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Lemma 1. Let \( \mathrm{V} \) be a vector subspace of \( \mathfrak{g} \) and \( \mathrm{R}\left( \mathrm{V}\right) \) the set of \( \alpha \in \mathrm{R} \) such that \( {\mathfrak{g}}^{\alpha } \subset \mathrm{V} \) . Then, \( \left( {\mathrm{V} \cap \mathfrak{h}}\right) + \mathop{\sum }\limits_{{\alpha \in \mathrm{R}... | A vector subspace \( \mathrm{W} \) of \( \mathrm{V} \) is stable under ad \( \mathfrak{h} \) if and only if\n\n\[ \mathrm{W} = \left( {\mathrm{W} \cap \mathfrak{h}}\right) + \mathop{\sum }\limits_{{\alpha \in \mathrm{R}}}\left( {\mathrm{W} \cap {\mathfrak{g}}^{\alpha }}\right) \]\n\n(Algebra, Chap. VII, §2, no. 2, Cor.... | Yes |
Lemma 2. Let \( {\mathfrak{h}}^{\prime } \) be a vector subspace of \( \mathfrak{h} \) and \( \mathrm{P} \) a subset of \( \mathrm{R} \) . Then \( {\mathfrak{h}}^{\prime } + {\mathfrak{g}}^{\mathrm{P}} \) is a subalgebra of \( \mathfrak{g} \) if and only if \( \mathrm{P} \) is a closed subset of \( \mathrm{R} \) and \[... | Indeed, \[ \left\lbrack {{\mathfrak{h}}^{\prime } + {\mathfrak{g}}^{\mathrm{P}},{\mathfrak{h}}^{\prime } + {\mathfrak{g}}^{\mathrm{P}}}\right\rbrack = \left\lbrack {{\mathfrak{h}}^{\prime },{\mathfrak{g}}^{\mathrm{P}}}\right\rbrack + \left\lbrack {{\mathfrak{g}}^{\mathrm{P}},{\mathfrak{g}}^{\mathrm{P}}}\right\rbrack = ... | Yes |
Then \( \mathrm{F} \mapsto \mathrm{P}\left( \mathrm{F}\right) \) is a bijection from \( \mathcal{F} \) to \( \mathcal{P} \) ; for all \( \mathrm{F} \in \mathcal{F},\overline{\mathrm{F}}\left( {\mathrm{P}\left( \mathrm{F}\right) }\right) \) is the closure of \( \mathrm{F} \) . | a) Let \( \mathrm{P} \in \mathcal{P} \) . There exists a chamber \( \mathrm{C} \) of \( \mathrm{S} \) and a subset \( \sum \) of the basis \( \mathrm{B}\left( \mathrm{C}\right) \) such that \( \mathrm{P} = {\mathrm{S}}_{ + }\left( \mathrm{C}\right) \cup \mathrm{Q} \) where \( \mathrm{Q} \) is the set of linear combinat... | Yes |
Lemma 1. For all \( \alpha ,\beta \in \mathrm{B} \), we have\n\n\[ \left\lbrack {{X}_{\alpha }^{0},{X}_{-\alpha }^{0}}\right\rbrack = - {H}_{\alpha }^{0} \]\n\n(10)\n\n\[ \left\lbrack {{H}_{\alpha }^{0},{H}_{\beta }^{0}}\right\rbrack = 0 \]\n\n(11)\n\n\[ \left\lbrack {{H}_{\alpha }^{0},{X}_{\beta }^{0}}\right\rbrack = ... | Indeed, relation (9) can be written\n\n\[ \left( {{X}_{\alpha }^{0}{X}_{-{\alpha }_{1}}^{0}}\right) \left( {{\alpha }_{2},\ldots ,{\alpha }_{n}}\right) = \left( {{X}_{-{\alpha }_{1}}^{0}{X}_{\alpha }^{0}}\right) \left( {{\alpha }_{2},\ldots ,{\alpha }_{n}}\right) - {\delta }_{\alpha ,{\alpha }_{1}}{H}_{\alpha }^{0}\lef... | Yes |
Lemma 4. There exists a unique involutive automorphism \( \theta \) of \( \mathfrak{a} \) such that\n\n\[ \theta \left( {x}_{\alpha }\right) = {x}_{-\alpha },\;\theta \left( {x}_{-\alpha }\right) = {x}_{\alpha },\;\theta \left( {h}_{\alpha }\right) = - {h}_{\alpha } \]\n\nfor all \( \alpha \in \mathrm{B} \) . | Indeed, there exists an involutive automorphism of the free Lie algebra \( \mathrm{L}\left( \mathrm{I}\right) \) satisfying these conditions. It leaves \( \mathcal{R} \cup \left( {-\mathcal{R}}\right) \) stable, and hence defines by passage to the quotient an involutive automorphism of \( \mathfrak{a} \) satisfying the... | Yes |
Lemma 5. Let \( z \in \mathfrak{a} \) . Then \( z \in {\mathfrak{a}}^{\mu } \) if and only if \( \left\lbrack {{h}_{\alpha }, z}\right\rbrack = \left\langle {\mu ,{\alpha }^{ \vee }}\right\rangle z \) for all \( \alpha \in \mathrm{B} \) . | For \( \mu \in \mathrm{Q} \), let \( {\mathfrak{a}}^{\left( \mu \right) } \) be the set of \( x \in \mathfrak{a} \) such that \( \left\lbrack {{h}_{\alpha }, x}\right\rbrack = \left\langle {\mu ,{\alpha }^{ \vee }}\right\rangle x \) for all \( \alpha \in \mathrm{B} \) . The sum of the \( {\mathfrak{a}}^{\left( \mu \rig... | Yes |
Lemma 7. The ideal \( \mathfrak{n} \) of \( {\mathfrak{a}}_{ + } \) generated by the \( {x}_{\alpha \beta }\left( {\alpha ,\beta \in \mathrm{B},\alpha \neq \beta }\right) \) is an ideal of \( \mathfrak{a} \) . The ideal of \( {\mathfrak{a}}_{ - } \) generated by the \( {y}_{\alpha \beta }\left( {\alpha ,\beta \in \math... | Let \( {\mathfrak{n}}^{\prime } = \mathop{\sum }\limits_{{\alpha ,\beta \in \mathrm{B},\alpha \neq \beta }}k{x}_{\alpha \beta } \) . Since each \( {x}_{\alpha \beta } \) is homogeneous in \( \mathfrak{a},\left\lbrack {{\mathfrak{a}}^{0},{\mathfrak{n}}^{\prime }}\right\rbrack \subset {\mathfrak{n}}^{\prime } \) (Lemma 5... | Yes |
Lemma 8. Let \( \alpha \in \mathrm{B} \cup \left( {-\mathrm{B}}\right) \) . Then ad \( {X}_{\alpha } \) is locally nilpotent. | Assume that \( \alpha \in \mathrm{B} \) . Let \( {\mathfrak{g}}^{\prime } \) be the set of \( z \in \mathfrak{g} \) such that \( {\left( \operatorname{ad}{X}_{\alpha }\right) }^{p}z = 0 \) for sufficiently large \( p \) . Since ad \( {X}_{\alpha } \) is a derivation of \( \mathfrak{g},{\mathfrak{g}}^{\prime } \) is a s... | Yes |
Lemma 10. Let \( \mu \in \mathrm{Q} \), and assume that \( \mu \) is not a multiple of a root. There exists \( w \in \mathrm{W}\left( \mathrm{R}\right) \) such that certain of the coordinates of \( {w\mu } \) with respect to the basis \( \mathrm{B} \) are \( > 0 \) and certain of them are \( < 0 \) . | Let \( {\mathrm{V}}_{\mathbf{R}} \) be the vector space \( \mathrm{Q}{ \otimes }_{\mathbf{Z}}\mathbf{R} \), in which \( \mathrm{R} \) is a root system. By the assumption, there exists \( f \in {\mathrm{V}}_{\mathbf{R}}^{ * } \) such that \( \langle f,\alpha \rangle \neq 0 \) for all \( \alpha \in \mathrm{R} \), and \( ... | Yes |
Lemma 11. Let \( \mu \in \mathrm{Q} \). If \( \mu \notin \mathrm{R} \cup \{ 0\} \), then \( {\mathfrak{g}}^{\mu } = 0 \). If \( \mu \in \mathrm{R} \), then \( \dim {\mathfrak{g}}^{\mu } = 1 \). | 1) If \( \mu \) is not a multiple of an element of \( \mathrm{R} \), there exists \( w \in \mathrm{W} \) such that \( {w\mu } \notin {\mathrm{Q}}_{ + } \cup {\mathrm{Q}}_{ - } \) (Lemma 10), so \( {\mathfrak{a}}^{w\mu } = 0,{\mathfrak{g}}^{w\mu } = 0 \), and hence \( {\mathfrak{g}}^{\mu } = 0 \) (Lemma 9).\n\n2) Let \(... | Yes |
Lemma 1. Let \( \alpha \in \mathrm{R} \) and \( t \in {k}^{ * } \) . Let \( \varphi \) be the homomorphism \( \lambda \mapsto {t}^{\lambda \left( {H}_{\alpha }\right) } \) from \( \mathrm{Q}\left( \mathrm{R}\right) \) to \( {k}^{ * } \) . Then \( f\left( \varphi \right) = {\theta }_{\alpha }\left( t\right) {\theta }_{\... | Let \( \rho \) be the representation of \( \mathfrak{{sl}}\left( {2, k}\right) \) on \( \mathfrak{g} \) associated to \( {X}_{\alpha } \) . Let \( \pi \) be the representation of \( \mathbf{{SL}}\left( {2, k}\right) \) compatible with \( \rho \) . Introduce the notations \( \theta \left( t\right), h\left( t\right) \) o... | Yes |
Lemma 2. Let \( \mathfrak{h} \) be a splitting Cartan subalgebra of \( \mathfrak{g} \), and \( s \in {\operatorname{Aut}}_{0}\left( {\mathfrak{g},\mathfrak{h}}\right) \) . Assume that the restriction of \( s \) to \( \mathop{\sum }\limits_{{\alpha \in \mathrm{R}}}{\mathfrak{g}}^{\alpha } \) does not have 1 as an eigenv... | By extension of \( k \), we are reduced to the case where \( s \in {\operatorname{Aut}}_{e}\left( {\mathfrak{g},\mathfrak{h}}\right) \) . The dimension of the nilspace of \( s - 1 \) is at least \( \dim \mathfrak{h} \) (Chap. VII,§4, no. 4, Prop. 9). Hence \( \left( {s - 1}\right) \mid \mathfrak{h} \) is nilpotent. Sin... | Yes |
