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Proposition 6. Let \( \mathrm{E} \) be a filtered bigebra and \( {\left( {\mathrm{E}}_{n}\right) }_{n \geq 0} \) its filtration. For every integer \( n \geq 0 \), let \( {\mathrm{E}}_{n}^{ + } = {\mathrm{E}}_{n} \cap {\mathrm{E}}^{ + } \) . Then \( {\mathrm{E}}_{0}^{ \mp } = \{ 0\} \) and\n\n\[ \n{c}^{ + }\left( {\math... | As \( {E}_{0} = \mathrm{K}.1,\;{E}_{0}^{ + } = 0.\; \) If \( x \in {\mathrm{E}}_{n},\;\pi \left( x\right) = x - \varepsilon \left( x\right) .1 \) (formula (1)), whence \( \pi \left( x\right) \in {\mathrm{E}}_{n}^{ + } \) and \( \pi \left( {\mathrm{E}}_{n}\right) \subset {\mathrm{E}}_{n}^{ + } \) . It follows that \( \p... | Yes |
Proposition 8. Let \( \mathrm{E} \) be a bigebra with coproduct denoted by \( {c}_{\mathrm{E}} \) and let \( h \) be a Lie algebra homomorphism of \( \mathfrak{g} \) into \( \mathrm{P}\left( \mathrm{E}\right) \) (no. 2, Proposition 4). The unital algebra \( \widetilde{\text{homomorphism}}f:\mathrm{U} \rightarrow \mathr... | We show that \( \left( {f \otimes f}\right) \circ c = {c}_{\mathrm{E}} \circ f \) . These are two unital algebra homomorphisms of \( \mathrm{U} \) into \( \mathrm{E} \otimes \mathrm{E} \) and, for \( a \in \sigma \left( \mathfrak{g}\right) \) ,\n\n\[ \left( {f \otimes f}\right) \left( {c\left( a\right) }\right) = f\lef... | Yes |
Proposition 9. For every integer \( n \geq 0 \), let \( {\mathbf{U}}^{n} \) be the vector subspace of \( \mathbf{U} \) generated by the \( \sigma {\left( x\right) }^{n} \) for \( x \in \mathfrak{g} \) . | (a) The sequence \( {\left( {\mathrm{U}}^{n}\right) }_{n \geq 0} \) is a graduation of the vector space \( \mathrm{U} \) compatible with its cogebra structure. Let \( \mathbf{U} \) be given the graduation \( \left( {\mathbf{U}}^{n}\right) \) . (b) The canonical mapping \( \eta : \mathrm{S}\left( \mathrm{g}\right) \righ... | Yes |
Proposition 10. The mapping \( \lambda \mapsto {f}_{\lambda } \) is an isomorphism of the algebra \( {\mathbf{U}}^{\prime } \) onto the algebra of formal power series \( \mathrm{K}{\left\lbrack \left\lbrack {\mathrm{X}}_{i}\right\rbrack \right\rbrack }_{i \in \mathrm{I}} \) . | Because \( \left( {e}_{\alpha }\right) \) is a basis of \( \mathrm{U} \), the mapping \( \lambda \mapsto {f}_{\lambda } \) is \( \mathrm{K} \) -linear and bijective. On the other hand, for \( \lambda ,\mu \) in \( {\mathrm{U}}^{\prime } \) ,\n\n\[ \n{f}_{\lambda \mu } = \mathop{\sum }\limits_{\alpha }\left\langle {{\la... | Yes |
Lemma 2. Let \( \mathrm{V} \) be a vector space, \( \mathrm{E} \) a cogebra and \( f:\mathrm{S}\left( \mathrm{V}\right) \rightarrow \mathrm{E} \) a cogebra morphism. If the restriction off to \( {\mathbf{S}}^{0}\left( \mathbf{V}\right) + {\mathbf{S}}^{1}\left( \mathbf{V}\right) \) is injective, then \( f \) is injectiv... | Let \( n \geq 0 \) ; we write \( {\mathbf{S}}_{n} = \mathop{\sum }\limits_{{i \geq n}}{\mathbf{S}}^{i}\left( \mathrm{\;V}\right) \) and \( {c}_{\mathrm{s}} \) for the coproduct of \( \mathbf{S}\left( \mathrm{V}\right) \) and show by induction on \( n \) that \( f \mid {\mathrm{S}}_{n} \) is injective. Since the asserti... | Yes |
Lemma 3. If a family of scalars \( \left( {\lambda }_{\alpha ,\beta }\right) \) of finite support (with \( \alpha ,\beta \) in \( {\mathbf{N}}^{\left( 1\right) } - \{ 0\} \) ) satisfies relations (20) and (21), there exists a family \( {\left( {\mu }_{\alpha }\right) }_{\left| \alpha \right| \geq 2} \) of finite suppor... | It suffices to prove that\n\n(23)\n\n\[ \alpha + \beta = \gamma + \delta \]\n\nimplies \( {\lambda }_{\alpha ,\beta } = {\lambda }_{\gamma ,\delta } \) for \( \alpha ,\beta ,\gamma ,\delta \) non-zero. By Riesz’s Decomposition Lemma (Algebra, Chapter VI,§ 1, no. 10, Theorem 1) there exist \( \pi ,\rho ,\sigma ,\tau \) ... | Yes |
Proposition 1. Let \( \psi \) be the canonical mapping of \( \operatorname{Lib}\left( \mathrm{X}\right) \) onto \( \mathrm{L}\left( \mathrm{X}\right) \) and \( \phi \) the restriction of \( \psi \) to \( \mathrm{X} \) . For every mapping \( f \) of \( \mathrm{X} \) into a Lie algebra \( \mathfrak{g} \), there exists on... | (a) Existence of \( \mathrm{F} \) : let \( h \) be the homomorphism of \( \operatorname{Lib}\left( \mathrm{X}\right) \) into \( \mathfrak{g} \) extending \( f \) (no. 1). For all \( a \) in \( \operatorname{Lib}\left( \mathrm{X}\right), h\left( {\mathrm{Q}\left( a\right) }\right) = h\left( {a.a}\right) = \left\lbrack {... | Yes |
Corollary 1. The family \( {\left( \phi \left( x\right) \right) }_{x \in \mathrm{X}} \) is free over \( \mathrm{K} \) in \( \mathrm{L}\left( \mathrm{X}\right) \) . | Let \( {x}_{1},{x}_{2},\ldots ,{x}_{n} \) be distinct elements of \( \mathrm{X} \) and \( {\lambda }_{1},\ldots ,{\lambda }_{n} \) be elements of \( \mathrm{K} \) such that\n\n(3)\n\n\[{\lambda }_{1} \cdot \phi \left( {x}_{1}\right) + \cdots + {\lambda }_{n} \cdot \phi \left( {x}_{n}\right) = 0.\]\n\nLet \( \mathfrak{g... | Yes |
Corollary 2. Let \( \mathfrak{a} \) be a Lie algebra. Every extension of \( \mathrm{L}\left( \mathrm{X}\right) \) by \( \mathfrak{a} \) is inessential. | Let \( \mathfrak{a} \rightarrow \mathfrak{g} \rightarrow \mathrm{L}\left( \mathrm{X}\right) \) be such an extension (Chapter I,§ 1, no. 7). As \( \mu \) is surjective, there exists a mapping \( f \) of \( \mathrm{X} \) into \( \mathfrak{g} \) such that \( \phi = \mu \circ f \) . Let \( \mathrm{F} \) be the homomorphism... | Yes |
Proposition 2. Let \( \mathrm{X} \) and \( \mathrm{Y} \) be two sets. Every mapping \( u : \mathrm{X} \rightarrow \mathrm{Y} \) can be extended uniquely to a Lie algebra homomorphism \( \mathrm{L}\left( u\right) : \mathrm{L}\left( \mathrm{X}\right) \rightarrow \mathrm{L}\left( \mathrm{Y}\right) \) . For every mapping \... | The existence and uniqueness of \( L\left( u\right) \) follow from Proposition 1 of no. 2. The homomorphisms \( \mathrm{L}\left( {v \circ u}\right) \) and \( \mathrm{L}\left( v\right) \circ \mathrm{L}\left( u\right) \) have the same restriction to \( \mathrm{X} \) and hence are equal (Proposition 1). | Yes |
Proposition 3. Let \( {\mathrm{K}}^{\prime } \) be a non-zero commutative ring and \( u : \mathrm{K} \rightarrow {\mathrm{K}}^{\prime } \) a ring homomorphism. For every set \( \mathrm{X} \) there exists one and only one Lie \( {\mathrm{K}}^{\prime } \) -algebra homomorphism\n\n\[ v : {\mathrm{L}}_{\mathrm{K}}\left( \m... | Applying Proposition 1 to \( \mathfrak{g} = {\mathrm{L}}_{{\mathrm{K}}^{\prime }}\left( \mathrm{X}\right) \) considered as a Lie K-algebra and the mapping \( x \mapsto x \) of \( \mathrm{X} \) into \( \mathfrak{g} \), we obtain a \( \mathrm{K} \) -homomorphism \( {\mathrm{L}}_{\mathrm{K}}\left( \mathrm{X}\right) \right... | Yes |
Lemma 1. The ideal \( \mathfrak{a} \) of Definition 1 is graded. | For \( a, b \) in \( \operatorname{Lib}\left( \mathrm{X}\right) \), let \( \mathrm{B}\left( {a, b}\right) = a.b + b.a \) . The formulae\n\n(10)\n\n\[ \mathrm{B}\left( {a, b}\right) = \mathrm{Q}\left( {a + b}\right) - \mathrm{Q}\left( a\right) - \mathrm{Q}\left( b\right) \]\n\n(11)\n\n\[ \mathrm{Q}\left( {{\lambda }_{1}... | Yes |
