\prime } \) be open neighbourhoods of \( e \) such that \( \phi \left( \mathrm{V}\right) = {\mathrm{V}}^{\prime } \) and \( \phi \mid \mathrm{V} \) is a local isomorphism of \( \mathrm{G} \) into \( \mathrm{G} \) . Then \[ {\phi }^{-1}\left( {{\;\operatorname{mod}\;{\left... | No |
Proposition 56. Let \( \\mathrm{G} \) be a Lie group of finite dimension \( n,\\mathrm{H} \) a p-dimensional Lie subgroup and \( \\mathrm{X} \) the Lie homogeneous space \( \\mathrm{G}/\\mathrm{H} \) . Suppose that\n\n\[ \n\\det {\\operatorname{Ad}}_{\\mathrm{L}\\left( \\mathrm{G}\\right) }h = \\det {\\operatorname{Ad}... | By \( §1 \), no. 8, Examples, \( {\\operatorname{Alt}}^{n - p}\\left( {\\mathrm{{TX}},\\mathrm{K}}\\right) \) is an analytic vector G-bundle. Let \( {x}_{0} \) be the canonical image of \( e \) in \( \\mathrm{X} \) ; its stabilizer is \( \\mathrm{H} \) . The fibre of \( {\\operatorname{Alt}}^{n - p}\\left( {\\mathrm{{T... | Yes |
Proposition 57. Let \( \mathrm{G} \) be a Lie group of finite dimension \( n \) . Choose a basis for \( \mathop{\bigwedge }\limits^{n}{\mathrm{\;T}}_{e}{\left( \mathrm{G}\right) }^{ * } \) ; by means of the right (resp. left) trivialization of \( \mathop{\bigwedge }\limits^{n}\mathrm{T}{\left( \mathrm{G}\right) }^{ * }... | We shall consider the case where \( \mathop{\bigwedge }\limits^{n}\mathrm{T}\left( \mathrm{G}\right) \) * has been trivialized using a right invariant form \( \omega \) . Suppose that the proposition has been proved for elements \( {u}_{1},{u}_{2} \) of \( \mathrm{U}\left( \mathrm{G}\right) \) . Then, \[ {}^{t}\left( {... | Yes |
Proposition 58. Let \( \\mathrm{G} \) and \( \\mathrm{H} \) be two Lie groups, \( \\mathrm{M} \) a manifold of class \( {\\mathrm{C}}^{r} \) , \( f \) a mapping of class \( {\\mathrm{C}}^{r} \) of \( \\mathrm{M} \) into \( \\mathrm{G} \) and \( h \) a morphism of \( \\mathrm{G} \) into \( \\mathrm{H} \) . Then\n\n\[ \n... | The latter expression is equal, on the one hand, to\n\n\[ \n\\mathrm{T}\\left( h\\right) \\left( {f{\\left( x\\right) }^{-1}.\\mathrm{T}\\left( f\\right) \\left( u\\right) }\\right) \\;\\left( {§2,\\text{ Proposition }5}\\right)\n\]\n\n\[ \n= {T}_{e}\\left( h\\right) \\left( {\\left( {{f}^{-1}.{df}}\\right) \\left( u\\... | Yes |
Proposition 59. Let \( \mathrm{G} \) be a Lie group, \( \mathrm{M} \) a manifold of class \( {\mathrm{C}}^{r} \) , \( f \) and \( g \) mappings of class \( {\mathrm{C}}^{r} \) of \( \mathrm{M} \) into \( \mathrm{G} \) and \( p \) the canonical surjection of \( \mathrm{{TM}} \) onto \( \mathrm{M} \) . (i) \( {\left( fg\... | Assertion (i) follows from \( §2 \), no. 2, Proposition 7. | No |
Corollary 1. Let \( s \in \mathrm{G} \) and \( {sg} \) be the mapping \( x \mapsto {sg}\left( x\right) \) of \( \mathrm{M} \) into \( \mathrm{G} \) . Then \( {\left( sg\right) }^{-1} \cdot d\left( {sg}\right) = {g}^{-1} \cdot {dg}. \) | This follows from Proposition 59 (i) taking \( f \) to be the constant mapping \( x \mapsto s \) of \( \mathrm{M} \) into \( \mathrm{G} \) . | Yes |