Lemma 3. (i) Let \( m = \\left( {\\mathrm{P}\\left( \\mathrm{R}\\right) : \\mathrm{Q}\\left( \\mathrm{R}\\right) }\\right) \) . If \( \\varphi \) is the \( m \) th power of an element of \( {\\mathrm{T}}_{\\mathrm{Q}} \), then \( \\varphi \\in q\\left( {\\;\\mathrm{T}}_{\\mathrm{P}}\\right) \) . | There exist a basis \( \\left( {{\\lambda }_{1},\\ldots ,{\\lambda }_{l}}\\right) \) of \( \\mathrm{P}\\left( \\mathrm{R}\\right) \) and integers \( {n}_{1} \\geq 1,\\ldots ,{n}_{l} \\geq 1 \) such that \( \\left( {{n}_{1}{\\lambda }_{1},\\ldots ,{n}_{l}{\\lambda }_{l}}\\right) \) is a basis of \( \\mathrm{Q}\\left( \\... | Yes |
Lemma 1. Let \( \mathrm{V} \) be a \( \mathfrak{g} \) -module and \( v \in \mathrm{V} \) . The following conditions are equivalent:\n\n(i) \( {\mathfrak{b}}_{ + }v \subset {kv} \) ;\n\n(ii) \( \mathfrak{h}v \subset {kv} \) and \( {\mathfrak{n}}_{ + }v = 0 \) ;\n\n(iii) \( \mathfrak{h}v \subset {kv} \) and \( {\mathfrak... | (i) \( \Rightarrow \) (ii): Assume that \( {\mathfrak{b}}_{ + }v \subset {kv} \) . There exists \( \lambda \in {\mathfrak{h}}^{ * } \) such that \( v \in {\mathrm{V}}^{\lambda } \) . Let \( \alpha \in {\mathrm{R}}_{ + } \) . Then \( {\mathfrak{g}}^{\alpha }.v \subset {\mathrm{V}}^{\lambda } \cap {\mathrm{V}}^{\lambda +... | Yes |
Lemma 2. Let \( \mathrm{V} \) be a simple \( \mathfrak{g} \) -module, \( \omega \) a weight of \( \mathrm{V} \) . The following conditions are equivalent:\n\n(i) every weight of \( \mathrm{V} \) is of the form \( \omega - \mu \) where \( \mu \) is a radical weight \( \geq 0 \) ;\n\n(ii) \( \omega \) is the highest weig... | (i) \( \Rightarrow \) (ii) \( \Rightarrow \) (iii): This is clear.\n\n(iii) \( \Rightarrow \) (iv): Assume that condition (iii) is satisfied. For all \( h \in \mathfrak{h} \),\n\n\( \operatorname{Ker}\left( {{h}_{\mathrm{V}} - \omega \left( h\right) }\right) \)\n\nis non-zero, contained in \( {\mathrm{V}}^{\omega } \),... | Yes |
Lemma 3. Put \( \mathrm{L} = \left( {{\mathfrak{n}}_{ - }\mathrm{U}}\right) \cap {\mathrm{U}}^{0} \). (i) We have \( \mathrm{L} = \left( {\mathrm{{Un}}}_{ + }\right) \cap {\mathrm{U}}^{0} \), and \( \mathrm{L} \) is a two-sided ideal of \( {\mathrm{U}}^{0} \). (ii) We have \( {\mathrm{U}}^{0} = \mathrm{V} \oplus \mathr... | It is clear that \( {\mathfrak{n}}_{ - }\mathrm{U} \) (resp. \( {\mathrm{{Un}}}_{ + } \) ) is the set of linear combinations of the elements \( u\left( {\left( {q}_{i}\right) ,\left( {m}_{i}\right) ,\left( {p}_{i}\right) }\right) \) such that \( \sum {q}_{i} > 0 \) (resp. \( \sum {p}_{i} > 0 \) ). On the other hand \[ ... | Yes |
Lemma 1. Let \( \mathrm{V} \) be a \( \mathfrak{g} \) -module and \( \rho \) the corresponding representation of \( \mathfrak{g} \) .\n\n(i) If \( a \) is a nilpotent element of \( \mathfrak{g} \), and if \( \rho \left( a\right) \) is locally nilpotent,\n\n\[ \rho \left( {{e}^{\operatorname{ad}a}b}\right) = {e}^{\rho \... | With the assumptions in (i), we have \( \rho \left( {{\left( \operatorname{ad}a\right) }^{n}b}\right) = {\left( \operatorname{ad}\rho \left( a\right) \right) }^{n}\rho \left( b\right) \) for all \( n \geq 0 \), so \( \rho \left( {{e}^{\operatorname{ad}a}b}\right) = {e}^{\operatorname{ad}\rho \left( a\right) }\rho \left... | Yes |
Lemma 2. Let \( {h}^{0} = \mathop{\sum }\limits_{{\alpha \in {\mathrm{R}}_{ + }}}{H}_{\alpha } \) . Then \( {h}^{0} = \mathop{\sum }\limits_{{\alpha \in \mathrm{B}}}{a}_{\alpha }{H}_{\alpha } \), where the \( {a}_{\alpha } \) are integers \( \geq 1 \) . Let \( {\left( {b}_{\alpha }\right) }_{\alpha \in \mathrm{B}},{\le... | The fact that the \( {a}_{\alpha } \) are integers \( \geq 1 \) follows from the fact that \( {\left( {H}_{\alpha }\right) }_{\alpha \in \mathrm{B}} \) is a basis of the root system \( {\left( {H}_{\alpha }\right) }_{\alpha \in \mathrm{B}} \) (cf. Chap. VI,§1, no. 5, Remark 5). We have:\n\n\[ \alpha \left( {h}^{0}\righ... | Yes |
Lemma 3. Let \( \mathrm{G} \) be an abelian group written additively, and \( \varphi : {\mathcal{F}}_{\mathfrak{a}} \rightarrow \mathrm{G} \) a map; by abuse of notation, we denote by \( \varphi \left( \mathrm{F}\right) \) the image under \( \varphi \) of the class of any finite dimensional \( \mathfrak{a} \) -module \... | There exists a unique homomorphism \( \theta \) from \( \mathcal{R}\left( \mathfrak{a}\right) \) to \( \mathrm{G} \) such that \( \theta \left( \left\lbrack \mathrm{E}\right\rbrack \right) = \varphi \left( \mathrm{E}\right) \) for every finite dimensional simple \( \mathfrak{a} \) -module \( \mathrm{E} \) . Let \( \mat... | Yes |
There exists on the additive group \( \mathcal{R}\left( \mathfrak{a}\right) \) a unique multiplication distributive over addition such that \( \left\lbrack \mathrm{E}\right\rbrack \left\lbrack \mathrm{F}\right\rbrack = \left\lbrack {\mathrm{E} \otimes \mathrm{F}}\right\rbrack \) for all finite dimensional \( \mathfrak{... | The uniqueness is clear. There exists a commutative multiplication on \( \mathcal{R}\left( \mathfrak{a}\right) = {\mathbf{Z}}^{\left( {\mathfrak{S}}_{\mathfrak{a}}\right) } \) that is distributive over addition and such that \( \left\lbrack \mathrm{E}\right\rbrack \left\lbrack \mathrm{F}\right\rbrack = \left\lbrack {\m... | Yes |
Lemma 5. There exists a unique involutive automorphism \( \mathrm{X} \mapsto {\mathrm{X}}^{ * } \) of the ring \( \mathcal{R}\left( \mathfrak{a}\right) \) such that \( {\left\lbrack \mathrm{E}\right\rbrack }^{ * } = \left\lbrack {\mathrm{E}}^{ * }\right\rbrack \) for every finite dimensional \( \mathfrak{a} \) -module ... | The uniqueness is clear. By Lemma 3, there exists a homomorphism \( \mathrm{X} \mapsto {\mathrm{X}}^{ * } \) from the additive group \( \mathcal{R}\left( \mathfrak{a}\right) \) to itself such that \( {\left\lbrack \mathrm{E}\right\rbrack }^{ * } = \left\lbrack {\mathrm{E}}^{ * }\right\rbrack \) for every finite dimensi... | Yes |
Lemma 6. If \( \lambda \in {\mathrm{P}}_{+ + } \), then \( \operatorname{ch}\left\lbrack \lambda \right\rbrack \in \mathbf{Z}{\left\lbrack \mathrm{P}\right\rbrack }^{\mathrm{W}} \) . The unique maximal term of \( \operatorname{ch}\left\lbrack \lambda \right\rbrack \) (Chap. VI, \( §3 \), no. 2, Def. 1) is \( {e}^{\lamb... | The first assertion follows from no. 1, Cor. 2 of Prop. 2, and the second from \( §6 \), no. 1, Prop. 1 (ii). | Yes |
Lemma 1. Let \( \Pi \) be a subgroup of \( {\mathrm{V}}^{ * } \) that generates the vector space \( {\mathrm{V}}^{ * } \), and \( m \) an integer \( \geq 0 \) . Then \( {\operatorname{pr}}_{m}\left( {\exp \Pi }\right) \) generates the vector space \( {\mathbf{S}}^{m}\left( {\mathrm{\;V}}^{ * }\right) \) . | By Algebra, Chap. I,§8, no. 2, Prop. 2, any product of \( m \) elements of \( {\mathrm{V}}^{ * } \) is a \( k \) -linear combination of elements of the form \( {x}^{m} \) where \( x \in \Pi \) . But \( {x}^{m} = m!{\operatorname{pr}}_{m}\left( {\exp x}\right) . \) Q.E.D. | No |
Lemma 2. Let \( \rho \) be a finite dimensional linear representation of \( \mathfrak{a} \), and \( m \) an integer \( \geq 0 \) . The function \( x \mapsto \operatorname{Tr}\left( {\rho {\left( x\right) }^{m}}\right) \) on \( \mathfrak{a} \) is an invariant polynomial function. | Put \( g\left( {{x}_{1},\ldots ,{x}_{m}}\right) = \operatorname{Tr}\left( {\rho \left( {x}_{1}\right) \ldots \rho \left( {x}_{m}\right) }\right) \) for \( {x}_{1},\ldots ,{x}_{m} \in \mathfrak{a} \) . If \( x \in \mathfrak{a} \), we have\n\n\[ \n- \left( {{\theta }^{ * }\left( x\right) g}\right) \left( {{x}_{1},\ldots ... | Yes |