Proposition 4. Let \( \\mathbf{S} \) be a subset of \( \\mathbf{X} \) . If \( {\\mathbf{N}}^{\\left( \\mathrm{S}\\right) } \) is identified with its canonical image in \( {\\mathbf{N}}^{\\left( \\mathrm{x}\\right) } \) (Algebra, Chapter I,§ 7, no. 7), then \( \\mathrm{L}\\left( \\mathrm{S}\\right) = \\mathop{\\sum }\\l... | Let \( \\alpha \\in {\\mathbf{N}}^{\\left( \\mathrm{S}\\right) } \) . The module \( {\\mathrm{L}}^{\\alpha }\\left( \\mathrm{S}\\right) \) is generated by the images in \( \\mathrm{L}\\left( \\mathrm{X}\\right) \) of the elements \( w \) in \( \\mathrm{M}\\left( \\mathrm{S}\\right) \) such that \( \\phi \\left( w\\righ... | Yes |
Proposition 5. Let \( \mathfrak{g} \) be a Lie algebra and \( \mathrm{P} \) a submodule of \( \mathfrak{g} \) . We define the submodules \( {\mathrm{P}}_{n} \) of \( \mathfrak{g} \) by the formulae \( {\mathrm{P}}_{1} = \mathrm{P} \) and \( {\mathrm{P}}_{n + 1} = \left\lbrack {\mathrm{P},{\mathrm{P}}_{n}}\right\rbrack ... | We prove (18) by induction on \( m \) . The case \( m = 1 \) is obvious. By the Jacobi identity,\n\n\[ \left\lbrack {\left\lbrack {\mathrm{P},{\mathrm{P}}_{m}}\right\rbrack ,{\mathrm{P}}_{n}}\right\rbrack \subset \left\lbrack {{\mathrm{P}}_{m},\left\lbrack {\mathrm{P},{\mathrm{P}}_{n}}\right\rbrack }\right\rbrack + \le... | Yes |
Proposition 7. Let \( \\mathrm{X} \) be a set and \( n \) an integer \( \\geq 1 \). (a) \( {\\mathrm{L}}^{n + 1}\\left( \\mathrm{X}\\right) = \\left\\lbrack {{\\mathrm{L}}^{1}\\left( \\mathrm{X}\\right) ,{\\mathrm{L}}^{n}\\left( \\mathrm{X}\\right) }\\right\\rbrack \) . | (a) We apply Proposition 5 with \( \\mathfrak{g} = \\mathrm{L}\\left( \\mathrm{X}\\right) \) and \( \\mathrm{P} = {\\mathrm{L}}^{1}\\left( \\mathrm{X}\\right) \). By induction on \( n \), we deduce from (12) (no. 6) and (19) the equality \( {\\mathrm{P}}_{n} = {\\mathrm{L}}^{n}\\left( \\mathrm{X}\\right) \). The desire... | Yes |
Proposition 8. Let \( \\mathrm{X} \) be a set, let \( \\mathrm{M} \) be an \( \\mathrm{L}\\left( \\mathrm{X}\\right) \) -module and let \( d \) be a mapping of \( \\mathrm{X} \) into \( \\mathrm{M} \) . There exists one and only one linear mapping \( \\mathrm{D} \) of \( \\mathrm{L}\\left( \\mathrm{X}\\right) \) into \... | We define a Lie algebra \( g \) with underlying module \( M \\times L\\left( X\\right) \) by means of the bracket\n\n(21)\n\n\[ \n\\left\\lbrack {\\left( {m, a}\\right) ,\\left( {{m}^{\\prime },{a}^{\\prime }}\\right) }\\right\\rbrack = \\left( {a \\cdot {m}^{\\prime } - {a}^{\\prime } \\cdot m,\\left\\lbrack {a,{a}^{\... | Yes |
Proposition 9. Let \( {\mathrm{S}}_{1} \) and \( {\mathrm{S}}_{2} \) be two disjoint sets and da mapping of \( {\mathrm{S}}_{1} \times {\mathrm{S}}_{2} \) into \( \mathrm{L}\left( {\mathrm{S}}_{2}\right) \) . Let \( \mathfrak{g} \) be the quotient Lie algebra of \( \mathrm{L}\left( {{\mathrm{S}}_{1} \cup {\mathrm{S}}_{... | For \( i = 1,2 \), let \( {\psi }_{i} \) denote the homomorphism of \( \widetilde{\mathrm{L}}\left( {\mathrm{S}}_{i}\right) \) into \( \mathfrak{g} \) which extends\n\n\( {\phi }_{i} \) and \( {\mathfrak{a}}_{i} \) its image. Clearly \( {\phi }_{i}\left( {\mathrm{\;S}}_{i}\right) \) generates \( {\mathfrak{a}}_{i} \) .... | Yes |
Proposition 11. There exists a Hall set relative to \( \mathrm{X} \) . | We shall construct by induction on the integer \( n \) sets \( {\mathrm{H}}_{n} \subset {\mathrm{M}}^{n}\left( \mathrm{X}\right) \) and a total ordering on these sets:\n\n(a) We write \( {\mathrm{H}}_{1} = \mathrm{X} \) and give it a total ordering.\n\n(b) The set \( {\mathrm{H}}_{2} \) consists of the products \( {xy}... | Yes |
Proposition 12. Let \( \mathrm{H} \) be a Hall set relative to \( \mathrm{X} \) and let \( x, y \) be in \( \mathrm{X} \) . (a) \( \mathrm{H} \cap \mathrm{M}\left( {\{ x\} }\right) = \{ x\} \) . (b) Suppose that \( x < y \) and let \( {d}_{y} \) be the homomorphism of \( \mathbf{M}\left( \mathbf{X}\right) \) into \( \m... | By Definition 2 (B), \( x \in \mathrm{H} \) and \( \mathrm{H} \cap {\mathrm{M}}^{2}\left( {\{ x\} }\right) = \varnothing \) . If \( w \in \mathrm{H} \cap \mathrm{M}\left( {\{ x\} }\right) \) , where \( n = l\left( w\right) \geq 3 \), the elements \( \alpha \left( w\right) \) and \( \beta \left( w\right) \) also belong ... | Yes |
For every integer \( p \geq 0 \), the module \( {\mathrm{L}}_{p} \) admits the family \( {\left( {\bar{w}}_{i}\right) }_{0 \leq i < p} \) as basis, the Lie algebra \( {\mathfrak{g}}_{p} \) admits \( {\left( \bar{u}\right) }_{u \in {\mathbf{P}}_{p}} \) as basic family and the module \( \mathrm{L}\left( \mathrm{X}\right)... | \( {\mathrm{L}}_{0} = \{ 0\} \) and \( {\mathfrak{g}}_{0} = \mathrm{L}\left( \mathrm{X}\right) \) and the lemma is true for \( p = 0 \) . We argue by induction on \( p \) . Suppose then that the lemma is true for some integer \( p \geq 0 \) . Let \( {u}_{i, w} = {\left( \operatorname{ad}{\bar{w}}_{p}\right) }^{i} \cdot... | Yes |
Corollary 1. There exists on the algebra \( \mathrm{A}\left( \mathrm{X}\right) \) a unique coproduct making \( \mathrm{A}\left( \mathrm{X}\right) \) into a bigebra such that the elements of \( \mathbf{X} \) are primitive. Further, \( \beta \) is an isomorphism of the bigebra \( \mathrm{U}\left( {\mathrm{L}\left( \mathr... | This follows from assertion (b) of the theorem and the fact that \( \mathbf{X} \) generates the unital algebra \( \mathrm{A}\left( \mathrm{X}\right) \) . | Yes |
Corollary 2. If \( \mathrm{K} \) is a field of characteristic \( 0,\mathrm{\;L}\left( \mathrm{X}\right) \) is the Lie algebra of primitive elements of \( \mathrm{A}\left( \mathrm{X}\right) \) . | This follows from Corollary 1 and the Corollary to Proposition 9 of § 1, no. 5. | No |
Proposition 1. (a) The restriction \( {\pi }_{0} \) of \( \pi \) to \( \mathrm{L}\left( \mathrm{X}\right) \) is a derivation of \( \mathrm{L}\left( \mathrm{X}\right) \) . | (a) Let \( \mathrm{E} \) be the endomorphism algebra of the module \( \mathrm{L}\left( \mathrm{X}\right) \) and \( \theta \) the homomorphism of \( \mathrm{A}\left( \mathrm{X}\right) \) into \( \mathrm{E} \) such that \( \theta \left( x\right) = \operatorname{ad}x \) for all \( x \in \mathrm{X} \) . The restriction of ... | Yes |
Lemma 1. Let \( n \) be an integer \( > 0,{\mathrm{\;T}}_{1},\ldots ,{\mathrm{T}}_{n} \) indeterminates and \( {u}_{1},\ldots ,{u}_{n} \) elements of \( \mathbf{Z} \) . Let \( {\left( c\left( \alpha \right) \right) }_{\alpha \in {\mathbf{N}}^{n} = \{ 0\} } \) be a family of elements of \( \mathbf{Z} \) such that\n\n(7)... | Formula (7) is equivalent, on taking logarithms on both sides (Algebra, Chapter IV, § 6, no. 9) to:\n\n(9)\n\n\[ \log \left( {1 - \mathop{\sum }\limits_{{i = 1}}^{n}{u}_{i}{\mathrm{\;T}}_{i}}\right) = \mathop{\sum }\limits_{{\alpha \neq 0}}c\left( \alpha \right) \log \left( {1 - {\mathrm{T}}^{\alpha }}\right) .\n\]\n\n... | Yes |
Proposition 2. The set \( \Gamma \) is a subgroup of \( {\mathrm{A}}^{ * } \) and \( \left( {\Gamma }_{\alpha }\right) \) is a central filtration on \( \Gamma \) . | \( \Gamma = \mathop{\bigcup }\limits_{{\alpha > 0}}{\Gamma }_{\alpha } \) by construction and the relation \( {\Gamma }_{\alpha } = \mathop{\bigcap }\limits_{{\beta < \alpha }}{\Gamma }_{\beta } \) follows from \( {A}_{\alpha } = \mathop{\bigcap }\limits_{{\beta < \alpha }}{A}_{\beta } \n\nWe show that \( {\Gamma }_{\... | Yes |