If the mappings f and g of \( \mathrm{M} \) into \( \mathrm{G} \) have the same left differential, the tangent mapping to \( f{g}^{-1} \) is everywhere zero. If further \( \mathrm{K} \) is of characteristic 0, then \( f{g}^{-1} \) is locally constant. | By Proposition 59,\n\n\[ \n{\left( f{g}^{-1}\right) }^{-1} \cdot d\left( {f{g}^{-1}}\right) = \left( {\operatorname{Ad} \circ g \circ p}\right) \circ \left( {{f}^{-1} \cdot {df}}\right) - \left( {\operatorname{Ad} \circ g \circ p}\right) \circ \left( {{g}^{-1} \cdot {dg}}\right) .\n\]\n\nIf \( {f}^{-1} \cdot {df} = {g}... | Yes |
Proposition 60. Let \( \mathrm{G} \) be a Lie group, of finite dimension if \( \mathrm{K} \) is of characteristic \( > 0,\mathrm{M} \) a manifold of class \( {\mathrm{C}}^{r},{fa} \) mapping of class \( {\mathrm{C}}^{r} \) of \( \mathrm{M} \) into \( \mathrm{G} \) and \( \propto \) the left differential of \( f \) . Th... | Let \( \omega \) be the canonical left differential form of G. Using Corollary 1 to Proposition 51, no. 14, we have\n\n\[ \n{d\alpha } = d\left( {{f}^{ * }\left( \omega \right) }\right) = {f}^{ * }\left( {d\omega }\right) = {f}^{ * }\left( {-{\left\lbrack \omega \right\rbrack }^{2}}\right) \n\]\n\n\[ \n= - {\left\lbrac... | Yes |
Lemma 1. Let \( \mathrm{G} \) be a Lie group germ and \( \mathfrak{h} \) a Lie subalgebra of \( \mathrm{L}\left( \mathrm{G}\right) \) admitting a topological supplement. The union of the \( g\mathfrak{h} \) (resp. \( \mathfrak{h}g \) ) for \( g \in \mathrm{G} \) is an integrable vector subbundle of \( \mathrm{T}\left( ... | By considering the left trivialization of \( \mathrm{T}\left( \mathrm{G}\right) \) (§ 2, no. 3), it is seen immediately that the \( g\mathfrak{h} \), for \( g \in \mathrm{G} \), are the fibres of a vector subbundle \( \mathrm{E} \) of \( \mathrm{T}\left( \mathrm{G}\right) \) . Let \( g \in \mathrm{G} \) . The set of \(... | Yes |
Corollary 1. Let \( \mathrm{G} \) and \( \mathrm{H} \) be two Lie group germs. If \( \mathrm{L}\left( \mathrm{G}\right) \) and \( \mathrm{L}\left( \mathrm{H}\right) \) are isomorphic, \( \mathrm{G} \) and \( \mathrm{H} \) are locally isomorphic. | This follows from Theorem 1 and \( §1 \), no. 10, Proposition 21. | No |
Corollary 2. Let \( \mathrm{G} \) be a Lie group germ. If \( \mathrm{L}\left( \mathrm{G}\right) \) is commutative, \( \mathrm{G} \) is locally isomorphic to the additive Lie group \( \mathbf{L}\left( \mathbf{G}\right) \) . | The Lie algebra of the additive group \( L\left( G\right) \) is isomorphic to \( L\left( G\right) \) . Hence it suffices to apply Corollary 1. | No |
Corollary 3. Let \( \mathrm{G} \) be a Lie group. If \( \mathrm{L}\left( \mathrm{G}\right) \) is commutative, \( \mathrm{G} \) contains a commutative open subgroup. | There exists an open Lie subgroup germ \( \mathrm{U} \) of \( \mathrm{G} \) which is commutative (Corollary 2). Let \( \mathrm{V} \) be a neighbourhood of \( e \) such that \( {\mathrm{V}}^{2} \subset \mathrm{U} \) . Then \( {xy} = {yx} \) for all \( x, y \) in V. Hence the subgroup of \( \mathrm{G} \) generated by \( ... | Yes |
Lemma 2. Let \( \mathbf{L} \) be a complete normed Lie algebra over \( \mathbf{R} \) or \( \mathbf{C} \) . Let \( \mathrm{G} \) be the set of \( x \in \mathrm{L} \) such that \( \parallel x\parallel < \frac{1}{3}\log \frac{3}{2} \) . Let \( \theta \) be the mapping \( x \mapsto - x \) of \( \mathrm{G} \) into \( \mathr... | (i) follows from Chapter II,§ 7, no. 2.\n\nAs \( \phi \) is a chart on \( G \), the differential \( \psi \) of \( \phi \) at 0 is an isomorphism of normable\n\nspaces. On the other hand, the expansion as an integral series \( \mathrm{H} = \mathop{\sum }\limits_{{i, j \geq 0}}{\mathrm{H}}_{ij} \) of the mapping \( H \) ... | Yes |