Lemma 3. Let \( \mathrm{E} \) be a finite dimensional \( \mathfrak{g} \) -module, and \( x \in \mathrm{E} \) . Then \( x \) is an invariant element of the \( \mathfrak{g} \) -module \( \mathrm{E} \) if and only if \( \left( {\exp {a}_{\mathrm{E}}}\right) .x = x \) for every nilpotent element \( a \) of \( \mathfrak{g} ... | The condition is clearly necessary. Assume now that it is satisfied. Let \( a \) be a nilpotent element of \( \mathfrak{g} \) . There exists an integer \( n \) such that \( {a}_{\mathrm{E}}^{n} = 0 \) . For all \( t \in k \), we have\n\n\[ 0 = \exp \left( {t{a}_{\mathrm{E}}}\right) .x - x = t{a}_{\mathrm{E}}x + \frac{1... | Yes |
Lemma 4. \( {}^{2} \) Let \( \mathrm{A} = {\bigoplus }_{n \geq 0}{\mathrm{\;A}}^{n} \) be a graded \( k \) -algebra, \( {k}^{\prime } \) an extension of \( k \), and \( {\mathrm{A}}^{\prime } = \mathrm{A}{ \otimes }_{k}{k}^{\prime } \) . Assume that \( {\mathrm{A}}^{\prime } \) is a graded polynomial algebra over \( {k... | We have \( {\mathrm{A}}^{\prime 0} = {k}^{\prime } \), so \( {\mathrm{A}}^{0} = k \) . Put \( {\mathrm{A}}_{ + } = {\bigoplus }_{n \geq 1}{\mathrm{\;A}}^{n} \) and \( \mathrm{P} = {\mathrm{A}}_{ + }/{\mathrm{A}}_{ + }^{2} \) . Then \( \mathrm{P} \) is a graded vector space, and there is a graded linear map \( f : \math... | No |
Lemma 5. A is a graded polynomial algebra if and only if \( \mathrm{P} \) is finite dimensional and \( g \) is bijective. | If \( \mathrm{P} \) is finite dimensional, \( \mathbf{S}\left( \mathrm{P}\right) \) is clearly a graded polynomial algebra, and so is A if \( g \) is bijective. Conversely, assume that A is generated by algebraically independent homogeneous elements \( {x}_{1},\ldots ,{x}_{m} \) of degrees \( {d}_{1},\ldots ,{d}_{m} \)... | Yes |
Lemma 6. Let \( \mathrm{V} \) be a finite dimensional vector space, \( \mathrm{G} \) a finite group of automorphisms of \( \mathrm{V} \), and \( v \) and \( {v}^{\prime } \) elements of \( \mathrm{V} \) such that \( {v}^{\prime } \notin \mathrm{G}v \) . There exists a G-invariant polynomial function \( f \) on \( \math... | Indeed, for each \( s \in \mathrm{G} \) there exists a polynomial function \( {g}_{s} \) on \( \mathrm{V} \) equal to 1 at \( v \) and to 0 at \( s{v}^{\prime } \) . Then the function \( g = 1 - \mathop{\prod }\limits_{{s \in \mathrm{G}}}{g}_{s} \) is equal to 0 at \( v \) and to 1 on \( \mathrm{G}{v}^{\prime } \) . Th... | Yes |
In the ring \( \mathbf{Z}\langle \mathrm{P}\rangle \), we have \( \mathrm{K} \cdot \mathop{\prod }\limits_{{\alpha \in {\mathrm{R}}_{ + }}}\left( {1 - {e}^{-\alpha }}\right) = \mathrm{K}{e}^{-\rho }d = 1 \) . | Indeed, \[ \mathrm{K} = \mathop{\prod }\limits_{{\alpha \in {\mathrm{R}}_{ + }}}\left( {{e}^{0} + {e}^{-\alpha } + {e}^{-{2\alpha }} + \cdots }\right) \] so \[ K{e}^{-\rho }d = \mathop{\prod }\limits_{{\alpha \in {\mathrm{R}}_{ + }}}\left( {1 + {e}^{-\alpha } + {e}^{-{2\alpha }} + \cdots }\right) \mathop{\prod }\limits... | Yes |
Lemma 2. Let \( \lambda \in {\mathfrak{h}}^{ * } \) . The module \( \mathrm{Z}\left( \lambda \right) \) (§6, no. 3) admits a character that is an element of \( \mathbf{Z}\langle \mathrm{P}\rangle \), and we have \( d.\operatorname{ch}\mathrm{Z}\left( \lambda \right) = {e}^{\lambda + \rho } \) . | Let \( {\alpha }_{1},\ldots ,{\alpha }_{q} \) be distinct elements of \( {\mathrm{R}}_{ + } \) . The \( {X}_{-{\alpha }_{1}}^{{n}_{1}}{X}_{-{\alpha }_{2}}^{{n}_{2}}\ldots {X}_{-{\alpha }_{q}}^{{n}_{q}} \otimes 1 \) form a basis of \( \mathrm{Z}\left( \lambda \right) \) (§6, Prop. 6 (iii)). For \( h \in \mathfrak{h} \),... | Yes |
Lemma 3. Let \( \mathrm{M} \) be a \( \mathfrak{g} \) -module which admits a character \( \mathrm{{ch}}\left( \mathrm{M}\right) \) whose support is contained in a finite union of the sets \( \mu - {\mathrm{P}}_{ + } \) . Let \( \mathrm{U} \) be the enveloping algebra of \( \mathfrak{g},\mathrm{Z} \) the centre of \( \m... | If \( \operatorname{Supp}\left( {\operatorname{ch}\mathrm{M}}\right) \) is empty, the lemma is clear. Assume that \( \operatorname{Supp}\left( {\operatorname{ch}\mathrm{M}}\right) \neq \varnothing \) . Let \( \lambda \) be a maximal element of this support, and put \( \dim {\mathrm{M}}^{\lambda } = m \) . There exists ... | Yes |
If \( \mathfrak{h} \) is splitting, the vector subspaces \( \mathfrak{s} \) and \( {\mathfrak{s}}^{\prime } \) of \( \mathfrak{h} \) are rational over \( \mathbf{Q} \) . | Assume that the Cartan subalgebra \( \mathfrak{h} \) is splitting. Let \( d \) be the dimension of \( \mathrm{D} \) ; put \( \mathrm{W} = \mathop{\bigwedge }\limits^{d}\left( \mathrm{\;V}\right) \) and \( \sigma = \mathop{\bigwedge }\limits^{d}\left( \rho \right) \) ; denote also by \( \left( {{e}_{1},\ldots ,{e}_{d}}\... | No |
Lemma 2. Let \( \mathfrak{n} \) be a subalgebra of \( \mathfrak{g} \) such that, for all \( n \in \mathfrak{n},{\operatorname{ad}}_{\mathfrak{g}}\left( n\right) \) is nilpotent. Let \( h \in \mathfrak{g} \) be such that \( \left\lbrack {h,\mathfrak{n}}\right\rbrack = \mathfrak{n} \) . Then \( {e}^{{\operatorname{ad}}_{... | It is clear that \( {e}^{{\operatorname{ad}}_{\mathfrak{g}}\left( \mathfrak{n}\right) } \cdot h \subset h + \mathfrak{n} \) . We shall prove that, if \( v \in \mathfrak{n} \), then \( h + v \in {e}^{{\operatorname{ad}}_{\mathfrak{g}}\left( \mathfrak{n}\right) } \) . \( h \) . It suffices to prove that \( h + v \in {e}^... | Yes |
Lemma 3. Let \( x \in \mathfrak{g},\mathfrak{p} = \operatorname{Ker}\left( {\operatorname{ad}x}\right) ,\mathfrak{q} = \operatorname{Im}\left( {\operatorname{ad}x}\right) \) . Then \( \left\lbrack {\mathfrak{p},\mathfrak{q}}\right\rbrack \subset \mathfrak{q} \), and \( \mathfrak{p} \cap \mathfrak{q} \) is a subalgebra ... | If \( u \in \mathfrak{p} \) and \( v \in \mathfrak{q} \), there exists \( w \in \mathfrak{g} \) such that \( v = \left\lbrack {x, w}\right\rbrack \), so\n\n\[ \left\lbrack {u, v}\right\rbrack = \left\lbrack {u,\left\lbrack {x, w}\right\rbrack }\right\rbrack = \left\lbrack {x,\left\lbrack {u, w}\right\rbrack }\right\rbr... | Yes |
Lemma 4. Let \( \\left( {x, h, y}\\right) \) and \( \\left( {x,{h}^{\\prime },{y}^{\\prime }}\\right) \) be \( {\\mathfrak{{sl}}}_{2} \)-triplets in \( \\mathfrak{g} \). There exists \( z \\in \\mathfrak{g} \) such that \( {\\operatorname{ad}}_{\\mathfrak{g}}z \) is nilpotent and such that\n\n\[ \n{e}^{{\\operatorname{... | Let \( \\mathfrak{n} = \\operatorname{Ker}\\left( {\\operatorname{ad}x}\\right) \\cap \\operatorname{Im}\\left( {\\operatorname{ad}x}\\right) \). For all \( p \\in \\mathbf{Z} \), let \( {\\mathfrak{g}}_{p} = \\operatorname{Ker}\\left( {\\operatorname{ad}h - p}\\right) \). By \( §1 \), no. 3 (applied to the adjoint rep... | Yes |
Lemma 5. Let \( \mathrm{V} \) be a finite dimensional vector space, \( \mathrm{A} \) and \( \mathrm{B} \) endomorphisms of \( \mathrm{V} \) . Assume that \( \mathrm{A} \) is nilpotent and that \( \left\lbrack {\mathrm{A},\left\lbrack {\mathrm{A},\mathrm{B}}\right\rbrack }\right\rbrack = 0 \) . Then \( \mathrm{{AB}} \) ... | \[ \text{Put}\mathrm{C} = \left\lbrack {\mathrm{A},\mathrm{B}}\right\rbrack \text{. Since}\left\lbrack {\mathrm{A},\mathrm{C}}\right\rbrack = 0\text{,} \]\n\[ \left\lbrack {\mathrm{A},{\mathrm{{BC}}}^{p}}\right\rbrack = \left\lbrack {\mathrm{A},\mathrm{B}}\right\rbrack {\mathrm{C}}^{p} = {\mathrm{C}}^{p + 1} \]\nfor ev... | Yes |
Lemma 7. Let \( \mathrm{K} \) be a commutative field with at least 4 elements. Let \( \mathrm{G} \) be the group of matrices \( \left( \begin{matrix} \alpha & \beta \\ 0 & {\alpha }^{-1} \end{matrix}\right) \) where \( \alpha \in {\mathrm{K}}^{ * },\beta \in \mathrm{K} \) . Let \( {\mathrm{G}}^{\prime } \) be the group... | If \( \alpha ,{\alpha }^{\prime } \in {\mathrm{K}}^{ * } \) and \( \beta ,{\beta }^{\prime } \in \mathrm{K} \) , \[ \left( \begin{matrix} \alpha & \beta \\ 0 & {\alpha }^{-1} \end{matrix}\right) \left( \begin{matrix} {\alpha }^{\prime } & {\beta }^{\prime } \\ 0 & {\alpha }^{\prime - 1} \end{matrix}\right) {\left( \beg... | Yes |