Proposition 3. (i) For all \( \alpha \in \mathbf{R} \), there exists a unique group homomorphism \( {g}_{\alpha } : {\operatorname{gr}}_{\alpha }\left( \mathrm{G}\right) \rightarrow {\operatorname{gr}}_{\alpha }\left( \mathrm{\;A}\right) \) which maps the class modulo \( {\mathrm{G}}_{\alpha }^{ + } \) of an element \(... | (i) Let \( \alpha > 0 \) . By hypothesis, for all \( a \) in \( {\mathrm{G}}_{\alpha },\rho \left( a\right) - 1 \in {\mathrm{A}}_{\alpha } \) ; let \( {p}_{\alpha }\left( a\right) \) denote the class of \( \rho \left( a\right) - 1 \) modulo \( {\mathrm{A}}_{\alpha }^{ + } \) . As \( {\mathrm{A}}_{2\alpha } \subset {\ma... | Yes |
Proposition 4. (ii) If \( {\left( {\mathrm{G}}_{n}\right) }_{n \in {\mathbf{N}}^{ * }} \) is an integral central filtration on \( \mathrm{G} \), then \( {\mathrm{C}}^{n}\mathrm{G} \subset {\mathrm{G}}_{n} \) for all \( n \in {\mathbf{N}}^{ * } \). | We prove (ii) by induction on \( n;{\mathrm{C}}^{1}\mathrm{G} = \mathrm{G} = {\mathrm{G}}_{1} \) ; for \( n > 1 \), \[ {\mathrm{C}}^{n}\mathrm{G} = \left( {\mathrm{G},{\mathrm{C}}^{n - 1}\mathrm{G}}\right) \subset \left( {\mathrm{G},{\mathrm{G}}_{n - 1}}\right) \subset {\mathrm{G}}_{n}. \] | Yes |
Proposition 5. Let \( \mathrm{G} \) be a group and \( \mathrm{{gr}}\left( \mathrm{G}\right) \) the graded Lie \( \mathbf{Z} \) -algebra associated with the lower central filtration on \( \mathrm{G} \) . Then \( \mathrm{{gr}}\left( \mathrm{G}\right) \) is generated by \( {\mathrm{{gr}}}_{1}\left( \overline{\mathrm{G}}\r... | Let \( \mathrm{L} \) be the Lie subalgebra of \( \operatorname{gr}\left( \mathrm{G}\right) \) generated by \( {\operatorname{gr}}_{1}\left( \mathrm{G}\right) \) ; we show that \( \mathrm{L} \supset {\operatorname{gr}}_{n}\left( \mathrm{G}\right) \) by induction on \( n \), the assertion being trivial for \( n = 1 \) . ... | Yes |
Proposition 1. Let \( \mathbf{B} \) be a unital associative algebra with a real filtration \( {\left( {\mathbf{B}}_{\mathbf{\alpha }}\right) }_{\mathbf{\alpha } \in \mathbf{R}} \) such that \( \mathrm{B} \) is Hausdorff and complete \( \left( {§4\text{, nos. 1 and 2}}\right) \) . Let \( f \) be a mapping of \( \mathrm{... | Let \( {f}^{\prime } \) be the unique unital algebra homomorphism of \( \mathrm{A}\left( \mathrm{X}\right) \) into B extending \( f \) (Algebra, Chapter III,§ 2, no. 7, Proposition 7). We show that \( {f}^{\prime } \) is continuous: \( {f}^{\prime }\left( {{\mathrm{A}}^{n}\left( \mathrm{X}\right) }\right) \subset {\mat... | Yes |
For an element a of \( \widehat{\mathrm{A}}\left( \mathrm{X}\right) \) to be invertible, it is necessary and sufficient that its constant term be invertible in \( \mathbf{K} \) . | If \( a \) is invertible in \( \widehat{\mathrm{A}}\left( \mathrm{X}\right) ,\varepsilon \left( a\right) \) is invertible in \( \mathrm{K} \) . Conversely, if \( \varepsilon \left( a\right) \) is invertible in \( \mathrm{K} \), there exists \( u \in {\widehat{\mathrm{A}}}_{1}\left( \mathrm{X}\right) \) such that \( a =... | Yes |
Lemma 2. Let \( n \) be a non-zero rational integer. In the ring of formal power series \( \mathrm{K}\left\lbrack \left\lbrack t\right\rbrack \right\rbrack \) we write \( {\left( 1 + t\right) }^{n} = \mathop{\sum }\limits_{{j \geq 0}}{c}_{j, n}{t}^{j} \) . There exists an integer \( j \geq 1 \) such that \( {c}_{j, n} ... | If \( n > 0 \), then \( {c}_{n, n} = 1 \) by the binomial formula.\n\nSuppose that \( n < 0 \) and let \( m = - n \) . If \( {c}_{j, n} = 0 \) for all \( j \geq 1 \), then \( {\left( 1 + t\right) }^{n} = 1 \), whence, taking the inverse, \( {\left( 1 + t\right) }^{m} = 1 \), which contradicts the formula \( {c}_{m, m} ... | Yes |
Lemma 3. Let \( {x}_{1},\ldots ,{x}_{s} \) be elements of \( \mathrm{X} \) such that \( s \geq 1 \) and \( {x}_{i} \neq {x}_{i + 1} \) for \( 1 \leq i \leq s - 1 \) ; let \( {n}_{1},\ldots ,{n}_{s} \) be non-zero rational integers. Then the element\n\n\( \mathop{\prod }\limits_{{i = 1}}^{s}{\left( 1 + {x}_{i}\right) }^... | Let \( \mathfrak{m} \) be a maximal ideal of \( \mathrm{K} \) and \( k \) the field \( \mathrm{K}/\mathfrak{m} \) ; let \( p : {\widehat{\mathrm{A}}}_{\mathrm{K}}\left( \mathrm{X}\right) \rightarrow {\widehat{\mathrm{A}}}_{k}\left( \mathrm{X}\right) \) be the unique continuous homomorphism of unital K-algebras such tha... | Yes |
Lemma 4. Let \( \sigma \) be the continuous endomorphism of \( \widehat{\mathrm{A}}\left( \mathrm{X}\right) \) such that \( \sigma \left( x\right) = x + r\left( x\right) \) for \( x \in \mathrm{X} \) (no. 1, Proposition 1). Then \( \sigma \) is an automorphism and \( \sigma \left( {{\underline{A}}_{m}\left( \mathrm{X}\... | \( \sigma \left( x\right) \equiv x{\;\operatorname{mod}\;.}{\widehat{\mathrm{A}}}_{2}\left( \mathrm{X}\right) \) for \( x \in \mathrm{X} \), whence, for \( n \geq 1 \) and \( {x}_{1},\ldots ,{x}_{n} \) in \( \mathrm{X} \) ,\n\n\[ \sigma \left( {x}_{1}\right) \ldots \sigma \left( {x}_{n}\right) \equiv {x}_{1}\ldots {x}_... | Yes |
Theorem 3. For all \( x \in \mathrm{X} \) , let \( c\left( x\right) \) be the canonical image of \( x \) in \( \mathrm{F}\left( \mathrm{X}\right) /\left( {\mathrm{F}\left( \mathrm{X}\right) ,\mathrm{F}\left( \mathrm{X}\right) }\right) \) . Let \( \mathfrak{g} \) be the graded Lie \( \mathbf{Z} \) -algebra associated wi... | (B) Surjectivity of \( \alpha \) . As \( \mathrm{X} \) generates the group \( \mathrm{F} = {\mathrm{C}}^{1} \), the set \( c\left( \mathrm{X}\right) \) generates the \( \mathbf{Z} \) -module \( {\mathfrak{g}}^{1} = {\mathrm{C}}^{1}/{\mathrm{C}}^{2} \) . But \( {\mathfrak{g}}^{1} \) generates the Lie \( \mathbf{Z} \) -a... | Yes |
Proposition 2. Suppose that \( \mathrm{X} \) is finite. For every integer \( n \geq 1 \), the group \( \mathrm{F}\left( \mathrm{X}\right) /{\mathrm{F}}_{n}^{\left( p\right) }\left( \mathrm{X}\right) \) is a finite \( p \) -group of nilpotency class \( \leq n \) . | Arguing by induction on \( n \), it suffices to prove that \( {\mathrm{F}}_{n}^{\left( p\right) }\left( \mathrm{X}\right) /{\mathrm{F}}_{n + 1}^{\left( p\right) }\left( \mathrm{X}\right) \) is a finite commutative \( p \) -group for all \( n \geq 1 \) . For all \( w \in {\mathrm{F}}_{n}^{\left( p\right) }\left( \mathrm... | Yes |
Proposition 3. For all \( w \neq 1 \) in \( \mathrm{F}\left( \mathrm{X}\right) \), there exist a finite p-group \( \mathrm{G} \) and a homomorphism \( f \) of \( \mathrm{F}\left( \mathrm{X}\right) \) into \( \mathrm{G} \) such that \( f\left( w\right) \neq 1 \) . | There exist elements \( {x}_{1},\ldots ,{x}_{r} \) of \( \mathrm{X} \) and integers \( {n}_{1},\ldots ,{n}_{r} \) such that \( w = {x}_{1}^{{n}_{1}}\ldots {x}_{r}^{{n}_{r}} \) . Let \( \mathrm{Y} = \left\{ {{x}_{1},\ldots ,{x}_{r}}\right\} \) . The canonical injection of \( \mathrm{Y} \) into \( \mathrm{X} \) extends t... | Yes |
Proposition 1. The exponential mapping is a homeomorphism of \( \mathfrak{m} \) onto \( 1 + \mathfrak{m} \) and the logarithmic mapping is the inverse homeomorphism. | For \( x \in {\mathrm{A}}_{\alpha },\frac{{x}^{n}}{n!} \in {\mathrm{A}}_{n\alpha } \) . It follows that the series defining the exponential converges uniformly on each of the sets \( {\mathrm{A}}_{\alpha } \) for \( \alpha > 0 \) ; as \( {\mathrm{A}}_{\alpha } \) is open in \( m \) and \( \mathfrak{m} = \mathop{\bigcup... | Yes |