Lemma 3. Let \( \mathrm{L} \) be a complete normed Lie algebra over \( \mathrm{K} \). Let \( \mathrm{G} \) be the set of \( x \in \mathrm{L} \) such that \( \parallel x\parallel < \lambda \). Let \( \mathrm{H} : \mathrm{G} \times \mathrm{G} \rightarrow \mathrm{G} \) be the Hausdorff function of \( \mathrm{L} \) (Chapte... | Assertions (i) and (iii) follow from Chapter II,§ 8, no. 3, Proposition 3 and (ii) can be proved as in Lemma 2. | No |
Corollary 1. Let \( \mathrm{G} \) be a Lie group. There exists a neighbourhood of e which contains no finite subgroup distinct from \{e\}. If \( \mathrm{K} = \mathbf{R} \) or \( \mathbf{C} \), there exists an open neighbourhood of e which contains no subgroup distinct from \( \{ e\} \) . | We write \( \mathrm{L}\left( \mathrm{G}\right) = \mathrm{L} \). Choose a norm on \( \mathrm{L} \) defining the topology on \( \mathrm{L} \) and such that \( \parallel \left\lbrack {x, y}\right\rbrack \parallel \leq \parallel x\parallel \parallel y\parallel \) for all \( x, y \) in L.\n\nSuppose that \( \mathrm{K} = \ma... | Yes |
Corollary 2. Let \( k \) be a non-discrete closed subfield of \( \mathrm{K},\mathrm{G} \) a Lie group over \( k \) and \( \mathrm{L} = \mathrm{L}\left( \mathrm{G}\right) \) . Suppose that \( \mathrm{L} \) has a normable Lie \( \mathrm{K} \) -algebra structure \( {\mathrm{L}}^{\prime } \), compatible with the normable L... | There exists a Lie group germ \( {\mathrm{G}}_{1} \) over \( \mathrm{K} \) such that \( \mathrm{L}\left( {\mathrm{G}}_{1}\right) = {\mathrm{L}}^{\prime } \) (Theorem 2). By Corollary 1 to Theorem 1 of no. 1, G and \( {\mathrm{G}}_{1} \), considered as Lie \( k \) - group germs, are locally isomorphic. Hence there exist... | Yes |
Proposition 1. Let \( \mathrm{G} \) be a Lie group germ and \( \mathrm{H} \) and \( {\mathrm{H}}^{\prime } \) two Lie subgroup germs. In order that \( \mathrm{L}\left( \mathrm{H}\right) \supset \mathrm{L}\left( {\mathrm{H}}^{\prime }\right) \), it is necessary and sufficient that \( \mathrm{H} \cap {\mathrm{H}}^{\prime... | If \( \mathrm{H} \cap {\mathrm{H}}^{\prime } \) is open in \( {\mathrm{H}}^{\prime } \), then \( \mathrm{L}\left( {\mathrm{H}}^{\prime }\right) = \mathrm{L}\left( {\overline{\mathrm{H}} \cap {\mathrm{H}}^{\prime }}\right) \subset \mathrm{L}\left( \mathrm{H}\right) \) . Suppose that \( \mathrm{L}\left( \mathrm{H}\right)... | Yes |
Proposition 2. Let \( \mathrm{G} \) be a Lie group over \( \mathrm{K}, k \) a non-discrete closed subfield of \( \mathrm{K} \) and \( \mathrm{H} \) a Lie subgroup of the Lie k-group \( \mathrm{G} \) . Suppose that \( \mathrm{L}\left( \mathrm{H}\right) \) is a vector sub- \( \mathrm{K} \) - space of \( \mathrm{L}\left( ... | There exists a Lie subgroup germ \( {\mathrm{H}}^{\prime } \) of the Lie K-group \( \mathrm{G} \) such that \( \mathrm{L}\left( {\mathrm{H}}^{\prime }\right) = \mathrm{L}\left( \mathrm{H}\right) \) (Theorem 3). Consider \( \mathrm{G},\mathrm{H},{\mathrm{H}}^{\prime } \) as Lie \( k \) -group germs; Theorem 3 then prove... | Yes |