Lemma 9. With the notations of Prop. 8, put \( {\mathfrak{g}}^{p} = \operatorname{Ker}\left( {\operatorname{ad}{h}^{0} - p}\right) \) for \( p \in \mathbf{Z} \) . Let \( {\mathfrak{g}}_{ * }^{2} \) be the set of elements of \( {\mathfrak{g}}^{2} = \mathop{\sum }\limits_{{\alpha \in \mathrm{B}}}{\mathfrak{g}}^{\alpha } ... | It is clear that \( {e}^{\text{ad }{\mathfrak{n}}_{ + }} \cdot x \subset x + \left\lbrack {{\mathfrak{n}}_{ + },{\mathfrak{n}}_{ + }}\right\rbrack \) . We prove that, if \( v \in \left\lbrack {{\mathfrak{n}}_{ + },{\mathfrak{n}}_{ + }}\right\rbrack \), then \( x + v \in {e}^{\text{ad }{\mathfrak{n}}_{ + }}.x \) . Put \... | Yes |
Lemma 1. Let \( \mathbf{n} \in {\mathbf{N}}^{\mathrm{I}} \) . If \( \left| \mathbf{n}\right| < r \), then \( {c}_{r}^{ + }\left( \left\lbrack \mathbf{n}\right\rbrack \right) = 0 \) . If \( \left| \mathbf{n}\right| = r \), then \[ {c}_{r}^{ + }\left( \left\lbrack \mathbf{n}\right\rbrack \right) = \mathop{\sum }\limits_{... | By Prop. 1, \[ {c}_{r}\left( {\mathbf{x}}^{\left( \mathbf{n}\right) }\right) = \sum {\mathbf{x}}^{\left( {\mathbf{p}}_{1}\right) } \otimes \cdots \otimes {\mathbf{x}}^{\left( {\mathbf{p}}_{r}\right) } \] where the summation extends over the set of sequences \( \left( {{\mathbf{p}}_{1},\ldots ,{\mathbf{p}}_{r}}\right) \... | Yes |
Lemma 2. If \( x, y \in \mathrm{A} \) and \( n \in \mathbf{N} \), \[ \frac{{\left( \operatorname{ad}x\right) }^{n}}{n!}y = \mathop{\sum }\limits_{{p + q = n}}{\left( -1\right) }^{q}\frac{{x}^{p}}{p!}y\frac{{x}^{q}}{q!} = \mathop{\sum }\limits_{{p + q = n}}{\left( -1\right) }^{q}{x}^{\left( p\right) }y{x}^{\left( q\righ... | Indeed, if we denote by \( {\mathrm{L}}_{x} \) and \( {\mathrm{R}}_{x} \) the maps \( z \mapsto {xz} \) and \( z \mapsto {zx} \) from \( \mathrm{A} \) to \( \mathrm{A} \), we have, since \( {\mathrm{L}}_{x} \) and \( {\mathrm{R}}_{x} \) commute, \[ \frac{1}{n!}{\left( \operatorname{ad}x\right) }^{n} = \frac{1}{n!}{\lef... | Yes |
Lemma 3. Let \( x, h \in \mathrm{A} \) and \( \lambda \in k \) be such that \( \left( {\operatorname{ad}h}\right) x = {\lambda x} \) . For all \( n \in \mathbf{N} \) , and all \( \mathrm{P} \in k\left\lbrack \mathrm{X}\right\rbrack \), we have\n\n\[ \mathrm{P}\left( h\right) {x}^{\left( n\right) } = {x}^{\left( n\right... | Since ad \( h \) is a derivation of \( \mathrm{A} \) and since \( \left( {\operatorname{ad}h}\right) x \) commutes with \( x \), we have\n\n\[ \left( {\operatorname{ad}h}\right) {x}^{n} = n{x}^{n - 1}\left( {\left( {\operatorname{ad}h}\right) x}\right) = {n\lambda }{x}^{n}, \]\n\nso\n\n\[ \left( {\operatorname{ad}h}\ri... | Yes |
Lemma 5. (i) \( {\mathcal{U}}_{ + } \) is a lattice in the vector space \( \mathrm{U}\left( {\mathfrak{n}}_{ + }\right) \) . | By definition, \( {\mathcal{U}}_{ + } \) is generated as a \( \mathbf{Z} \) -module by the elements\n\n\[ \n{x}_{\varphi }^{\left( \mathbf{n}\right) } = \mathop{\prod }\limits_{{1 \leq i \leq r}}{x}_{\varphi \left( i\right) }^{\left( n\left( i\right) \right) }\n\]\n\nwhere \( r \in \mathbf{N},\varphi = \left( {\varphi ... | Yes |
Lemma 6. Let \( \mathrm{M} \) be a free \( \mathbf{Z} \) -module of finite type, \( u \) an endomorphism of \( \mathrm{M} \) , and \( v \) the endomorphism \( u \otimes 1 \) of \( \mathrm{M} \otimes \mathbf{z}\mathbf{Q} \) . Assume that \( \left( \begin{array}{l} v \\ n \end{array}\right) \left( \mathrm{M}\right) \subs... | a) For any polynomial \( \mathrm{P} \in \mathbf{Q}\left\lbrack \mathrm{T}\right\rbrack \) such that \( \mathrm{P}\left( \mathbf{Z}\right) \subset \mathbf{Z} \), we have \( \mathrm{P}\left( v\right) \left( \mathrm{M}\right) \subset \mathrm{M} \) (no. 4, Cor. of Prop. 2), so \( \det \mathrm{P}\left( v\right) \in \mathbf{... | Yes |
Lemma 1. Let \( \mathrm{V} \) be a finite dimensional vector space, \( \mathrm{Q} \) a non-degenerate quadratic form on \( \mathrm{V},\Psi \) the symmetric bilinear form associated to \( \mathrm{Q},\mathrm{C}\left( \mathrm{Q}\right) \) the Clifford algebra of \( \mathrm{V} \) relative to \( \mathrm{Q},{f}_{0} \) the co... | Assertion (i) is clear. If \( a, b \in \mathfrak{o}\left( \Psi \right) \), we have (putting \( \Psi \left( {x, y}\right) = \langle x, y\rangle \) ): \[ \mathop{\sum }\limits_{r}\left( {a{e}_{r}}\right) \left( {b{e}_{r}^{\prime }}\right) = \mathop{\sum }\limits_{{r, s, t}}\left\langle {a{e}_{r},{e}_{s}^{\prime }}\right\... | Yes |
For \( 1 \leq r \leq l \), let \( {\mathrm{F}}_{r} \) be the subspace of \( \mathop{\bigwedge }\limits^{r}\mathrm{V} \) generated by the isotropic r-vectors. Then | \[ \mathop{\bigwedge }\limits^{r}\mathrm{\;V} = {\mathrm{F}}_{r} + {X}_{ - }\left( {\mathop{\bigwedge }\limits^{{r - 2}}\mathrm{\;V}}\right) = {\mathrm{F}}_{r} + \left( {\mathop{\sum }\limits_{{i = 1}}^{l}{e}_{i} \land {e}_{-i}}\right) \land \mathop{\bigwedge }\limits^{{r - 2}}\mathrm{\;V}. \] | No |
Lemma 1. Let \( \mathrm{G} \) be a Lie group, \( \mathrm{K} \) a compact subgroup of \( \mathrm{G} \), and \( \mathrm{F} \) an invariant bilinear form on \( \mathrm{L}\left( \mathrm{G}\right) \) . Let \( x, y \in \mathrm{L}\left( \mathrm{G}\right) \) . There exists an element \( k \) of \( \mathrm{K} \) such that \( \m... | The function \( v \mapsto \mathrm{F}\left( {\left( {\operatorname{Ad}v}\right) \left( x\right), y}\right) \) from \( \mathrm{K} \) to \( \mathbf{R} \) is continuous, and hence has a minimum at some point \( k \in \mathrm{K} \) . Let \( u \in \mathrm{L}\left( \mathrm{K}\right) \) and put\n\n\[ h\left( t\right) = \mathrm... | Yes |
Lemma 2. Let \( \mathrm{H} \) be a Lie group, \( {f}_{\mathrm{T}} : \mathrm{T} \rightarrow \mathrm{H} \) and \( \widetilde{f} : \widetilde{\mathrm{D}}\left( \mathrm{G}\right) \rightarrow \mathrm{H} \) morphisms of Lie groups such that \( {f}_{\mathrm{T}}\left( {p\left( t\right) }\right) = \widetilde{f}\left( t\right) \... | Put \( \mathrm{Z} = \mathrm{C}{\left( \mathrm{G}\right) }_{0} \) ; by \( §1 \), no. 4, Cor. 1 of Prop. 4, the morphism of Lie groups \( g : \mathrm{Z} \times \widehat{\mathrm{D}}\left( \mathrm{G}\right) \rightarrow \mathrm{G} \) such that \( g\left( {z, x}\right) = {z}^{-1}p\left( x\right) \) is a covering; its kernel ... | Yes |
Lemma 1. If \( \mathrm{G} \) is commutative, \( \widetilde{\mathrm{V}} \) is the direct sum of the \( {\widetilde{\mathrm{V}}}_{\lambda }\left( \mathrm{G}\right) \) for \( \lambda \in \mathrm{X}\left( \mathrm{G}\right) \) . | Since \( \rho \) is semi-simple ( \( §1 \), no. 1), it suffices to prove the lemma in the case in which \( \rho \) is simple. In that case, the commutant \( \mathrm{Z} \) of \( \rho \left( \mathrm{G}\right) \) in \( \operatorname{End}\left( \widetilde{\mathrm{V}}\right) \) reduces to homotheties (Algebra, Chap. VIII, §... | Yes |
Lemma 2. Let \( \mathrm{S} \) be a closed subgroup of \( \mathrm{T} \) and \( \mathrm{Z}\left( \mathrm{S}\right) \) its normalizer in \( \mathrm{G} \) . (i) \( \mathrm{R}\left( {\mathrm{Z}{\left( \mathrm{S}\right) }_{0},\mathrm{\;T}}\right) \) is the set of \( \alpha \in \mathrm{R}\left( {\mathrm{G},\mathrm{T}}\right) ... | The Lie algebra \( \mathrm{L}{\left( \mathrm{Z}\left( \mathrm{S}\right) \right) }_{\left( \mathrm{C}\right) } \) consists of the invariants of \( \mathrm{S} \) on \( {\mathfrak{g}}_{\mathbf{C}} \) (Chap. III, §9, no. 3, Prop. 8), and hence is the direct sum of \( {\mathfrak{t}}_{\mathbf{C}} \) and the \( {\mathfrak{g}}... | Yes |