Lemma 1. Let \( \mathfrak{g} \) be a filtered Lie algebra \( \left( {§4,\text{no. 1}}\right) ,{\left( {\mathfrak{g}}_{\alpha }\right) }_{\alpha \in \mathbf{R}} \) its filtration and let \( \alpha \in \mathbf{R} \) . Let \( \mathrm{P} \) be a homogeneous Lie polynomial of degree \( n \) in the indeterminates \( {\left( ... | Every Lie polynomial of degree \( n \geq 2 \) is a finite sum of terms of the form \( \left\lbrack {\mathrm{Q},\mathrm{R}}\right\rbrack \) where \( \mathrm{Q} \) and \( \mathrm{R} \) are of degree \( < n \) and the sum their degrees is equal to \( n \) (§ 2, no. 7, Proposition 7). The lemma follows by induction on \( n... | Yes |
Proposition 2. The homomorphism \( {f}_{t} : \mathrm{L}\left( \mathrm{I}\right) \rightarrow \mathrm{g} \) such that \( {f}_{t}\left( {\mathrm{\;T}}_{t}\right) = {t}_{t}\left( {§2\text{, no. 4}}\right) \) can be extended by continuity to one and only one continuous homomorphism \( {\widehat{f}}_{t} \) of \( \widehat{\ma... | There exists \( \alpha > 0 \) such that \( {t}_{i} \in {\mathfrak{g}}_{\alpha } \) for all \( i \in \mathbf{I} \) ; hence \( {f}_{t}\left( {{\mathrm{L}}^{\mathrm{v}}\left( \mathbf{I}\right) }\right) \subset {\mathfrak{g}}_{\left| \mathrm{v}\right| \alpha } \) for all \( v \) (Lemma 1), which implies the continuity of \... | Yes |
Proposition 3. If \( a \in \mathfrak{m}, b \in \mathfrak{m} \), then \( \exp \mathrm{H}\left( {a, b}\right) = \exp a.\exp b \) . | Let \( a, b \) be in \( \mathfrak{m} \) ; there exists \( \alpha > 0 \) such that \( a \in {\mathrm{A}}_{\alpha } \) and \( b \in {\mathrm{A}}_{\alpha } \) . Then there exists a continuous homomorphism \( \theta \) of the Magnus algebra \( \widehat{\mathrm{A}}\left( {\{ \mathrm{U},\mathrm{V}\} }\right) \) into A mappin... | Yes |
Proposition 4. Let \( \\mathfrak{g} \) be a complete Hausdorff filtered Lie algebra such that \( \\mathfrak{g} = \\mathop{\\bigcup }\\limits_{{\\alpha > 0}}{\\mathfrak{g}}_{\\alpha } \). The mapping \( \\left( {a, b}\\right) \\mapsto \\mathrm{H}\\left( {a, b}\\right) \) is a group law on \( \\mathfrak{g} \) compatible ... | The mapping \( \\left( {a, b}\\right) \\mapsto \\mathrm{H}\\left( {a, b}\\right) \) of \( \\mathfrak{g} \\times \\mathfrak{g} \) into \( \\mathfrak{g} \) is continuous (no. 3); as the mapping \( a \\mapsto - a \) is obviously continuous, it suffices to prove the relations\n\n(18)\n\n\[ \n\\mathrm{H}\\left( {\\mathrm{H}... | Yes |
Lemma 1. \[ \begin{Vmatrix}{\widetilde{\mathrm{H}}}_{r, s}\end{Vmatrix} \leq {\mathrm{M}}^{r + s - 1}{\eta }_{r, s}. \] | \[ \begin{Vmatrix}{\widetilde{\mathrm{H}}}_{r, s}\end{Vmatrix} < {\eta }_{r, s}\frac{{\mathrm{M}}^{r + s - 1}}{r + s} \leq {\eta }_{r, s}{\mathrm{M}}^{r + s - 1}, \] which proves the lemma. | Yes |
Proposition 1. The formal power series \( \widetilde{\mathrm{H}} \) is a convergent series (Differentiable and Analytic Manifolds, R, 3.1.1); its domain of absolute convergence (Differentiable and Analytic Manifolds, R, 3.1.4) contains the open set\n\n\[ \Omega = \left\{ {\left( {x, y}\right) \in \mathfrak{g} \times \m... | Let \( u, v \) be two real numbers \( > 0 \) such that \( u + v < \frac{1}{\mathrm{M}}\log 2 \) ; then (Lemma\n\n(12) \( \mathrm{M}\mathop{\sum }\limits_{{r, s \geq 0}}\begin{Vmatrix}{\widetilde{\mathrm{H}}}_{r, s}\end{Vmatrix}{u}^{r}{v}^{s} \)\n\n\[ \leq \mathop{\sum }\limits_{{r, s \geq 0}}{\eta }_{r, s}{\mathrm{M}}^... | Yes |
If \( \left( {x, y, z}\right) \in {\Omega }^{\prime } \), then\n\n(16)\n\n\[)\;\left( {x, y}\right) \in \Omega ,\;\left( {h\left( {x, y}\right) ,\;z}\right) \in \Omega ,\;\left( {y,\;z}\right) \in \Omega ,\;\left( {x, h\left( {y,\;z}\right) }\right) \in \Omega\n\]\n\nand\n\n(17)\n\n\[h\left( {h\left( {x, y}\right), z}\... | Let \( \left( {x, y, z}\right) \in {\Omega }^{\prime } \) ; clearly \( \left( {x, y}\right) \in \Omega \) and \( \left( {y, z}\right) \in \Omega \) . Moreover:\n\n\[ \parallel h\left( {x, y}\right) \parallel \leq \mathop{\sum }\limits_{{r, s}}\begin{Vmatrix}{\widetilde{\mathrm{H}}}_{r, s}\end{Vmatrix}\parallel x\parall... | Yes |
Proposition 3. For \( \parallel x\parallel + \parallel y\parallel < \frac{1}{2}\log 2 \) , | \[ {\exp }_{\mathrm{A}}x \cdot {\exp }_{\mathrm{A}}y = {\exp }_{\mathrm{A}}h\left( {x, y}\right) . \] It follows from (18) and the relation \( {e}^{\mathrm{U}}{e}^{\mathrm{V}} = {e}^{\mathrm{H}\left( {\mathrm{U},\mathrm{V}}\right) } \) that \[ m \circ \left( {1 + \widetilde{e},1 + \widetilde{e}}\right) = \left( {1 + \w... | Yes |
Lemma 1. Let \( n \) be an integer \( \geq 0 \) and let \( n = {n}_{0} + {n}_{1}p + \cdots + {n}_{k}{p}^{k} \), with \( 0 \leq {n}_{i} \leq p - 1 \), be the p-adic expansion of \( n \) . Let \( \mathrm{S}\left( n\right) = {n}_{0} + {n}_{1} + \cdots + {n}_{k} \) . Then\n\n\[ {v}_{p}\left( {n!}\right) = \frac{n - \mathrm... | \n\n\( {v}_{p}\left( {n!}\right) = \mathop{\sum }\limits_{{i = 1}}^{n}{v}_{p}\left( i\right) \) and the number of integers \( i \) between 1 and \( n \) for which \( {v}_{p}\left( i\right) \geq j \) is equal to the integral part \( \left\lbrack {n/{p}^{j}}\right\rbrack \) of \( n/{p}^{j} \) . Then\n\n\[ {v}_{p}\left( {... | Yes |
Lemma 2. \( v\left( n\right) \leq v\left( {n!}\right) \leq \left( {n - 1}\right) \theta \) and \( v\left( n\right) \leq \left( {\log n}\right) /\left( {\log p}\right) \) for every integer \( n \geq 1 \) . | \( v\left( {n!}\right) = {v}_{p}\left( {n!}\right) = \left( {n - \mathrm{S}\left( n\right) }\right) \theta \leq \left( {n - 1}\right) \theta \) by Lemma \( 1. \)\n\nOn the other hand, \( n \geq {p}^{v\left( n\right) } \), whence \( v\left( n\right) \leq \left( {\log n}\right) /\left( {\log p}\right) \). | Yes |
Proposition 1. Let \( r \) and \( s \) be two integers \( \geq 0 \) . If \( {\mathrm{H}}_{r, s} = \mathop{\sum }\limits_{{b \in \mathrm{B}}}{\lambda }_{b}{e}_{b} \), where \( {\lambda }_{b} \in \mathbf{Q} \) , is the decomposition of \( \mathrm{H} \) with respect to the basis \( {\left( {e}_{b}\right) }_{b \in \mathbf{... | The ring \( {A}_{{\mathbf{Z}}_{\left( p\right) }}\left( I\right) \) is identified with the sub- \( {\mathbf{Z}}_{\left( p\right) } \) -module of \( {A}_{\mathbf{Q}}\left( I\right) \) generated by the words \( w \in \operatorname{Mo}\left( \mathrm{I}\right) \) . As \( {\mathrm{L}}_{{\mathbf{Z}}_{\left( p\right) }}\left(... | Yes |
Lemma 3.\n\[ \n\begin{Vmatrix}{\widetilde{\mathrm{H}}}_{r, s}\end{Vmatrix} \leq {a}^{-\left( {r + s - 1}\right) \theta }. \n\] | Let B be a Hall set relative to I and let \( {\mathrm{H}}_{r, s} = \mathop{\sum }\limits_{{b \in \mathrm{B}}}{\lambda }_{b}{e}_{b} \) be the decomposition of \( {\mathrm{H}}_{r, s} \) with respect to the corresponding basis of \( \mathrm{L}\left( {\{ \mathrm{U},\mathrm{V}\} }\right) \) . Then\n\n(11)\n\n\[ \n\left| {\l... | Yes |
Proposition 2. The formal power series \( \widetilde{\text{ H }} \) is a convergent series (Differentiable and Analytic Manifolds, R,4.1.1). If \( \mathrm{G} \) is the ball \( \left\{ {x \in \mathfrak{g} \mid \parallel x\parallel < {a}^{\mathfrak{g}}}\right\} \), the domain of absolute convergence of \( \widetilde{\mat... | If \( u \) and \( v \) are two real numbers \( > 0 \) such that \( u < {a}^{\theta } \) and \( v < {a}^{\theta } \), then (Lemma 3)\n\n(14)\n\n\[ \begin{Vmatrix}{\widetilde{\mathrm{H}}}_{r, s}\end{Vmatrix}{u}^{r}{v}^{s} \leq {a}^{\theta }{\left( u{a}^{-\theta }\right) }^{r}{\left( v{a}^{-\theta }\right) }^{s} \]\n\nand... | Yes |