Proposition 3. Let \( \mathrm{G} \) be a Lie group germ, \( \mathrm{L} \) its Lie algebra and \( \phi \) an exponential mapping of \( \mathrm{G} \) . Let \( {\mathrm{L}}_{1},\ldots ,{\mathrm{L}}_{n} \) be vector subspaces of \( \mathrm{L} \) such that \( \mathrm{L} \) is the topological direct sum of \( {\mathrm{L}}_{1... | Let \( {k}_{i} \) be the canonical injection of \( {\mathrm{L}}_{i} \) into \( {\mathrm{L}}_{1} \times {\mathrm{L}}_{2} \times \cdots \times {\mathrm{L}}_{n} \) . Then, for all \( b \in {\mathbf{L}}_{i},\left( {{\mathbf{T}}_{\left( 0,\ldots ,0\right) }\theta }\right) \left( {{\mathbf{T}}_{0}{k}_{i}}\right) \left( b\rig... | Yes |
Proposition 4. Let \( \mathrm{G} \) be a Lie group germ and \( \phi \) an injective exponential mapping of \( \mathrm{G} \) . For all \( x, y \) in \( \mathrm{L}\left( \mathrm{G}\right) \) ,\n\n(1)\n\n\[ x + y = \mathop{\lim }\limits_{{\lambda \in {\mathrm{K}}^{ * },\lambda \rightarrow 0}}{\lambda }^{-1}{\phi }^{-1}\le... | Let \( \mathrm{L} = \mathrm{L}\left( \mathrm{G}\right) \) be given a norm defining the topology of \( \mathrm{L} \) and such that \( \parallel \left\lbrack {x, y}\right\rbrack \parallel \leq \parallel x\parallel \parallel y\parallel \) for all \( x, y \) in L. Using Theorems 2 and 4, it can be assumed that \( \mathrm{G... | Yes |
Proposition 5. Let \( \mathrm{G} \) be a Lie group, \( k \) a non-discrete closed subfield of \( \mathrm{K},{\mathrm{G}}^{\prime } \) the group \( \mathrm{G} \) considered as a Lie group over \( k \) and \( \phi \) (resp. \( {\phi }^{\prime } \) ) an exponential mapping of \( \mathrm{G} \) (resp. \( {\mathrm{G}}^{\prim... | \( \phi \) satisfies hypothesis (i) of Theorem 4 relative to \( {\mathrm{G}}^{\prime } \) and is therefore an exponential mapping of \( {\mathrm{G}}^{\prime } \) . | No |
Proposition 6. Let \( \mathrm{G} \) be a Lie group germ, \( \mathrm{L} \) its Lie algebra and \( \phi : \mathrm{V} \rightarrow \mathrm{G} \) an exponential mapping of \( \mathrm{G} \) . For all \( x \in \mathrm{V} \), let \( {\mathrm{T}}_{x}\left( \mathrm{\;L}\right) \) be identified with \( \mathrm{L} \), so that the ... | \[ \varpi \left( x\right) = \mathop{\sum }\limits_{{n \geq 0}}\frac{1}{\left( {n + 1}\right) !}{\left( \operatorname{ad}x\right) }^{n} \] Let \( L \) 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 a... | Yes |
Proposition 7. (i) If \( t \in \mathbf{Z} \), a \( t \) -th power mapping coincides on a neighbourhood of e with the mapping \( g \mapsto {g}^{t} \) . | It suffices to prove the proposition when \( \mathrm{G} \) is the Lie group germ defined by a complete normed Lie algebra and when the \( t \) -th power mappings considered are constructed using the exponential mapping \( \phi = {\operatorname{Id}}_{\mathrm{G}} \) . But in that case everything is obvious. | No |
Proposition 8. Let \( \mathrm{G} \) and \( \mathrm{H} \) be Lie group germs, \( h \) a morphism of \( \mathrm{G} \) into \( \mathrm{H} \) and \( {\phi }_{\mathrm{G}} \) and \( {\phi }_{\mathrm{H}} \) exponential mappings of \( \mathrm{G} \) and \( \mathrm{H} \) . There exists a neighbourhood \( \mathrm{V} \) of 0 in \(... | Let \( \mathrm{L}\left( \mathrm{G}\right) \) and \( \mathrm{L}\left( \mathrm{H}\right) \) be given norms defining their topologies and such that \( \parallel \left\lbrack {x, y}\right\rbrack \parallel \leq \parallel x\parallel \parallel y\parallel \) for all \( x \) and \( y \) . It can be assumed that \( \mathrm{G} \)... | Yes |