Lemma 3. Let \( {\mathrm{B}}_{0} \) be a basis of \( \mathrm{R}\left( {\mathrm{G},\mathrm{T}}\right) \) . The group \( {\operatorname{Int}}_{\mathrm{G}}\left( \mathrm{T}\right) \) operates simply-transitively on the set of framings of \( \left( {\mathrm{G},\mathrm{T}}\right) \) of the form \( \left( {{\mathrm{B}}_{0},{... | For all \( \alpha \in {\mathrm{B}}_{0} \), denote by \( \mathrm{K}\left( \alpha \right) \) the restriction of the quadratic form \( \mathrm{K} \) to \( \mathrm{V}\left( \alpha \right) \) ; the operation of \( \mathrm{T} \) on \( \mathrm{V}\left( \alpha \right) \) defines a morphism \( {\iota }_{\alpha } : \mathrm{T} \r... | Yes |
Lemma 1. Let \( u \) be an automorphism of \( \mathrm{G} \), and \( \mathrm{H} \) the set of its fixed points.\n\na) \( \mathrm{H} \) is a closed subgroup of \( \mathrm{G} \).\n\nb) If \( {\mathrm{H}}_{0} \) is central in \( \mathrm{G} \), then \( \mathrm{G} \) is commutative (so \( \mathrm{G} = \mathrm{T} \)). | Assertion \( a \) ) is clear. To prove \( b \) ), we can replace \( \mathrm{G} \) by \( \mathrm{D}\left( \mathrm{G}\right) \) ( \( §1 \), Cor. 1 of Prop. 4), and hence can assume that \( \mathrm{G} \) is semi-simple. Then, if \( {\mathrm{H}}_{0} \) is central in G, we have \( \mathrm{L}\left( \mathrm{H}\right) = \{ 0\}... | Yes |
Lemma 2. Let \( x \) be an element of \( \mathrm{T} \) and \( \mathrm{S} \) a subtorus of \( \mathrm{T} \). If the identity component of \( \mathrm{Z}\left( x\right) \cap \mathrm{Z}\left( \mathrm{S}\right) \) reduces to \( \mathrm{T} \), there exists an element \( s \) of \( \mathrm{S} \) such that \( {xs} \) is regula... | For all \( \alpha \in \mathrm{R}\left( {\mathrm{G},\mathrm{T}}\right) \), let \( {\mathrm{S}}_{\alpha } \) be the submanifold of \( \mathrm{S} \) consisting of the elements \( s \) of \( \mathrm{S} \) such that \( {\left( xs\right) }^{\alpha } = 1 \). If there is no element \( s \) of \( \mathrm{S} \) such that \( {xs}... | Yes |
Lemma 3. Assume that \( \mathrm{G} \) is simply-connected. Let \( \mathrm{C} \) be a chamber of \( \mathrm{t} \), and \( u \) an automorphism of \( \mathrm{G} \) such that \( \mathrm{T} \) and \( \mathrm{C} \) are stable under \( u \) . Then the set of points of \( \mathrm{T} \) fixed by \( u \) is connected. | Since \( \mathrm{G} \) is simply-connected, \( \Gamma \left( \mathrm{T}\right) \) is generated by the nodal vectors \( {K}_{\alpha } \) \( \left( {\alpha \in \mathrm{R}\left( {\mathrm{G},\mathrm{T}}\right) }\right) \), and hence has a basis consisting of the family of the \( {K}_{\alpha } \) for which \( \alpha \) belo... | Yes |
Lemma 4. Let \( g \in \mathrm{G}, t \in \mathrm{T} \), and let \( \bar{g} \) be the image of \( g \) in \( \mathrm{G}/\mathrm{T} \) . Identify the tangent space of G/T (resp. T, resp. G) at \( \bar{g} \) (resp. \( t \), resp. \( {gt}{g}^{-1} \) ) with \( \mathfrak{g}/\mathfrak{t} \) (resp. \( \mathfrak{t} \), resp. \( ... | Let \( \mathrm{F} \) be the map from \( \mathrm{G} \times \mathrm{T} \) to \( \mathrm{T} \) such that \( \mathrm{F}\left( {g, t}\right) = {gt}{g}^{-1} \) . Since \( \mathrm{F} \circ \left( {\gamma \left( g\right) ,{\operatorname{Id}}_{\mathrm{T}}}\right) = \operatorname{Int}g \circ \mathrm{F} \), we have \( {\mathrm{T}... | Yes |
There exists an A-bilinear map from \( {\mathrm{{Alt}}}^{s}\left( {\mathrm{M}}^{\prime \prime }\right) \times {\mathrm{{Alt}}}^{r}\left( {\mathrm{M}}^{\prime }\right) \) to \( {\mathrm{{Alt}}}^{s + r}\left( \mathrm{M}\right) \), denoted by \( \left( {u, v}\right) \mapsto u \cap v \), and characterized by either of the ... | The existence of a form \( {v}_{1} \) satisfying condition \( a \) ) follows from the fact that \( {\Lambda }^{r}\left( i\right) \) induces an isomorphism from \( {\Lambda }^{r}\left( {\mathrm{M}}^{\prime }\right) \) to a direct factor submodule of \( {\Lambda }^{r}\left( \mathrm{M}\right) \) (Algebra, Chap. III, \( §7... | Yes |
Lemma 2. If \( {\omega }_{\mathrm{G}} = {\omega }_{\mathrm{G}/\mathrm{T}} \cap {\omega }_{\mathrm{T}} \), then\n\n\[{\int }_{\mathrm{G}}{\omega }_{\mathrm{G}} = {\int }_{\mathrm{G}/\mathrm{T}}{\omega }_{\mathrm{G}/\mathrm{T}} \cdot {\int }_{\mathrm{T}}{\omega }_{\mathrm{T}}\] | Denote by \( \pi \) the canonical morphism from \( \mathrm{G} \) to \( \mathrm{G}/\mathrm{T} \) . Let \( g \in \mathrm{G} \), and let \( {t}_{1},\ldots ,{t}_{n - r} \) be elements of \( {\mathrm{T}}_{\pi \left( g\right) }\left( {\mathrm{G}/\mathrm{T}}\right) \) . Identify the fibre \( {\pi }^{-1}\left( {\pi \left( g\ri... | Yes |
Lemma 3. The inverse image on \( \left( {\mathrm{G}/\mathrm{T}}\right) \times {\mathrm{T}}_{r} \) of the measure \( {dg} \) on \( {\mathrm{G}}_{r} \) under the local homeomorphism \( {f}_{r} \) (Integration, Chap. V,§6, no. 6) is the measure \( \mu \otimes {\delta }_{\mathrm{G}}{dt} \), where \( \mu \) is the unique \(... | Choose an invariant differential form \( {\omega }_{\mathrm{T}} \) (resp. \( {\omega }_{\mathrm{G}/\mathrm{T}} \) ) on \( \mathrm{T} \) (resp. \( \mathrm{G}/\mathrm{T} \) ) of maximal degree, such that the measure defined by \( {\omega }_{\mathrm{T}} \) (resp. \( {\omega }_{\mathrm{G}/\mathrm{T}} \) ) is equal to \( {d... | Yes |
Lemma 2. Let \( \varphi \) be a linear representation of the complex Lie algebra \( {\mathfrak{g}}_{\mathbf{C}} \) on a finite dimensional complex vector space \( \mathrm{V} \). There exists a representation \( \tau \) of \( \mathrm{G} \) on \( \mathrm{V} \) such that \( \mathrm{L}{\left( \tau \right) }_{\left( \mathbf... | If there exists a representation \( \tau \) of \( \mathrm{G} \) such that \( \mathrm{L}{\left( \tau \right) }_{\left( \mathbf{C}\right) } = \varphi \), then \( \varphi \) is semi-simple because \( \mathrm{G} \) is connected and \( \tau \) is semi-simple (Chap. III,§6, no. 5, Cor. 2 of Prop. 13), and the weights of \( {... | Yes |
Lemma 3. For each \( \lambda \in {\mathrm{X}}_{+ + } \), let \( {\mathrm{C}}_{\lambda } \) be an element of \( \mathbf{Z}{\left\lbrack \mathrm{X}\left( \mathrm{T}\right) \right\rbrack }^{\mathrm{W}} \) having unique maximal term \( {e}^{\lambda } \) . Then the family \( {\left( {\mathrm{C}}_{\lambda }\right) }_{\lambda... | The proof is identical to that of Prop. 3 of Chap. VI, §3, no. 4, replacing \( \mathrm{A} \) by \( \mathbf{Z},\mathrm{P} \) by \( \mathrm{X}\left( \mathrm{T}\right) \) and \( \mathrm{P} \cap \overline{\mathrm{C}} \) by \( {\mathrm{X}}_{+ + } \) . | Yes |
THEOREM 2 (H. Weyl). For all \( \lambda \in {\mathrm{X}}_{+ + } \), we have \( \mathrm{J}\left( \rho \right) \cdot {\chi }_{\lambda } \mid \mathrm{T} = \mathrm{J}\left( {\lambda \rho }\right) \). | The function \( \mathrm{J}\left( \rho \right) \cdot {\chi }_{\lambda } \mid \mathrm{T} \) is anti-invariant under \( \mathrm{W} \), and is a linear combination with integer coefficients of elements of \( \mathrm{X}\left( \mathrm{T}\right) \) . Thus, by Chap. VI, \( §3 \), no. 3, Prop. 1, it can be written as \( \mathop... | Yes |
Lemma 5. We have\n\n\[ \mathrm{J}\left( \mu \right) \left( {\exp \left( {z\gamma }\right) }\right) = {e}^{{z\delta }\left( \mu \right) \left( \gamma \right) }\mathop{\prod }\limits_{{\alpha > 0}}\left( {1 - {e}^{-{z\delta }\left( \mu \right) \left( {K}_{\alpha }\right) }}\right) . \] | We now prove Lemma 5. Let \( z \in \mathbf{C} \) ; denote by \( {\varphi }_{z} \) the map from \( \mathfrak{t} \) to the \( \mathbf{C} \) -algebra \( \operatorname{Map}\left( {\mathrm{X}\left( \mathrm{T}\right) ,\mathbf{C}}\right) \) of maps from \( \mathrm{X}\left( \mathrm{T}\right) \) to \( \mathbf{C} \) that associa... | Yes |