Proposition 4. Let \( \\mathrm{R} \) be a real number such that \( 0 < \\mathrm{R} \\leq {a}^{\\theta } \) . The mapping \( {\\exp }_{\\mathrm{A}} \) defines an analytic isomorphism of \( {\\mathrm{G}}_{\\mathrm{R}} \) onto \( 1 + {\\mathrm{G}}_{\\mathrm{R}} \) and the inverse isomorphism is the restriction of \( {\\lo... | \[ e\\left( {l\\left( \\mathbf{X}\\right) }\\right) = l\\left( {e\\left( \\overline{\\mathbf{X}}\\right) }\\right) = \\mathbf{X} \]. By (20),(21) and Differentiable and Analytic Manifolds, \( \\mathrm{R},{4.1.5} \), we deduce that \( {e}_{\\mathrm{A}}\\left( {{l}_{\\mathrm{A}}\\left( x\\right) }\\right) = {l}_{\\mathrm... | Yes |
Proposition 5. For \( x, y \) in \( \mathrm{G} \) , \n\n(26) \n\n\[ \n{\exp }_{\mathrm{A}} \cdot {\exp }_{\mathrm{A}}y = {\exp }_{\mathrm{A}}h\left( {x, y}\right) . \n\] | \[ \n{e}^{\mathrm{U}}{e}^{\mathrm{V}} = {e}^{\mathrm{H}\left( {\mathrm{U},\mathrm{V}}\right) } \n\] \nand hence \n\n\[ \nm \circ \left( {1 + \widetilde{e},1 + \widetilde{e}}\right) = \left( {1 + \widetilde{e}}\right) \circ \widetilde{\mathrm{H}} \n\] \nin \( \widetilde{\mathrm{H}}\left( {\mathrm{A} \times \mathrm{A};\m... | No |
Proposition 1. A Lie group is a complete metrizable topological group. | Since \( e \) admits an open neighbourhood homeomorphic to an open ball of a normed space, \( e \) admits a countable fundamental system of neighbourhoods whose intersection is \( \{ e\} \) . Hence G is metrizable (General Topology, Chapter III, § 1, Corollary to Proposition 2 and Chapter IX, § 3, Proposition 1). By Le... | No |
Proposition 2. Let \( \mathrm{G} \) be a Lie group.\n\n(i) If \( \mathrm{K} = \mathbf{R} \) or \( \mathbf{C},\mathrm{G} \) is locally connected. | Let \( \mathrm{U} \) be a neighbourhood of \( e \) . There exists an open neighbourhood \( {\mathrm{U}}_{1} \) of \( e \) contained in \( \mathrm{U} \) and homeomorphic to an open ball of a normed space \( \mathrm{E} \) over K. If \( \mathrm{K} = \mathbf{R} \) or \( \mathbf{C},{\mathrm{U}}_{\mathbf{1}} \) is connected,... | Yes |
Lemma 2. Let \( \mathrm{X} \) be a manifold of class \( {\mathrm{C}}^{r} \), e a point of \( \mathrm{X} \), \( \mathrm{U} \) and \( \mathrm{V} \) open neighbourhoods of e and \( m \) a mapping of class \( {\mathrm{C}}^{r} \) of \( \mathrm{U} \times \mathrm{U} \) into \( \mathrm{X} \) satisfying the following conditions... | \n\( m\left( {e, y}\right) = y \) for all \( y \in \mathrm{U} \) and hence, by the implicit function theorem, there exists an open neighbourhood \( {\mathrm{W}}_{1} \) of \( e \) in \( \mathrm{V} \) and a mapping \( {\theta }_{1} \) of class \( {\mathrm{C}}^{r} \) of \( {\mathrm{W}}_{1} \) into \( \mathrm{V} \) such th... | Yes |
Proposition 3. Let \( \mathrm{X} \) be an analytic manifold and \( m \) an analytic associative law of composition on \( \mathrm{X} \) admitting an identity element. The set \( \mathrm{G} \) of invertible elements of \( \mathrm{X} \) is open in \( \mathrm{X} \) and \( \mathrm{G} \) is a Lie group with \( m \mid \left( ... | By Lemma 2, \( \mathrm{G} \) is a neighbourhood of the identity element. For all \( g \in \mathrm{G} \) , the mapping \( x \mapsto m\left( {g, x}\right) \) is an automorphism of the manifold \( \mathrm{X} \) . Hence the image of \( \mathrm{G} \) under this mapping is a neighbourhood of \( g \), obviously contained in G... | Yes |
Proposition 4. Let \( \mathrm{G} \) and \( \mathrm{H} \) be Lie groups and \( f \) a homomorphism of the group \( \mathrm{G} \) into the group \( \mathrm{H} \) . For \( f \) to be analytic, it is necessary and sufficient that there exist a nonempty open subset \( \mathbf{U} \) of \( \mathbf{G} \) such that \( f \mid \m... | The condition is obviously necessary. Suppose that it holds. For all \( {x}_{0} \in \mathrm{G} \) , \( f\left( {{x}_{0}x}\right) = f\left( {x}_{0}\right) f\left( x\right) \) for all \( x \in \mathrm{U} \) and hence \( f \mid {x}_{0}\mathrm{U} \) is analytic. But the sets \( {x}_{0}\mathrm{U} \), where \( {x}_{0} \in \m... | Yes |
Proposition 5. Let \( \mathrm{G} \) be a Lie group and \( \mathrm{H} \) a Lie subgroup of \( \mathrm{G} \). (i) \( \mathrm{H} \) is closed in \( \mathrm{G} \). (ii) The canonical injection of \( \mathrm{H} \) into \( \mathrm{G} \) is a Lie group morphism. (iii) Let \( \mathrm{L} \) be a Lie group and fa mapping of \( \... | By Differentiable and Analytic Manifolds, R, 5.8.3, H is locally closed. Hence H is closed (General Topology, Chapter III, § 2, Proposition 4). Assertion (ii) is obvious. Assertion (iii) follows from Differentiable and Analytic Manifolds, R, 5.8.5. | No |
Proposition 6. Let \( \mathrm{G} \) be a Lie group and \( \mathrm{H} \) a subgroup of \( \mathrm{G} \). For \( \mathrm{H} \) to be a Lie subgroup of \( \mathrm{G} \), it is necessary and sufficient that there exist a point \( h \in \mathrm{H} \) and an open neighbourhood \( \mathrm{U} \) of \( h \) in \( \mathrm{G} \) ... | The condition is obviously necessary. Suppose that it holds. For all \( {h}^{\prime } \in \mathrm{H} \) , the translation \( \gamma \left( {{h}^{\prime }{h}^{-1}}\right) \) is an automorphism of the manifold \( \mathrm{G} \) and maps the submanifold \( \mathrm{H} \cap \mathrm{U} \) of \( \mathrm{U} \) into the submanif... | Yes |
Proposition 7. If the mapping \( \left( {m, l}\right) \mapsto \sigma \left( l\right) m \) of \( \mathbf{M} \times \mathbf{L} \) into \( \mathbf{M} \) is analytic, the group \( \mathrm{S} \), with the product manifold structure of \( \mathrm{M} \) and \( \mathrm{L} \), is a Lie group. | For \( l,{l}^{\prime } \) in \( \mathrm{L} \) and \( m,{m}^{\prime } \) in \( \mathrm{M} \) ,\n\n\[ \left( {m, l}\right) {\left( {m}^{\prime },{l}^{\prime }\right) }^{-1} = {ml}{l}^{-1}{m}^{\prime - 1} = m\left( {\sigma \left( {l{l}^{\prime - 1}}\right) {m}^{\prime - 1}}\right) l{l}^{\prime - 1} \]\n\n\[ = \left( {m\le... | Yes |
Proposition 8. Let \( \\mathrm{G} \) and \( \\mathrm{H} \) be Lie groups, \( p : \\mathrm{G} \\rightarrow \\mathrm{H} \) and \( s : \\mathrm{H} \\rightarrow \\mathrm{G} \) Lie group morphisms such that \( p \\circ s = {\\mathrm{{id}}}_{\\mathrm{H}} \) and \( \\mathrm{N} = \\operatorname{Ker}p \) . Then \( \\mathrm{N} \... | \n\( {\\mathrm{T}}_{e}\\left( p\\right) \\circ {\\mathrm{T}}_{e}\\left( s\\right) = {\\mathrm{{id}}}_{{\\mathrm{T}}_{e}\\left( \\mathrm{H}\\right) } \) and hence \( p \) (resp. \( s \) ) is a submersion (resp. an immersion). By Differentiable and Analytic Manifolds, R, 5.10.5, N is a Lie subgroup of G. On the other han... | Yes |
Proposition 9. Let \( x \in \mathrm{X} \) and \( {g}_{0} \in \mathrm{G} \) . (i) If \( \rho \left( x\right) \) is an immersion (resp. a submersion, a subimmersion) at \( {g}_{0} \), then, for all \( g \in \mathrm{G},\mathrm{p}\left( {gx}\right) \) is an immersion (resp. a submersion, a subimmersion). (ii) If \( \rho \l... | This follows immediately from formulae (4) and (5) since \( {\mathrm{T}}_{g}\left( {\delta \left( g\right) }\right) ,{\mathrm{T}}_{x}\left( {\tau \left( g\right) }\right) \) and \( {\mathrm{T}}_{g}\left( {\gamma \left( {g}^{-1}\right) }\right) \) are isomorphisms. | Yes |