Corollary 1. Let \( \mathrm{G} \) be a Lie group germ, \( {\mathrm{G}}^{\prime } \) a Lie subgroup germ of \( \mathrm{G} \) and \( \phi \) an exponential mapping of \( \mathrm{G} \). (i) There exists an open neighbourhood \( \mathrm{V} \) of 0 in \( \mathrm{L}\left( {\mathrm{G}}^{\prime }\right) \) such that \( \phi \m... | (i) is obtained by applying Proposition 8 to the canonical injection of \( {\mathrm{G}}^{\prime } \) into \( \mathrm{G} \) and (ii) follows from (i). | No |
Corollary 2. Let \( \mathrm{G} \) be a Lie group, \( \mathrm{p} \) an analytic linear representation of \( \mathrm{G} \) and \( \phi \) an exponential mapping of \( \mathrm{G} \) . There exists a neighbourhood \( \mathrm{V} \) of 0 in \( \mathrm{L}\left( \mathrm{G}\right) \) such that\n\n\[ \rho \left( {\phi \left( x\r... | This follows from Proposition 8 and Example 2 of no. 3. | No |
Corollary 3. Let \( \mathrm{G} \) be a Lie group and \( \mathrm{o} \) an exponential mapping of \( \mathrm{G} \) .\n\n(i) There exists a neighbourhood \( \mathrm{V} \) of \( 0 \) in \( \mathrm{L}\left( \mathrm{G}\right) \) such that\n\n\[ \operatorname{Ad}\left( {\phi \left( x\right) }\right) = \exp \operatorname{ad}x.... | (i) follows from Corollary 2 and § 3, no. 12, Proposition 44. | No |
Let \( \mathrm{G} \) be a finite-dimensional Lie group, \( \Omega \) a symmetric open neighbourhood of \( e \) in \( \mathrm{G} \) and \( \mathrm{H} \) a subset of \( \Omega \) containing \( e \) such that the conditions \( x \in \mathrm{H}, y \in \mathrm{H} \) , \( x{y}^{-1} \in \Omega \) imply \( x{y}^{-1} \in \mathr... | (i) Let \( {\mathfrak{h}}_{e} = \mathfrak{h} \) . Then \( \mathfrak{h} \) is a Lie subalgebra of \( \mathrm{L}\left( \mathrm{G}\right) \) which is invariant under \( {\operatorname{Ad}}_{\mathrm{L}\left( \mathrm{G}\right) }\left( \mathrm{H}\right) \) . | Yes |
Proposition 9. Let \( \mathrm{G} \) be a finite-dimensional Lie group and \( \mathrm{H} \) a subgroup of \( \mathrm{G} \). (i) There exists on \( \mathrm{H} \) one and only one analytic manifold structure with the following property: for all \( r \) between 1 and \( \omega \), for every manifold \( \mathrm{V} \) of cla... | In (i) the uniqueness is obvious. We prove the existence. Let \( \mathfrak{h} \) be the Lie algebra tangent at \( e \) to \( \mathrm{H} \). Let \( {\mathrm{H}}^{\prime } \) be a Lie subgroup germ of \( \mathrm{G} \) with Lie algebra \( \mathfrak{h} \). By replacing \( {\mathrm{H}}^{\prime } \) by an open subgroup germ ... | Yes |
Lemma 5. Let \( \mathrm{X} \) be a manifold of class \( {\mathrm{C}}^{r},\mathrm{\;F} \) and \( {\mathrm{F}}^{\prime } \) vector bundles of class \( {\mathrm{C}}^{r} \) with base space \( \mathrm{X} \) and \( \mathrm{o} \) a morphism of \( \mathrm{F} \) into \( {\mathrm{F}}^{\prime } \) . For all \( x \in \mathrm{X} \)... | Let \( \theta \) and \( {\theta }^{\prime } \) be the mappings of \( \mathrm{F} \oplus {\mathrm{F}}^{\prime } \) into itself defined as follows: if \( \left( {u, v}\right) \in {\mathrm{F}}_{x} \oplus {\mathrm{F}}_{x}^{\prime } \), then\n\n\[ \theta \left( {u, v}\right) = \left( {u, v + \phi \left( u\right) }\right) ,\;... | Yes |