Lemma 1. Let \( \mathrm{T} \) and \( {\mathrm{T}}^{\prime } \) be two topological spaces, \( \mathrm{A} \) and \( {\mathrm{A}}^{\prime } \) compact subsets of \( \mathrm{T} \) and \( {\mathrm{T}}^{\prime } \), respectively, \( \mathrm{W} \) a neighbourhood of \( \mathrm{A} \times {\mathrm{A}}^{\prime } \) in \( \mathrm... | Let \( x \in \mathrm{A} \) ; there exist open subsets \( {\mathrm{U}}_{x} \) of \( \mathrm{T} \) and \( {\mathrm{U}}_{x}^{\prime } \) of \( {\mathrm{T}}^{\prime } \) such that \( \{ x\} \times {\mathrm{A}}^{\prime } \subset {\mathrm{U}}_{x} \times {\mathrm{U}}_{x}^{\prime } \subset \mathrm{W} \) : indeed, the compact s... | Yes |
Lemma 2. Let \( x \) be a point of \( \mathrm{X},\varphi : \mathrm{X} \rightarrow \mathrm{Y} \) a morphism of class \( {\mathrm{C}}^{1} \) that is an immersion at \( x \) . There exists a neighbourhood \( \Omega \) of \( \varphi \) in \( {\mathcal{C}}^{1}\left( {\mathrm{X};\mathrm{Y}}\right) \) and a neighbourhood \( \... | Let \( \mathrm{U} \) be a relatively compact open neighbourhood of \( x \) isomorphic to a finite dimensional vector space, and such that \( \varphi \left( \overline{\mathrm{U}}\right) \) is contained in the domain \( \mathrm{V} \) of a chart. The set \( {\Omega }_{0} \) of \( \psi \in {\mathcal{C}}^{1}\left( {\mathrm{... | Yes |
Lemma 3. Let \( \mathrm{G} \) be a compact topological group operating continuously on a topological space \( \mathrm{X} \) ; let \( \mathrm{A} \) be a subset of \( \mathrm{X} \), stable under \( \mathrm{G} \), and \( \mathrm{W} \) a neighbourhood of \( \mathrm{A} \) . Then, there exists an open neighbourhood \( \mathr... | Put \( \mathrm{F} = \mathrm{X} - \overset{ \circ }{\mathrm{W}} \) and \( \mathrm{V} = \mathrm{X} - \mathrm{{GF}} \) . Then \( \mathrm{V} \) is open (General Topology, Chap. III, \( §4 \), no. 1, Cor. 1 of Prop. 1), stable under G, and A \( \subset \) V \( \subset \) W. | Yes |
Lemma 4. Let \( \mathrm{H} \) be a compact Lie group, \( \rho : \mathrm{H} \rightarrow \mathbf{{GL}}\left( \mathrm{V}\right) \) a continuous (hence analytic) representation of \( \mathrm{H} \) on a finite dimensional real vector space, and \( \mathrm{W} \) a neighbourhood of the origin in \( \mathrm{V} \) . There exist... | Choose a scalar product on V invariant under \( \mathrm{H}\left( {§1\text{, no. 1}}\right) \) . There exists a real number \( r > 0 \) such that the open ball B of radius \( r \) is contained in \( \mathrm{W} \) ; it is clearly stable under \( \mathrm{H} \) . Put \( u\left( v\right) = r{\left( {r}^{2} + \parallel v{\pa... | Yes |
Lemma 5. Let \( \mathrm{Z} \) be a separated manifold of class \( {\mathrm{C}}^{r} \), equipped with a law of left operation \( m : \mathrm{G} \times \mathrm{Z} \rightarrow \mathrm{Z} \) of class \( {\mathrm{C}}^{r} \), and \( \mu : \mathrm{Z} \rightarrow \mathrm{X} \) a morphism (of class \( \left. {\mathrm{C}}^{r}\ri... | Since \( \mu \) is compatible with the operations of \( \mathrm{G} \), it is étale at every point of \( \mathrm{G}z \) ; since the canonical map \( \mathrm{G}/{\mathrm{G}}_{x} \rightarrow \mathrm{G}x \) is a homeomorphism, so is the map from \( \mathrm{G}z \) to \( \mathrm{G}x \) induced by \( \mu \) . Hence, it follow... | Yes |
a) Every decreasing sequence of compact subgroups of \( \mathrm{G} \) is stationary. | a) Let \( {\left( {\mathrm{H}}_{i}\right) }_{i \geq 1} \) be a decreasing sequence of compact subgroups of \( \mathrm{G} \) ; these are Lie subgroups of G (Chap. III, §8, no. 2, Th. 2). The sequence of integers \( {\left( \dim {\mathrm{H}}_{i}\right) }_{i \geq 1} \) is decreasing, hence stationary, so there exists an i... | Yes |
Lemma 1. Let \( \left( {{\mathrm{G}}_{\alpha },{f}_{\alpha \beta }}\right) \) be a projective system of topological groups relative to a filtered index set \( \mathrm{I} \), and \( \mathrm{G} \) its limit. Assume that the canonical maps \( {f}_{\alpha } : \mathrm{G} \rightarrow {\mathrm{G}}_{\alpha } \) are surjective.... | Let \( \alpha ,\beta \) be two elements of \( \mathrm{I} \), with \( \alpha \leq \beta \) . Then \( {f}_{\alpha \beta }\left( {\mathrm{D}\left( {\mathrm{G}}_{\beta }\right) }\right) \subset \mathrm{D}\left( {\mathrm{G}}_{\alpha }\right) \) , and \( {f}_{\alpha \beta }\left( {\mathrm{C}\left( {\mathrm{G}}_{\beta }\right... | Yes |
Lemma 2. Let \( {\left( {\mathrm{S}}_{a}\right) }_{a \in \mathrm{A}},{\left( {\mathrm{T}}_{b}\right) }_{b \in \mathrm{B}} \) be two finite families of almost simple, simply-connected Lie groups (Chap. III,§ 9, no. 8, Def. 3), \( u : \mathop{\prod }\limits_{{a \in \mathrm{A}}}{\mathrm{S}}_{a} \rightarrow \mathop{\prod }... | Denote by \( {\mathfrak{s}}_{a} \) (resp. \( {\mathfrak{t}}_{b} \) ) the Lie algebra of \( {\mathrm{S}}_{a} \) (resp. \( {\mathrm{T}}_{b} \) ) for \( a \in \mathrm{A} \) (resp. \( b \in \mathrm{B}) \), and consider the homomorphism \( \mathrm{L}\left( u\right) : \mathop{\prod }\limits_{{a \in \mathrm{A}}}{\mathfrak{s}}... | Yes |
Lemma 3. Under the hypotheses of Lemma 1, assume that the \( {\mathrm{G}}_{\alpha } \) are simply-connected compact Lie groups. Then, the topological group \( \mathrm{G} \) is isomorphic to the product of a family of almost simple, simply-connected compact Lie groups. | For all \( \alpha \in \mathrm{I} \), the group \( {\mathrm{G}}_{\alpha } \) is the direct product of a finite family of almost simple, simply-connected subgroups \( {\left( {\mathrm{S}}_{\alpha }^{\lambda }\right) }_{\lambda \in {\mathrm{L}}_{\alpha }} \) (Chap. III, \( §9 \), no. 8, Prop. 28). Let \( \beta \in \mathrm... | Yes |
Lemma 1. Let \( {\mathrm{W}}^{\mathrm{G}} \) be the subspace of \( \mathrm{W} \) consisting of the elements invariant under \( \mathrm{G} \). The endomorphism \( {\int }_{\mathrm{G}}\rho \left( g\right) {dg} \) of \( \mathrm{W} \) is a projection with image \( {\mathrm{W}}^{\mathrm{G}} \), compatible with the operation... | \[ \text{Put}p = {\int }_{\mathrm{G}}\rho \left( g\right) {dg}\text{; for}h \in \mathrm{G}\text{,} \] \[ \rho \left( h\right) \circ p = {\int }_{\mathrm{G}}\rho \left( {hg}\right) {dg} = {\int }_{\mathrm{G}}\rho \left( g\right) {dg} = p \] and similarly \( p \circ \rho \left( h\right) = p \) . Thus, \( p \) is compatib... | Yes |
Lemma 2. Let \( u \) be an endomorphism of a finite dimensional vector space \( \mathrm{E} \) over a field \( \mathrm{K} \) . Then\n\n\[ \operatorname{Tr}{u}^{2} = \operatorname{Tr}{\mathbf{S}}^{2}\left( u\right) - \operatorname{Tr}{\Lambda }^{2}\left( u\right) \] | Let \( {\chi }_{u}\left( \mathrm{X}\right) = \mathop{\prod }\limits_{{i = 1}}^{n}\left( {\mathrm{X} - {\alpha }_{i}}\right) \) be a decomposition of the characteristic polynomial of \( u \) into linear factors in a suitable extension of \( \mathrm{K} \) . We have \( \operatorname{Tr}{u}^{2} = \) \( \mathop{\sum }\limit... | Yes |
Theorem 3 If \( {R}_{0} \leq 1,{E}_{0} \) is globally asymptotically stable in the region \( \Omega \) . | Proof Let\n\n\[ F = \left\lbrack \begin{matrix} \frac{{a}_{1}\lambda }{\mu + d} & \frac{{a}_{2}\lambda }{\mu + d} \\ 0 & 0 \end{matrix}\right\rbrack ,\;V = \left\lbrack \begin{matrix} \mu + d + \gamma & 0 \\ - \gamma & \mu + d + \alpha + \delta \end{matrix}\right\rbrack .\n\]\n\nWrite \( y = {\left( L, S\right) }^{T} \... | Yes |