Proposition 10. Let \( \mathrm{G} \) be a Lie group and \( \mathrm{X} \) a manifold of class \( {\mathrm{C}}^{r} \) with a law of left operation of class \( {\mathrm{C}}^{r} \) of \( \mathrm{G} \) on \( \mathrm{X} \) . Suppose that:\n\n(a) the group \( \mathrm{G} \) operates properly and freely on \( \mathrm{X} \) ;\n\... | Let \( \theta \) be the mapping \( \left( {g, x}\right) \mapsto \left( {x,{gx}}\right) \) of \( \mathrm{G} \times \mathrm{X} \) into \( \mathrm{X} \times \mathrm{X} \) . This mapping is of class \( {\mathrm{C}}^{r} \) . We show that it is an immersion. For \( u \in {\mathrm{T}}_{g}\left( \mathrm{G}\right) \) and \( v \... | Yes |
Proposition 11. Let \( \mathrm{X} \) be a Lie group and \( \mathrm{G} \) a Lie subgroup of \( \mathrm{X} \). (i) There exists on the homogeneous set \( \mathrm{X}/\mathrm{G} \) one and only one analytic manifold structure such that the canonical projection \( \pi \) of \( \mathrm{X} \) onto \( \mathrm{X}/\mathrm{G} \) ... | By General Topology, Chapter III, § 4, no. 1, Example 1, G operates properly and freely on \( \mathrm{X} \) by right translation. Hence the first assertion of (i) follows from Proposition 10 of no. 5. The second follows from the Remark of no. 5. Since \( \pi \) is a submersion, the kernel of \( {\mathrm{T}}_{x}\left( \... | Yes |
Proposition 12. Let \( \mathrm{X} \) be a Lie group and \( \mathrm{Y} \) a non-empty analytic manifold with a law of analytic left operation of \( \mathrm{X} \) on \( \mathrm{Y} \) . For all \( y \in \mathrm{Y} \), let \( \mathrm{p}\left( y\right) \) be the orbital mapping by \( y \) and \( {\mathrm{X}}_{y} \) the stab... | As the canonical mapping of \( \mathrm{X} \) onto \( \mathrm{X}/{\mathrm{X}}_{y} \) is a submersion, the equivalences (i) \( \Leftrightarrow \) (ii), \( \left( {\mathrm{i}}^{\prime }\right) \Leftrightarrow \left( {\mathrm{{ii}}}^{\prime }\right) \) are immediate. (i) \( \Leftrightarrow \left( {\mathrm{i}}^{\prime }\rig... | Yes |
Proposition 13. Let \( \mathrm{G} \) be a Lie group, \( \mathrm{H} \) a normal Lie subgroup of \( \mathrm{G},\mathrm{X} \) a manifold of class \( {\mathrm{C}}^{r} \) and \( \left( {g, x}\right) \mapsto {gx} \) a law of left operation of class \( {\mathrm{C}}^{r} \) of \( \mathrm{G} \) on \( \mathrm{X} \) . Suppose that... | Clearly \( \mathrm{H} \) operates freely on \( \mathrm{X} \) ; it operates properly by General Topology, Chapter III, \( §4 \), no. 1, Example 1. The orbital mappings of \( \mathrm{H} \) on \( \mathrm{X} \) are immersions since the canonical injection of \( \mathrm{H} \) into \( \mathrm{G} \) is an immersion. This prov... | Yes |
Proposition 14. Let \( \mathrm{G} \) be a Lie group, \( \mathrm{X} \) an analytic manifold and \( \left( {g, x}\right) \mapsto {gx} \) a law of analytic left operation of \( \mathrm{G} \) on \( \mathrm{X} \) . Let \( x \in \mathrm{X} \) . Suppose that the corresponding orbital mapping \( \wp \left( x\right) \) is a sub... | The inverse image of \( x \) under \( \rho \left( x\right) \) is \( {\mathrm{G}}_{x} \) . As \( \rho \left( x\right) \) is a subimmersion, \( {\mathrm{G}}_{x} \) is a submanifold and, for all \( g \in \mathrm{G} \), the tangent space \( \mathrm{J} \) to \( g{\mathrm{G}}_{x} = {\rho }_{\left( x\right) }^{-1}\left( {gx}\... | Yes |
Proposition 16. Let \( \mathrm{G} \) be a Lie group, \( \mathrm{X} \) a manifold of class \( {\mathrm{C}}^{r}\left( {r \geq 2}\right) \) and \( \left( {g, x}\right) \mapsto {gx} \) a law of left operation of class \( {\mathrm{C}}^{r} \) of \( \mathrm{G} \) on \( \mathrm{X} \), whence, by transporting the structure, the... | Let \( {\mathrm{{pr}}}_{1} \) (resp. \( {\mathrm{{pr}}}_{2} \) ) be the canonical projection of \( \mathrm{G} \times \mathrm{X} \) onto \( \mathrm{G} \) (resp. \( \mathrm{X} \) ) and let \( {\mathrm{E}}_{1} \) (resp. \( {\mathrm{E}}_{2} \) ) be the inverse image of TG (resp. TX) relative to \( {\mathrm{{pr}}}_{1} \) (r... | Yes |
Proposition 17. Let \( \mathrm{G} \) be a Lie group, \( \mathrm{X} \) a left Lie homogeneous space of \( \mathrm{G},{x}_{0} \) a point of \( \mathrm{X},{\mathrm{G}}_{0} \) the stabilizer of \( {x}_{0} \) in \( \mathrm{G},\mathrm{E} \) and \( {\mathrm{E}}^{\prime } \) left vector \( \mathrm{G} \) -bundles of class \( {\... | The uniqueness of this morphism is obvious. We prove its existence. Let \( g \) , \( {g}^{\prime } \) elements of \( \mathrm{G} \) and \( u \in {\mathrm{E}}_{0} \) be such that \( {gu} = {g}^{\prime }u \) . Then \( {g}^{\prime - 1}g \in {\mathrm{G}}_{0} \) and \( {g}^{\prime - 1}{gu} = u \) and hence \( {g}^{\prime - 1... | Yes |
For all \( u \in {\mathrm{E}}_{0}^{{\mathrm{G}}_{0}} \), let \( {\sigma }_{u} \) be the mapping of \( \mathrm{X} \) into \( \mathrm{E} \) defined by \( {\sigma }_{u}\left( {g{x}_{0}}\right) = {gu} \) for all \( g \in \mathrm{G} \). (i) The G-invariant sections \( \dagger \) of \( \mathrm{E} \) are of class \( {\mathrm{... | To prove (i) it is sufficient to prove that each section \( {\sigma }_{u} \) is of class \( {\mathrm{C}}^{r} \) . Let \( {\mathrm{E}}^{\prime } \) be the trivial \( \mathrm{G} \) -bundle of base \( \mathrm{X} \) and fibre \( {\mathrm{E}}_{0}^{{\mathrm{G}}_{0}} \) . Let \( f \) be the canonical injection of \( {\mathrm{... | Yes |
Proposition 18. Let \( \mathrm{G} \) be a group and \( \mathrm{U} \) and \( \mathrm{V} \) two subsets of \( \mathrm{G} \) containing e. Suppose that \( \mathbf{U} \) has an analytic manifold structure satisfying the following conditions:\n\n(i) \( \mathrm{V} = {\mathrm{V}}^{-1},{\mathrm{\;V}}^{2} \subset \mathrm{U},\ma... | (a) Let \( \mathrm{A} \) be an open subset of \( \mathrm{V} \) and \( {v}_{0} \) an element of \( \mathrm{V} \) such that \( {v}_{0}\mathrm{\;A} \subset \mathrm{V} \) . Then \( {v}_{0}\mathrm{\;A} \) is the set of \( v \in \mathrm{V} \) such that \( {v}_{0}^{-1}v \in \mathrm{A} \) and hence is an open subset of \( V \)... | Yes |
Proposition 20. Let \( \mathrm{G} \) be a Lie group germ and \( g \in \mathrm{G} \). There exist an open neighbourhood \( \mathrm{U} \) of \( e \) and an open neighbourhood \( \mathrm{V} \) of \( g \) with the following properties:\n\n(a) ug is defined for all \( u \in \mathrm{U} \);\n\n(b) \( v{g}^{-1} \) is defined f... | As the set of definition of the product is open in \( \mathrm{G} \times \mathrm{G} \), there exist an open neighbourhood \( \mathrm{U} \) of \( e \) and an open neighbourhood \( \mathrm{V} \) of \( g \) with properties (a) and (b). Let \( \eta \left( u\right) = {ug} \) for \( u \in \mathrm{U},{\eta }^{\prime }\left( v\... | Yes |
Proposition 21. Let \( \mathrm{G},\mathrm{H} \) be two Lie group germs and \( \phi \) a morphism of \( \mathrm{G} \) into \( \mathrm{H} \) . The following conditions are equivalent:\n\n(i) \( \phi \) is étale at \( e \) ;\n\n(ii) there exist open Lie subgroup germs \( {\mathrm{G}}^{\prime },{\mathrm{H}}^{\prime } \) of... | The implication (ii) \( \Rightarrow \) (i) is obvious. Suppose that \( \phi \) is étale at \( e \) . There exists an open Lie subgroup gcrm \( {\mathrm{G}}_{1} \) of \( \mathrm{G} \) such that \( \phi \left( {\mathrm{G}}_{1}\right) \) is open in \( \mathrm{H} \) and \( \phi \mid {\mathrm{G}}_{1} \) is an isomorphism of... | Yes |