Lemma 6. Let \( \mathrm{G} \) be a Lie group germ, \( \omega \) the canonical left differential form of \( \mathrm{G} \) (§ 3, no. 18.9), 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 va... | If \( \left( {x, g}\right) \in \mathrm{M} \times \mathrm{G} \) and \( \left( {u, v}\right) \in {\mathrm{T}}_{x}\left( \mathrm{M}\right) \times {\mathrm{T}}_{g}\left( \mathrm{G}\right) \), then \[ {\theta }_{\left( x, g\right) }\left( {u, v}\right) = \alpha \left( u\right) - {g}^{-1}v. \] Hence the kernel of \( {\theta ... | Yes |
Proposition 11. Let \( {\mathrm{G}}_{1} \) and \( {\mathrm{G}}_{2} \) be Lie group germs and \( {\mathrm{X}}_{1} \) and \( {\mathrm{X}}_{2} \) manifolds of class \( {\mathrm{C}}^{r}\left( {r \geq 2}\right) \) . For \( i = 1,2 \), let \( {\psi }_{i} \) be a law chunk of left operation of class \( {\mathrm{C}}^{r} \) of ... | Let \( {p}_{1} : {\mathrm{G}}_{1} \times {\mathrm{X}}_{2} \rightarrow {\mathrm{G}}_{1},{p}_{2} : {\mathrm{G}}_{1} \times {\mathrm{X}}_{2} \rightarrow {\mathrm{X}}_{2} \) be the canonical projections. For all \( \left( {{g}_{1},{x}_{2}}\right) \in {\mathrm{G}}_{1} \times {\mathrm{X}}_{2} \), let \( {f}_{{g}_{1},{x}_{2}}... | Yes |
Corollary 1. Let \( \mathrm{G} \) be a Lie group germ and \( \mathrm{X} \) a paracompact manifold of class \( {\mathrm{C}}^{r}\left( {r \geq 2}\right) \) . Let \( a \mapsto {\mathrm{D}}_{a} \) be a law of left infinitesimal operation of class \( {\mathrm{C}}^{r - 1} \) of \( \mathrm{L}\left( \mathrm{G}\right) \) on \( ... | Assertion (ii) follows from the Corollary to Proposition 11. By Theorem 6 (i), there exist an open covering \( {\left( {\mathrm{X}}_{i}\right) }_{i \in \mathrm{I}} \) of \( \mathrm{X} \) and, for all \( i \in \mathrm{I} \), a law chunk of left operation \( {\psi }_{i} \) of Class \( {\mathrm{C}}^{r - 1} \) of \( \mathr... | Yes |
Corollary 2. Let \( \mathrm{X} \) be a paracompact manifold of class \( {\mathrm{C}}^{r}\left( {r \geq 2}\right) \) and \( \xi \) a vector field of class \( {\mathrm{C}}^{r - 1} \) on \( \mathrm{X} \) . There exists a law chunk of operation \( \psi \) of class \( {\mathrm{C}}^{r - 1} \) of \( \mathrm{K} \) on \( \mathr... | This is a special case of Corollary 1. | No |
Proposition 2. (i) \( {\psi }_{j} \equiv 0{\;\operatorname{mod}\;\deg }j \) . (ii) If \( t \in \mathrm{K} \), then \[ {x}^{\left\lbrack t\right\rbrack } = \mathop{\sum }\limits_{{i = 1}}^{\infty }\left( \begin{array}{l} t \\ i \end{array}\right) {\psi }_{i}\left( x\right) \] where the formal power series on the right i... | Assertion (i) is obvious from the definition of the \( {\psi }_{j} \) . We prove (ii) for \( t \) an integer \( \geq 0 \) . By (14), \[ {x}^{\left\lbrack t\right\rbrack } = \mathop{\sum }\limits_{{\alpha \left( {1,\ldots ,\alpha \left( t\right) \in {\mathbf{N}}^{n}}\right. }}{a}_{\alpha \left( 1\right) ,\ldots ,\alpha ... | Yes |
\[ \mathrm{E}\left( {tx}\right) = {\left( \mathrm{E}\left( x\right) \right) }^{\left\lbrack t\right\rbrack } \] | or, by (24), \[ \mathrm{E}\left( {tx}\right) = \mathop{\sum }\limits_{{m \geq 0}}\mathop{\sum }\limits_{{1 \leq r \leq m}}{t}^{r}{\phi }_{r, m}\left( {\mathrm{E}\left( x\right) }\right) . \] The two sides are formal power series in \( t \) and \( x \) . Equating the terms of first degree in \( t \), we obtain \[ x = \m... | Yes |