Proposition 1 If \( {R}_{0} > 1 \), then the \( S \) - coordinate of the unique smoking equilibrium of model (1), \( {S}^{ * } \) is strictly decreasing with the increase of \( {b}_{i}\left( {i = 1,2}\right) \) due to behavior change. | Proof Since \( {R}_{0} > 1 \), there exists a unique smoking equilibrium \( \left( {{P}^{ * },{L}^{ * },{S}^{ * },{Q}^{ * }}\right) \) of the system (1), by equations (5), we have\n\n\[ \Phi \left( {S}^{ * }\right) = \Psi \left( {S}^{ * }\right) \Leftrightarrow \frac{\lambda }{\mu + d} - \frac{\left( {\mu + d + \gamma ... | Yes |
Lemma 1 Let \( {A}_{k} = \left( {a}_{ij}\right) \) be a \( k \times k\left( {k \geq 2}\right) \) symmetric matrix, \( {r}_{i} > 0 \) for all \( 1 \leq i \leq k \) and \( \mathop{\sum }\limits_{{i = 1}}^{k}{r}_{i} = 1 \), where \( {a}_{ii} = {r}_{i}^{-1} - 1 \) for \( 1 \leq i \leq k \) and \( {a}_{ij} = - 1 \) for \( i... | Proof Let \( {R}_{k} = {B}_{k}^{2} \) and \( {\mathbf{1}}_{k} = {\left( 1,1,\cdots ,1\right) }^{\prime } \) . Then \( {A}_{k} = {R}_{k}^{-1} - {\mathbf{1}}_{k}{\mathbf{1}}_{k}^{\prime } \) . It can be shown that\n\n\[ \n{R}_{k}^{1/2}{\mathbf{1}}_{k}{\mathbf{1}}_{k}^{\prime }{R}_{k}^{1/2} = {\left( {\left( {r}_{i}{r}_{j... | Yes |
Theorem 1 Assume that the coefficients \( \sigma ,\bar{\theta } \) and \( \mu \) satisfy\n\n\[ \n{\sigma }^{2} > \bar{\theta },\;{\sigma }^{2} > \frac{3}{2}\mu \n\]\n\n(11)\n\nthen the semi-discrete matrix \( \mathbf{S} \) in (10) is an \( {H}_{ + } \) -matrix. | Proof First we show that the off-diagonal \( {a}_{i} \) in \( \mathbf{S} \) are nonpositive. From the definition of \( {a}_{i} \), when \( i = 2 \), we have\n\n\[ \n{a}_{2} = - {\sigma }^{2} + \frac{1}{2}\bar{\theta } - \frac{3}{4}\mu < 0, \n\]\n\nbecause \( {\sigma }^{2} > \bar{\theta } \) . Consequently, when \( i \g... | Yes |
Theorem 2 The fully implicit finite volume scheme (16) is stable, i.e.\n\n\[ \n{\begin{Vmatrix}{\mathbf{V}}^{n}\end{Vmatrix}}_{\infty } \leq \max \left\{ {{\begin{Vmatrix}{\mathbf{V}}^{0}\end{Vmatrix}}_{\infty },{c}_{1},{c}_{2}}\right\} \n\]\n\nwhere\n\n\[ \n{c}_{1} = \mathop{\max }\limits_{{1 \leq n \leq N}}\left| {V}... | Proof It follows from (12) (15) that\n\n\[ \n\left( {1 + {\Delta \tau }{b}_{i}}\right) \left| {V}_{i}^{n + 1}\right| \leq \left| {V}_{i}^{n}\right| - {\Delta \tau }{a}_{i}\left| {V}_{i - 1}^{n + 1}\right| - {\Delta \tau }{c}_{i}\left| {V}_{i + 1}^{n + 1}\right| \n\]\n\n\[ \n\leq {\begin{Vmatrix}{\mathbf{V}}^{n}\end{Vma... | Yes |
Theorem 3 The finite volume scheme (16) is unconditionally monotone. | Proof For \( i = 0 \) or \( J \), the theorem is trivially true. When \( 0 < i < J \), it follows from Remark 1 that the matrix \( \mathbf{I} + {\Delta \tau }\mathbf{S} \) is an \( M \) -matrix, hence \( {\left\lbrack \left( \mathbf{I} + \Delta \tau \mathbf{S}\right) {\mathbf{V}}^{n + 1}\right\rbrack }_{i} \) is a stri... | Yes |
Theorem 6 Let \( \mathbf{S} \) and \( \mathbf{B} = \mathbf{I} + {\Delta \tau }\mathbf{S} \) be the semi-discrete and the full discrete matrix of (4), respectively. If \( {\sigma }^{2} > \bar{\theta } \) and \( {\sigma }^{2} > \frac{3}{2}\mu \), and the parameter matrix \( \Omega \) satisfies \( \Omega \geq \frac{1}{2}\... | Proof According to Corollary 1, the full discrete matrix \( \mathbf{B} \) is an \( {H}_{ + } \) -matrix under the assumptions \( {\sigma }^{2} > \bar{\theta } \) and \( {\sigma }^{2} > \frac{3}{2}\mu \) . Also, \( \mathbf{B} = D - L - U \) is an \( H \) -compatible splitting of the matrix \( \mathbf{B} \) . Then from T... | Yes |
Proposition 2.1 Let the hypotheses (A1) and (A2) hold. If the initial data \( {u}_{0} \in \) \( {H}_{{\Gamma }_{0}}^{1}\left( \Omega \right) \cap {H}^{2}\left( \Omega \right) ,{u}_{1} \in {H}_{{\Gamma }_{0}}^{1}\left( \Omega \right) \) and \( {f}_{0}\left( {x, t}\right) \in {H}_{{\Gamma }_{0}}^{1}\left( {\Omega \times ... | \[ u \in {L}^{\infty }\left( {\lbrack 0, T);{H}_{{\Gamma }_{0}}^{1}\left( \Omega \right) \cap {H}^{2}\left( \Omega \right) }\right) ,\] \[ {u}_{t} \in {L}^{\infty }\left( {\lbrack 0, T);{H}_{{\Gamma }_{0}}^{1}\left( \Omega \right) }\right) ,\] \[ {u}_{tt} \in {L}^{\infty }\left( {\lbrack 0, T);{L}^{2}\left( \Omega \rig... | Yes |
Proposition 2.2 Let the hypotheses (A1) and (A2) hold. If the initial data \( {u}_{0} \in \) \( {H}_{{\Gamma }_{0}}^{1}\left( \Omega \right) ,{u}_{1} \in {L}^{2}\left( \Omega \right) \) and \( {f}_{0}\left( {x, t}\right) \in {L}^{2}\left( {\Omega \times \left( {-\tau ,0}\right) }\right) \) are small enough, then there ... | Remark 2. 1 Proposition 2. 1 can be obtained by Faedo-Galerkin's method. And Proposition 2.2 can be obtained by standard arguments of density. | No |
Proposition 2.1 For a BGWPFRE, it holds that\n\n\[ \n{\psi }_{n + 1}\left( {s, t}\right) = \mathrm{E}{g}_{{Z}_{n}}\left( {{\psi }_{{\xi }_{n}}\left( {s, t}\right) }\right) ,0 \leq s, t \leq 1. \n\] | Proof By definition of \( {\psi }_{n + 1}\left( {s, t}\right) \), for \( 0 \leq s, t \leq 1 \), we obtain\n\n\[ \n{\psi }_{n + 1}\left( {s, t}\right) = \mathrm{E}{s}^{{F}_{n + 1}}{t}^{{M}_{n + 1}} = \mathrm{E}\left( {\mathrm{E}{s}^{{F}_{n + 1}}{t}^{{M}_{n + 1}} \mid {\xi }_{n}}\right) ) \n\]\n\n\[ \n= \mathrm{E}\left( ... | Yes |
Proposition 2.2 For a BGWPFRE, it holds that\n\n\[ \mathrm{E}{F}_{n + 1} = \mathrm{E}{\phi }_{n}\left( {Z}_{n}\right) \mathrm{E}{f}_{{\xi }_{n}, i};\mathrm{E}{M}_{n + 1} = \mathrm{E}{\phi }_{n}\left( {Z}_{n}\right) \mathrm{E}{m}_{{\xi }_{n}, i}. \] | Proof By the definition of \( {\varphi }_{{\xi }_{n}}\left( {s, t}\right) \) and \( {\psi }_{n + 1}\left( {s, t}\right) \), we can find\n\n\[ {\varphi }_{{\xi }_{n}}\left( {s, t}\right) = \mathop{\sum }\limits_{{k = 0}}^{\infty }\mathop{\sum }\limits_{{j = 0}}^{\infty }{s}^{k}{t}^{j}{p}_{k, j}\left( {\xi }_{n}\right) .... | Yes |
Theorem 2.2 For a superadditive BGWPFRE, when random control functions are superadditive, it holds that\n\n\[ \mathrm{E}\left\lbrack {{s}^{{Z}_{n + 1}} \mid {Z}_{0} = N}\right\rbrack \geq \mathrm{E}{\left( {\Phi }_{{\xi }_{0}}\left( {\Phi }_{{\xi }_{1}}\left( \cdots {\Phi }_{{\xi }_{n}}\left( s\right) \cdots \right) \r... | Proof Let \( F\left( \bar{\xi }\right) = \sigma \left( {{\xi }_{0},{\xi }_{1},\cdots ,{\xi }_{n},\cdots }\right), n = 0,1,2,\cdots \) . By the definition of \( {\lambda }_{k,\theta }\left( s\right) \) and \( {\Phi }_{k,\theta }\left( s\right) \), and using Theorem 2.1, we have\n\n\[ \mathrm{E}\left\lbrack {{s}^{{Z}_{1}... | Yes |
Theorem 3.1 For a superadditive BGWPFRE, when random control functions are superadditive, it holds that\n\n\[ \n{r}_{\theta } \mathrel{\text{:=}} \mathop{\lim }\limits_{{k \rightarrow \infty }}{r}_{k,\theta } = \mathop{\sup }\limits_{{k \geq 0}}{r}_{k,\theta }.\n\] | Proof Since the mating function \( L\left( {\cdot , \cdot }\right) \) and \( {\phi }_{n}\left( k\right) \) are superadditive, we have\n\n\[ \n\left( {k + j}\right) {r}_{k + j,\theta } = \mathrm{E}\left( {{Z}_{n + 1} \mid {Z}_{n} = k + j,{\xi }_{n} = \theta }\right)\n\]\n\n\[ \n= \mathrm{E}\left( {L\left( {\mathop{\sum ... | Yes |
For a superadditive BGWPFRE, when random control functions are superadditive, there exists a non-negative finite random variable \( W \) such that\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}{W}_{n} = W\text{. a. s. . } \] | Proof For \( n = 0,1,\cdots \), by the definition of \( {W}_{n} \), we have\n\n\[ \mathrm{E}\left( {{W}_{n + 1} \mid {F}_{n, Z}\left( \bar{\xi }\right) }\right) = \mathrm{E}\left( {\left. \frac{{Z}_{n + 1}}{\mathop{\prod }\limits_{{k = 0}}^{n}{r}_{{\xi }_{k}}}\right| \;{F}_{n, Z}\left( \bar{\xi }\right) }\right) = {\le... | Yes |