Proposition 22. Let \( \mathrm{H} \) be a Lie group, \( \mathrm{U} \) a Lie subgroup germ of \( \mathrm{H} \) and \( \mathrm{N} \) the set of \( g \in \mathrm{H} \) such that \( \mathrm{U} \) and \( g\mathrm{U}{g}^{-1} \) have the same germ at \( e \) (General Topology, Chapter I, \( §6 \), no. 10). Then \( N \) is a s... | Clearly \( \mathrm{N} \) is a subgroup of \( \mathrm{H} \) . If \( g \in \mathrm{U} \), then \( {ge} \in \mathrm{U} \) and \( {ge}{g}^{-1} \in \mathrm{U} \), hence \( {gu} \in \mathrm{U} \) and \( {gu}{g}^{-1} \in \mathrm{U} \) for \( u \) sufficiently close to \( e \) in \( \mathrm{U} \) and hence the germ of \( g\mat... | Yes |
Proposition 23. Let \( \\left( {\\mathrm{G}, e,\\theta, m}\\right) \) be a Lie group germ, \( \\mathrm{X} \) a manifold of class \( {\\mathrm{C}}^{r} \) , \( {x}_{0} \) a point of \( \\mathrm{X},\\Omega \) an open neighbourhood of \( \\left( {e,{x}_{0}}\\right) \) in \( \\mathrm{G} \\times \\mathrm{X} \) and \( \\psi \... | There exist an open neighbourhood \( {\\mathrm{X}}^{\\prime } \) of \( {x}_{0} \) in \( \\mathrm{X} \) and an open neighbourhood \( {\\mathrm{G}}^{\\prime } \) of \( e \) in \( \\mathrm{G} \) such that \( \\psi \\left( {e, x}\\right) = x \) for all \( x \\in \\mathrm{X} \), and\n\n\[ \n\\psi \\left( {g,\\psi \\left( {{... | Yes |
Lemma 3. Let \( \mathrm{X} \) be a normal space and \( {\left( {\mathrm{X}}_{i}\right) }_{i \in \mathrm{I}} \) a locally finite open covering of \( \mathrm{X} \). For all \( \left( {i, j}\right) \in \mathrm{I} \times \mathrm{I} \) and all \( x \in {\mathrm{X}}_{i} \cap {\mathrm{X}}_{j} \), let \( {\mathrm{V}}_{ij}\left... | There exists an open covering \( {\left( {\mathrm{X}}_{i}^{\prime }\right) }_{i \in \mathrm{I}} \) of \( \mathrm{X} \) such that \( {\overline{\mathrm{X}}}_{i}^{\prime } \subset {\mathrm{X}}_{i} \) for all \( i \in \mathrm{I} \) (General Topology, Chapter IX,§ 4, Theorem 3). Let \( x \in \mathrm{X} \). Let \( {\mathrm{... | Yes |
Proposition 24. Let \( \mathrm{G} \) be a Lie group germ, \( \mathrm{X} \) a manifold of class \( {\mathrm{C}}^{r} \) and \( {\left( {\mathrm{X}}_{i}\right) }_{i \in \mathbf{I}} \) a locally finite open covering of \( \mathrm{X} \) . For all \( i \in \mathrm{I} \), let \( {\psi }_{i} \) be a law chunk of left operation... | For all \( \left( {i, j}\right) \in \mathrm{I} \times \mathrm{I} \) and all \( x \in {\mathrm{X}}_{i} \cap {\mathrm{X}}_{j} \) choose an open neighbourhood \( {\mathrm{V}}_{ij}\left( x\right) \) of \( x \) in \( {\mathrm{X}}_{i} \cap {\mathrm{X}}_{j} \) such that \( {\psi }_{i} \) and \( {\psi }_{j} \) are defined and ... | Yes |
Proposition 1. Let \( \mathrm{X} \) be a manifold of class \( {\mathrm{C}}^{r} \) and \( m \) a law of composition of class \( {\mathrm{C}}^{r} \) on \( \mathrm{X} \) . If \( m \) is associative (resp. commutative), then \( \mathrm{T}\left( m\right) \) is associative (resp. commutative). | If \( m \) is associative, then \( m \circ \left( {m \times {\operatorname{Id}}_{\mathrm{x}}}\right) = m \circ \left( {{\operatorname{Id}}_{\mathrm{x}} \times m}\right) \), whence\n\n\[ \mathrm{T}\left( m\right) \circ \left( {\mathrm{T}\left( m\right) \times {\mathrm{{Id}}}_{\mathrm{T}\left( \mathrm{X}\right) }}\right)... | Yes |
Proposition 2. Let \( \mathrm{X} \) be a manifold of class \( {\mathrm{C}}^{r}, m \) a law of composition of class \( {\mathrm{C}}^{r} \) on \( \mathrm{X} \) and e an identity element for \( m \) . (i) The vector \( {0}_{e} \) is an identity element for \( \mathrm{T}\left( m\right) \) . | Properties (3) and (4) show that \( \mathrm{T}\left( m\right) \left( {{0}_{e}, u}\right) = \mathrm{T}\left( m\right) \left( {u,{0}_{e}}\right) = u \) for all \( u \in \mathrm{T}\left( \mathrm{X}\right) \), whence (i). | Yes |
Proposition 3. Let \( {\mathrm{X}}_{1},{\mathrm{X}}_{2},\ldots ,{\mathrm{X}}_{p},\mathrm{Y} \) be manifolds of class \( {\mathrm{C}}^{r} \), i an integer of \( \left( {1, p}\right) ,{m}_{i} \) (resp. \( n \) ) a law of composition of class \( {\mathrm{C}}^{r} \) on \( {\mathrm{X}}_{i} \) (resp. Y) and \( u \) a mapping... | The proof is analogous to that of Proposition 1. | No |
Proposition 4. Let \( \mathrm{G} \) be a Lie group. Then \( \mathrm{T}\left( \mathrm{G}\right) \), with the law of composition tangent to the multiplication of \( \mathrm{G} \), is a Lie group. The identity element of \( \mathrm{T}\left( \mathrm{G}\right) \) is the vector \( {0}_{e} \) . | This follows from Propositions 1 and 2. | No |
Proposition 5. Let \( \mathrm{G} \) and \( \mathrm{H} \) be Lie groups and \( f \) a morphism of \( \mathrm{G} \) into \( \mathrm{H} \) . Then \( \mathrm{T}\left( f\right) \) is a morphism of the Lie group \( \mathrm{T}\left( \mathrm{G}\right) \) into the Lie group \( \mathrm{T}\left( \mathrm{H}\right) \) . | We know that \( \mathrm{T}\left( f\right) \) is analytic. On the other hand, let \( m \) (resp. \( n \) ) denote the multiplication on \( \mathrm{G} \) (resp. \( \mathrm{H} \) ). Then \( f \circ m = n \circ \left( {f \times f}\right) \), whence\n\n\[ \mathrm{T}\left( f\right) \circ \mathrm{T}\left( m\right) = \mathrm{T... | Yes |
Proposition 6. Let \( \mathrm{G} \) be a Lie group.\n\n(i) The canonical projection \( p : \mathrm{T}\left( \mathrm{G}\right) \rightarrow \mathrm{G} \) is a Lie group morphism. | Assertion (i) follows from (5). | No |
Proposition 7. Let \( \mathrm{G} \) be a Lie group, \( \mathrm{M} \) a manifold of class \( {\mathrm{C}}^{r} \) and \( f \) and \( g \) mappings of class \( {\mathrm{C}}^{r} \) of \( \mathrm{M} \) into \( \mathrm{G} \), so that \( {fg} \) is a mapping of class \( {\mathrm{C}}^{r} \) of \( \mathrm{M} \) into \( \mathrm{... | \[ \left( {\mathrm{T}{fg}}\right) u = \mathrm{T}\left( f\right) u \cdot y + x \cdot \mathrm{T}\left( g\right) u. \] Let \( m \) be the multiplication of \( \mathrm{G} \). Then \( {fg} = m \circ \left( {f, g}\right) \). Now \[ \mathrm{T}\left( {f, g}\right) \left( u\right) = \left( {\mathrm{T}\left( f\right) u,\mathrm{\... | Yes |
Proposition 8. Let \( {\mathrm{G}}_{1} \) and \( {\mathrm{G}}_{2} \) be Lie groups, \( {\mathrm{X}}_{1} \) and \( {\mathrm{X}}_{2} \) manifolds of class \( {\mathrm{C}}^{r} \) and \( {f}_{i} \) a law of left operation of class \( {\mathrm{C}}^{r} \) of \( {\mathrm{G}}_{i} \) on \( {\mathrm{X}}_{i}\left( {i = 1,2}\right... | \( {f}_{2} \circ \left( {\phi \times \psi }\right) = \psi \circ {f}_{1} \), whence\n\n\[ \mathrm{T}\left( {f}_{2}\right) \circ \left( {\mathrm{T}\left( \phi \right) \times \mathrm{T}\left( \psi \right) }\right) = \mathrm{T}\left( \psi \right) \circ \mathrm{T}\left( {f}_{1}\right) \] | Yes |
Proposition 2. The algebra \( {\mathcal{T}}^{\left( \infty \right) }\left( \mathrm{G}\right) \) is associative. It is commutative if and only if \( \mathrm{G} \) is commutative. | Let \( t \in {\mathcal{T}}^{\left( \infty \right) }\left( \mathrm{G}\right) ,{t}^{\prime } \in {\mathcal{T}}^{\left( \infty \right) }\left( \mathrm{G}\right) ,{t}^{\prime \prime } \in {\mathcal{T}}^{\left( \infty \right) }\left( \mathrm{G}\right) \) . Then \( t * \left( {{t}^{\prime } * {t}^{\prime \prime }}\right) \) ... | Yes |
Proposition 3. If \( t \in {\mathcal{T}}^{\left( \infty \right) }\left( \mathrm{G}\right) \) and \( g \in \mathrm{G} \), then \( \gamma {\left( g\right) }_{ * }t = {\varepsilon }_{g} * t,\;\delta {\left( g\right) }_{ * }t = t * {\varepsilon }_{{g}^{-1}} \) , \( {\left( \operatorname{Int}g\right) }_{ * }t = {\varepsilon... | Consider the diagram\n\n\[\n\mathrm{G}\overset{\Phi }{ \rightarrow }\mathrm{G} \times \mathrm{G}\overset{\Psi }{ \rightarrow }\mathrm{G}\n\]\n\nwhere \( \phi \) is the mapping \( h \mapsto \left( {g, h}\right) \) and \( \psi \) is the mapping \( \left( {{h}^{\prime }, h}\right) \mapsto {h}^{\prime }h \) . Then \( \gamm... | Yes |