Proposition 4. For the chart of \( \mathrm{G} \) used to be canonical, it is necessary and sufficient that \( {\psi }_{j} = 0 \) for \( j \geq 2 \) . | This is sufficient by Proposition 3. Suppose that the chart is canonical and that \( {\psi }_{i} = 0 \) for \( 2 \leq i < n \) . Then \( {nx} = {x}^{\left\lbrack n\right\rbrack } = \mathop{\sum }\limits_{{i = 0}}^{n}\left( \begin{array}{l} n \\ i \end{array}\right) {\psi }_{i}\left( x\right) = {nx} + {\psi }_{n}\left( ... | Yes |
Theorem 2. Let \( \mathrm{G} \) be a Lie group.\n\n(i) The mapping \( \mathrm{H} \mapsto \mathrm{L}\left( \bar{\mathrm{H}}\right) \) is a bijection of the set of integral subgroups of \( \mathrm{G} \) onto the set of Lie subalgebras of \( \mathrm{L}\left( \mathrm{G}\right) \) admitting a topological supplement. | (a) Let \( \mathfrak{h} \) be a Lie subalgebra of \( \mathrm{L}\left( \mathrm{G}\right) \) admitting a topological supplement. Let \( {\mathrm{H}}_{1} \) be a Lie subgroup germ of \( \mathrm{G} \) such that \( \mathrm{L}\left( {\mathrm{H}}_{1}\right) = \mathfrak{h}\left( {§4\text{, Theorem 3}}\right) \) . \( {\mathrm{H... | Yes |
Proposition 1. Let \( \mathrm{G} \) and \( \mathrm{M} \) be Lie groups, \( \mathrm{H} \) an integral subgroup of \( \mathrm{G} \) and \( \phi \) a morphism of \( \mathrm{M} \) into \( \mathrm{G} \) such that \( \mathrm{L}\left( \phi \right) \left( {\mathrm{L}\left( \mathrm{M}\right) }\right) \subset \mathrm{L}\left( \m... | In the notation of Remark 1, \( \phi \) is a morphism of \( \mathbf{M} \) into \( \mathbf{Y} \) (Differentiable and Analytic Manifolds, R,9.3.2) and hence \( \phi \left( \mathrm{M}\right) \subset \mathrm{H} \) since \( \mathrm{M} \) is connected. | No |
Corollary 1. Let \( \mathrm{G} \) and \( \mathrm{H} \) be Lie groups, \( \phi \) a Lie group morphism of \( \mathrm{G} \) into \( \mathrm{H},\mathrm{N} \) the kernel of \( \phi \) and \( h = \mathrm{L}\left( \phi \right) \) . Suppose that \( \widetilde{\mathrm{G}} \) is connected and that \( \mathrm{H} \) is finite-dim... | (i) has already been proved \( \left( {§3\text{, no. 8, Proposition 28}}\right) \) . | No |
Corollary 2. Let \( \mathrm{G} \) be a Lie group and \( {\mathrm{H}}_{1} \) and \( {\mathrm{H}}_{2} \) integral subgroups of \( \mathrm{G} \) . If \( \mathrm{L}\left( {\mathrm{H}}_{2}\right) \subset \mathrm{L}\left( {\mathrm{H}}_{1}\right) \), then \( {\mathrm{H}}_{2} \) is an integral subgroup of \( {\mathrm{H}}_{1} \... | Let \( {i}_{1} : {\mathrm{H}}_{1} \rightarrow \mathrm{G},{i}_{2} : {\mathrm{H}}_{2} \rightarrow \mathrm{G} \) be the canonical injections. Then\n\n\[ \mathrm{L}\left( {i}_{2}\right) \left( {\mathrm{L}\left( {\mathrm{H}}_{2}\right) }\right) = \mathrm{L}\left( {\mathrm{H}}_{2}\right) \subset \mathrm{L}\left( {\mathrm{H}}... | Yes |
Corollary 3. Let \( \mathrm{G} \) be a finite-dimensional Lie group and \( {\left( {\mathrm{H}}_{i}\right) }_{i \in 1} \) a family of Lie subgroups of \( \mathrm{G} \). Then \( \mathrm{H} = {\bigcap }_{i \in 1}{\mathrm{H}}_{i} \) is a Lie subgroup of \( \mathrm{G} \) and \[ \mathrm{L}\left( \mathrm{H}\right) = \mathop{... | There exists a finite subset \( \mathrm{J} \) of \( \mathrm{I} \) such that \( \mathop{\bigcap }\limits_{{i \in \mathrm{J}}}\mathrm{L}\left( {\mathrm{H}}_{i}\right) \) is equal to the intersection \( \mathrm{M} \) of all the \( \mathrm{L}\left( {\mathrm{H}}_{i}\right) \). We know that \( {\mathrm{H}}^{ * } = \mathop{\b... | Yes |