For a superadditive BGWPFRE, when random control functions are superadditive, we have guaranteed the existence sequence of \( {\left\{ {\sigma }_{{\xi }_{n}}\right\} }_{n = 0}^{\infty } \), such that \( {\sigma }_{k,{\xi }_{n}} \leq {\sigma }_{{\xi }_{n}}, k = 1,2,\cdots \) . | Proof Since the mating function \( L\left( {\cdot , \cdot }\right) \) and \( {\phi }_{n}\left( k\right) \) are superadditive, we have\n\n\[ \mathrm{E}\left( {{Z}_{n + 1}^{2} \mid {Z}_{n} = k + j,{\xi }_{n} = \theta }\right) \]\n\n\[ = \mathrm{E}\left( {\left. {\left( L\left( \mathop{\sum }\limits_{{i = 1}}^{{\phi }_{n}... | Yes |
Lemma 3. \( {\mathbf{2}}^{\left\lbrack {10}\right\rbrack } \) Let \( \left\{ {{X}_{n};n \geq 1}\right\} \) be an ND sequence with \( \mathrm{E}{X}_{n} = 0 \) and \( \mathrm{E}{\left| {X}_{n}\right| }^{p} < \infty, p \) \( \geq 2 \) . Then for \( {B}_{n} = \mathop{\sum }\limits_{{i = 1}}^{n}\mathrm{E}{X}_{i}^{2} \) , | \[ \mathrm{E}{\left| {S}_{n}\right| }^{p} \leq {c}_{p}\left\{ {\mathop{\sum }\limits_{{i = 1}}^{n}\mathrm{E}{\left| {X}_{i}\right| }^{p} + {B}_{n}^{p/2}}\right\} ,\] (3.1) \[ \mathrm{E}\left( {\mathop{\max }\limits_{{1 \leq i \leq n}}{\left| {S}_{i}\right| }^{p}}\right) \leq {c}_{p}{\log }^{p}n\left\{ {\mathop{\sum }\l... | Yes |
Lemma 2.4 \( f : {H}_{1} \times \left\lbrack {{t}_{0},{t}_{0} + \sigma }\right\rbrack \rightarrow {\mathbf{R}}^{n}, t \rightarrow f\left( {{y}_{t}, t}\right) \) is for \( y \in P{C}_{1} \) Lebesgue integrable over \( \left\lbrack {{t}_{0},{t}_{0} + \sigma }\right\rbrack \) and (i) (ii) \( \left( {\mathrm{i}}^{\prime }\... | Proof We can prove it easily by using Theorem 4.3 in [1]. | No |
Lemma 2. \( {\mathbf{5}}^{\left\lbrack 3\right\rbrack } \) Let \( \bar{O} \subset X \), assume that \( F : \bar{O} \times \left\lbrack {a, b}\right\rbrack \rightarrow X \) belongs to the class \( \mathrm{F}(\bar{O} \times \lbrack a \) , \( b\rbrack, h,\omega ) \) . If \( x, y : \left\lbrack {a, b}\right\rbrack \rightar... | \[ \begin{Vmatrix}{{\int }_{a}^{b}D\left\lbrack {F\left( {x\left( \tau \right), t}\right) - F\left( {y\left( \tau \right), t}\right) }\right\rbrack }\end{Vmatrix} \leq {\int }_{a}^{b}\omega \left( {\parallel x\left( t\right) - y\left( t\right) \parallel }\right) \mathrm{d}h\left( t\right) . \] | Yes |
Theorem 4.2 Assume that \( T > 0,{\varepsilon }_{0} > 0, L > 0, X = {H}_{1} \subset {PC}\left( {\left\lbrack {-r,0}\right\rbrack ,{\mathbf{R}}^{n}}\right) \) . Consider functions \( f : X \times \lbrack 0, + \infty ) \rightarrow {\mathbf{R}}^{n} \) and \( g : X \times \lbrack 0, + \infty ) \times \left( {0,{\varepsilon... | Proof By Lemma 2. 4, (1.1) is equivalent to\n\n\[ \frac{\mathrm{d}x}{\mathrm{\;d}\tau } = D\left\lbrack {{\varepsilon F}\left( {x, t}\right) }\right\rbrack ,\]\n\nwhere \( F\left( {x, t}\right) = {\int }_{0}^{t}f\left( {{x}_{s}, s}\right) \mathrm{d}s + \mathop{\sum }\limits_{{k = 1}}^{\infty }{I}_{k}\left( x\right) {H}... | Yes |
Lemma 2. \( {\mathbf{1}}^{\left\lbrack 3,9\right\rbrack } \) Let \( {X}_{0} \) denote the closed subspace of all constant functions in \( {W}^{1, p}\left( \Omega \right) \) . Let \( X \) be the quotient space \( {W}^{1, p}\left( \Omega \right) /{X}_{0} \) . For \( u \in {W}^{1, p}\left( \Omega \right) \), define the ma... | \[ \parallel u - {Pu}{\parallel }_{p} \leq C\parallel \nabla u{\parallel }_{{\left( {L}^{p}\left( \Omega \right) \right) }^{N}}. \] | Yes |
Lemma 2. \( {\mathbf{3}}^{\left\lbrack 3\right\rbrack } \) The mapping \( {\Phi }_{p} : {W}^{1, p}\left( \Omega \right) \rightarrow \mathbf{R} \) defined by \( {\Phi }_{p}\left( u\right) = {\int }_{\Gamma }{\varphi }_{x}\left( {u{\left. \right| }_{\Gamma }\left( x\right) }\right) \mathrm{d}\Gamma \left( x\right) \), fo... | From Lemma 1.5, we know that \( \partial {\Phi }_{p} : {W}^{1, p}\left( \Omega \right) \rightarrow {\left( {W}^{1, p}\left( \Omega \right) \right) }^{ * } \) is maximal monotone. | No |
Lemma 2.4 Define the mapping \( F : {W}^{1, p}\left( \Omega \right) \rightarrow {\left( {W}^{1, p}\left( \Omega \right) \right) }^{ * } \) by\n\n\[ \left( {v,{Fu}\left( x\right) }\right) = {\int }_{\Omega }g\left( {x, u\left( x\right) ,\varepsilon \nabla u\left( x\right) }\right) v\left( x\right) \mathrm{d}x, \]\n\nfor... | Proof Similar to the proof of Lemma 3. 3 in [10], we can obtain the result. | No |
Theorem 2.2 For \( f \in {L}^{p}\left( \Omega \right) \), there exists a unique \( u \in {L}^{p}\left( \Omega \right) \) satisfying \( f = \bar{H}u + {A}_{p}u \) . And, this \( u\left( x\right) \) is just the unique solution of (1.4). | Proof From (b), we know that \( \parallel {Hu}{\parallel }_{2}^{2} = {\int }_{\Omega }{\left| g\left( x, u,\varepsilon \nabla u\right) \right| }^{2}\mathrm{\;d}x \leq \) Const. \( (\parallel u{\parallel }_{2}^{2} + \) \( \parallel h\left( x\right) {\parallel }_{2}^{2} \) ), which implies that \( H \) is bounded.\n\nFro... | Yes |
If we set \( \alpha \left( t\right) = 1 + t{\left( 1 + {t}^{2}\right) }^{-1/2}, t \geq 0 \), then it is obvious that \( \alpha : {\mathbf{R}}^{ + }\bigcup \{ 0\} \) \( \rightarrow {\mathbf{R}}^{ + } \) is a continuous nonlinear mapping and \( \alpha \left( t\right) \leq 2 \) . And \( \mathop{\lim }\limits_{{t \rightarr... | \[ {pt}{\alpha }^{\prime }\left( t\right) + \left( {p - 1}\right) \alpha \left( t\right) = \frac{pt}{{\left( 1 + {t}^{2}\right) }^{3/2}} + \left( {p - 1}\right) + \left( {p - 1}\right) \frac{t}{\sqrt{1 + {t}^{2}}} > 0. \] | Yes |
For \( 1 < p \leq 2 \), if we set \( \alpha \left( t\right) = {\left( C + {t}^{2/p}\right) }^{\left( {p - 2}\right) /2}{t}^{\left( {2 - p}\right) /p}, t > 0, C \geq 0 \), then it is obvious that \( \alpha : {\mathbf{R}}^{ + } \rightarrow {\mathbf{R}}^{ + } \) is a continuous nonlinear mapping and \( \alpha \left( t\rig... | \[ {pt}{\alpha }^{\prime }\left( t\right) + \left( {p - 1}\right) \alpha \left( t\right) = {\left( C + {t}^{2/p}\right) }^{p/2 - 2}{t}^{2/p - 2}\left\lbrack {{tC}\left( {2 - p}\right) + \left( {p - 1}\right) t\left( {C + {t}^{2/p}}\right) }\right\rbrack \] \[ = {\left( C + {t}^{2/p}\right) }^{p/2 - 2}{t}^{2/p - 2}\left... | Yes |
Example 3.4 For \( s \leq 0 \), if we set \( \alpha \left( t\right) = {\left( 1 + {t}^{2/p}\right) }^{s/2}{t}^{\left( {m - p + 1}\right) /p}, t > 0, m \geq 0, m + s + 1 = \) \( p \), then it is obvious that \( \alpha : {\mathbf{R}}^{ + } \rightarrow {\mathbf{R}}^{ + } \) is a continuous nonlinear mapping and \( \alpha ... | \[ {pt}{\alpha }^{\prime }\left( t\right) + \left( {p - 1}\right) \alpha \left( t\right) = {t}^{\left( {m - p + 1}\right) /p}{\left( 1 + {t}^{2/p}\right) }^{s/2 - 1}\left\lbrack {m + \left( {p - 1}\right) {t}^{2/p}}\right\rbrack > 0. \] | Yes |
Lemma 1. \( {\mathbf{2}}^{\left\lbrack 3\right\rbrack } \) Let \( \mathrm{A} \) be a Banach algebra with a unit \( e, x \in \mathrm{A} \), then \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{\begin{Vmatrix}{x}^{n}\end{Vmatrix}}^{1/n} \) exists and the spectral radius \( \rho \left( x\right) \) satisfies | \[ \rho \left( x\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}{\begin{Vmatrix}{x}^{n}\end{Vmatrix}}^{1/n} = \inf {\begin{Vmatrix}{x}^{n}\end{Vmatrix}}^{1/n}. \] | Yes |
Lemma 2.1 Let \( \mathrm{A} \) be a Banach algebra and \( k \in \mathrm{A} \) . If \( \rho \left( k\right) < 1 \), then \( \mathop{\lim }\limits_{{n \rightarrow \infty }}\begin{Vmatrix}{k}^{n}\end{Vmatrix} = 0 \) . | Proof Since \( \rho \left( k\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}{\begin{Vmatrix}{k}^{n}\end{Vmatrix}}^{1/n} < 1 \), then there exists \( \alpha > 0 \) such that \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{\begin{Vmatrix}{k}^{n}\end{Vmatrix}}^{1/n} < \) \( \alpha < 1 \) . Letting \( n \) be b... | Yes |
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