Proposition 6. Let \( \mathrm{G} \) , \( \mathrm{H} \) be Lie groups and \( \mathrm{\phi } \) a morphism of \( \mathrm{G} \) into \( \mathrm{H} \) . For \( t,{t}^{\prime } \) in \( {\mathcal{T}}^{\left( \infty \right) }\left( \mathrm{G}\right) ,{\phi }_{ * }\left( {t * {t}^{\prime }}\right) = {\phi }_{ * }\left( t\righ... | Consider the diagram\n\n\n\nwhere \( m\left( {g,{g}^{\prime }}\right) = g{g}^{\prime }, n\left( {h,{h}^{\prime }}\right) = h{h}^{\prime } \) . This diagram is commutative. Hence\n\n\[ \n{\phi }_{ * }\left( {t * {t}^{... | Yes |
Proposition 7. Let \( \mathrm{G} \) be a Lie group. Let \( t,{t}^{\prime } \) be in \( {\mathcal{T}}^{\left( \infty \right) }\left( \mathrm{G}\right) \) . (i) The product \( t * {t}^{\prime } \) calculated relative to \( {\mathrm{G}}^{ \vee } \) is equal to the product \( {t}^{\prime } * t \) calculated relative to \( ... | Consider the diagram where \( s\left( {g,{g}^{\prime }}\right) = \left( {{g}^{\prime }, g}\right), m\left( {g,{g}^{\prime }}\right) = g{g}^{\prime }, n\left( {g,{g}^{\prime }}\right) = {g}^{\prime }g \) for all \( g,{g}^{\prime } \) in G. This diagram is commutative. Hence \( {n}_{ * }\left( {t \otimes {t}^{\prime }}\r... | Yes |
Proposition 8. Let \( \mathrm{G},\mathrm{H} \) be Lie groups and \( \phi \) a morphism of \( \mathrm{G} \) into \( \mathrm{H} \) . If \( t \in {\mathcal{T}}^{\left( \infty \right) }\left( \mathrm{G}\right) \) , then \( {\phi }_{ * }\left( {t}^{ \vee }\right) = {\left( {\phi }_{ * }\left( t\right) \right) }^{ \vee } \) ... | Let \( \theta \) (resp. \( {\theta }^{\prime } \) ) be the mapping \( g \mapsto {g}^{-1} \) of \( \mathrm{G} \) into \( \mathrm{G} \) (resp. of \( \mathrm{H} \) into \( \mathrm{H} \) ). Then \( \phi \circ \theta = {\theta }^{\prime } \circ \phi \), whence \( {\phi }_{ * }\left( {{\theta }_{ * }\left( t\right) }\right) ... | Yes |
Proposition 9. Let \( {\mathrm{G}}_{1},\cdots ,{\mathrm{G}}_{n} \) be Lie groups and \( \mathrm{G} = {\mathrm{G}}_{1} \times \cdots \times {\mathrm{G}}_{n} \) . If the vector spaces \( {\mathcal{T}}^{\left( \infty \right) }\left( \mathrm{G}\right) \) and \( {\mathcal{T}}^{\left( \infty \right) }\left( {\mathrm{G}}_{1}\... | \[ {\left( {t}_{1} \otimes \cdots \otimes {t}_{n}\right) }^{ \vee } = {t}_{1}^{ \vee } \otimes \cdots \otimes {t}_{n}^{ \vee }.\]\n\nIt suffices to consider the case \( n = 2 \) . Let \( {t}_{1},{t}_{1}^{\prime } \) be in \( {\mathcal{T}}^{\left( \infty \right) }\left( {\mathrm{G}}_{1}\right) ,{t}_{2},{t}_{2}^{\prime }... | Yes |
Proposition 10. Let \( \mathrm{H} \) be a Lie subgroup of \( \mathrm{G} \) and \( i : \mathrm{H} \rightarrow \mathrm{G} \) the canonical injection. Then \( {i}_{ * } \) is an injective homomorphism of the algebra \( {\mathcal{T}}^{\left( \infty \right) }\left( \mathrm{H}\right) \) into the algebra \( {\mathcal{T}}^{\le... | This follows from Propositions 6 and 8 and Differentiable and Analytic Manifolds, R, 13.2.3. | No |
Proposition 11. Let \( \mathrm{G} \) be a Lie group.\n\n(i) The cogebra \( {\mathcal{T}}^{\left( \infty \right) }\left( \mathrm{G}\right) \), with convolution, is a bigebra (Algebra, Chapter III, § 11, no. 4). | We prove (i). In the definition of bigebras referred to, condition (1) follows from Propositions 2 and 3 and condition (2) follows from Differentiable and Analytic Manifolds, R,13.5.1. Let \( d \) be the mapping \( g \mapsto \left( {g, g}\right) \) of \( \mathrm{G} \) into \( \mathrm{G} \times \mathrm{G} \) . Then \( c... | Yes |
Proposition 12. Let \( \mathrm{G} \) , \( \mathrm{H} \) be two Lie groups and \( \phi \) a morphism of \( \mathrm{G} \) into \( \mathrm{H} \) . Then \( {\phi }_{ * } \) is a bigebra morphism of \( {\mathcal{T}}^{\left( \infty \right) }\left( \mathrm{G}\right) \) into \( {\mathcal{T}}^{\left( \infty \right) }\left( \mat... | This follows from Proposition 6 and Differentiable and Analytic Manifolds, R, 13.5.1. | No |
Proposition 13. Let \( t \in {\mathcal{T}}^{\left( s\right) }\left( \mathrm{G}\right) ,{t}^{\prime } \in {\mathcal{T}}^{\left( {s}^{\prime }\right) }\left( \mathrm{G}\right), u \in {\mathcal{T}}^{\left( {s}^{\prime \prime }\right) }\left( \mathrm{X}\right) \), such that\n\n\[ s + {s}^{\prime } + {s}^{\prime \prime } \l... | This can be proved as is Proposition 2 of no. 1. | No |
Proposition 14. (i) Let \( {g}_{0} \in \mathrm{G} \) and \( \tau \left( {g}_{0}\right) \) be the mapping \( x \mapsto f\left( {{g}_{0}, x}\right) \) of \( \mathrm{X} \) into \( \mathrm{X} \) . If \( u \in {\mathcal{T}}^{\left( r\right) }\left( \mathrm{X}\right) \), then \( \tau {\left( {g}_{0}\right) }_{ * }u = {\varep... | This can be proved as is Proposition 3 of no. 1. | No |
Proposition 15. Let \( \mathrm{G} \) (resp. \( {\mathrm{G}}^{\prime } \) ) be a Lie group and \( \mathrm{X} \) (resp. \( {\mathrm{X}}^{\prime } \) ) a manifold of class \( {\mathrm{C}}^{r} \) . Suppose that a law of left operation of class \( {\mathrm{C}}^{r} \) of \( \mathrm{G} \) (resp. \( {\mathrm{G}}^{\prime } \) )... | This can be proved as is Proposition 6 of no. 2. | No |
Proposition 17. Let \( t \in {\mathcal{T}}^{\left( \mathrm{s}\right) }\left( \mathrm{G}\right) ,{t}^{\prime } \in {\mathcal{T}}^{\left( {\mathrm{s}}^{\prime }\right) }\left( \mathrm{X}\right) \) and \( f : X \rightarrow \mathrm{F} \) a function of class \( {\mathrm{C}}^{r} \) with \( s + {s}^{\prime } \leq r \) . Then\... | \n\n\[ \left\langle {{t}^{\prime }, t * f}\right\rangle = \left\langle {{t}^{\prime }, x \mapsto \left\langle {t, g \mapsto f\left( {{g}^{-1}x}\right) }\right\rangle }\right\rangle \;\text{by (4)} \]\n\n\[ = \left\langle {t \otimes {t}^{\prime },\left( {g, x}\right) \mapsto f\left( {{g}^{-1}x}\right) }\right\rangle \;\... | Yes |
Proposition 18. Let \( t \in {\mathcal{T}}^{\left( \mathrm{s}\right) }\left( \mathrm{G}\right) ,{t}^{\prime } \in {\mathcal{T}}^{\left( {\mathrm{s}}^{\prime }\right) }\left( \mathrm{G}\right) \) and \( f : \mathrm{X} \rightarrow \mathrm{F} \) a function of class \( {\mathrm{C}}^{r} \), with \( s + {s}^{\prime } \leq r ... | For all \( x \in \mathrm{X} \) ,\n\n\[ \langle {\varepsilon }_{x},\left( {t * {t}^{\prime }}\right) * f\rangle = \langle {\left( t * {t}^{\prime }\right) }^{ \vee } * {\varepsilon }_{x}, f\rangle \;\text{by (5)} \]\n\n\[ = \langle {{t}^{\prime }}^{ \vee } * \left( {{t}^{ \vee } * {\varepsilon }_{x}}\right), f\rangle \;... | Yes |
Proposition 19. Let \( t \in {\mathcal{T}}^{\left( s\right) }\left( \mathrm{G}\right) \), with \( s \leq r \). Let \( f \) (resp. \( {f}^{\prime } \)) be a function of class \( {\mathrm{C}}^{r} \) on \( \mathrm{X} \) with values in a Hausdorff polynormed space \( \mathrm{F} \) (resp. \( {\mathrm{F}}^{\prime } \)). Let ... | Let \( x \in \mathrm{X} \) and let \( \rho \left( x\right) \) denote the orbital mapping of \( x \). Then \[ \left\langle {{\varepsilon }_{x}, t * \left( {f{f}^{\prime }}\right) }\right\rangle = \left\langle {{t}^{ \vee },\left( {f{f}^{\prime }}\right) \circ \rho \left( x\right) }\right\rangle \] by (4) \[ = \left\lang... | Yes |
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