Corollary 4. Let \( \mathrm{G} \) be a finite-dimensional connected Lie group. The following conditions are equivalent:\n\n(i) G is unimodular (Integration, Chapter VII,§ 1, no. 3, Definition 3);\n\n(ii) \( \det \operatorname{Ad}g = 1 \) for all \( g \in \mathrm{G} \) ;\n\n(iii) Tr ad \( a = 0 \) for all \( a \in \math... | The mapping \( g \mapsto \det \) Ad \( g \) is a morphism \( \phi \) of G into K*. By \( §3 \), Propositions 35 (no. 10) and 44 (no. 12), \( \mathrm{L}\left( \bar{\phi }\right) a = \operatorname{Tr} \) ad \( a \) for all \( a \in \mathrm{L}\left( \mathrm{G}\right) \) . Clearly Im \( \mathcal{L}\left( \phi \right) = \{ ... | Yes |
Proposition 2. Let \( \mathrm{G} \) be a finite-dimensional Lie group and \( \mathrm{H} \) an integral subgroup of G. The following conditions are equivalent:\n\n(i) \( \mathrm{H} \) is closed;\n\n(ii) the topology on \( \mathrm{H} \) is induced by that on \( \mathrm{G} \) ;\n\n(iii) \( \mathrm{H} \) is a Lie subgroup ... | (i) \( \Rightarrow \) (iii): this follows from \( §1 \), Propositions 2 (iv) (no. 1) and 14 (iii) (no. 7).\n\n(iii) \( \Rightarrow \) (ii): obvious.\n\n(ii) \( \Rightarrow \) (i): if the topology on \( \mathrm{H} \) is induced by that on \( \mathrm{G} \) , \( \mathrm{H} \) is closed because \( \mathrm{H} \) is complete... | No |
Proposition 3. Let \( \mathrm{G} \) be a Lie group, \( \mathrm{H} \) an integral subgroup of \( \mathrm{G},\mathrm{M} \) a non-empty connected analytic manifold, \( f \) a mapping of \( \mathbf{M} \) into \( \overline{\mathbf{G}} \) and \( r \in {\mathbf{N}}_{\mathbf{K}} \) . Consider the following conditions:\n\n(i) \... | (ii) \( \Rightarrow \) (i) and (ii) \( \Rightarrow \) (iii): obvious.\n\n(iii) \( \Rightarrow \) (ii): suppose that condition (iii) holds. By Differentiable and Analytic Manifolds, R,9.2.8, \( f \) is a morphism of class \( {\mathrm{C}}^{r} \) of \( \mathrm{M} \) into the left foliation associated with \( \mathrm{L}\le... | Yes |
Corollary 2. Let \( \mathrm{G} \) be a Lie group and \( {\mathrm{H}}_{1} \) and \( {\mathrm{H}}_{2} \) integral subgroups of \( \mathrm{G} \) . Suppose that the topology on \( {\mathrm{H}}_{1} \) admits a countable base. Then \[ {\mathrm{H}}_{2} \subset {\mathrm{H}}_{1} \Leftrightarrow \mathrm{L}\left( {\mathrm{H}}_{2}... | The last assertion and the implication \( \mathrm{L}\left( {\mathrm{H}}_{2}\right) \subset \mathrm{L}\left( {\mathrm{H}}_{1}\right) \Rightarrow {\mathrm{H}}_{2} \subset {\mathrm{H}}_{1} \) follow from Corollary 2 to Proposition 1. The converse implication follows from Proposition 3. | No |
Corollary 3. Let \( \mathrm{G} \) be a Lie group and \( {\mathrm{H}}_{1} \) and \( {\mathrm{H}}_{2} \) integral subgroups of \( \mathrm{G} \) whose topology admits a countable base. If \( {\mathrm{H}}_{1} \) and \( {\mathrm{H}}_{2} \) have the same underlying set, the Lie group structures on \( {\mathrm{H}}_{1} \) and ... | This follows from Corollary 2. | No |
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