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Lemma 3.1 \( {}^{\left\lbrack 1\right\rbrack }\;\{ 1,2,\cdots \} \smallsetminus {\mathcal{T}}_{6} \) consists of exactly 21 integers, namely | \( {\mathcal{T}}_{6} = \{ 1,2,\cdots \} \smallsetminus \{ 1,2,3,4,5,7,8,9,{11},{12},{15},{17},{20},{21},{23},{29},{36},{39},{41},{44},{84}\} . \) | Yes |
Theorem 3.1 \( {\mathcal{T}}_{6} = \mathbb{N} \smallsetminus \{ 1,2,3,4,5,7\} \) . | Proof From Lemma 3.1 we have known \( \{ {p}^{2} - {q}^{2} - {r}^{2} - {s}^{2} : p > \max \{ q + r, q + s, r + \) \( s\}, p, q, r, s \in \mathbb{N}\} = \mathbb{N} \smallsetminus \{ 1,2,3,4,5,7,8,9,{11},{12},{15},{17},{20},{21},{23},{29},{36},{41},{44},{84}\} \) . By Lemma \( {3.2} \) we find \( \{ 8,9,{11},{12},{15},{1... | Yes |
Theorem 5.1 \( {\mathcal{T}}_{8} = \left\{ {{p}^{2} - {q}^{2} - {r}^{2} - {s}^{2} - {u}^{2} - {v}^{2} : p > \max \{ s + u + v, q + r + v, q + s\}, q > }\right. \) \( r, s > u\} \), where \( p, q, r, s, u, v \in \mathbb{N} \) . | Proof If an octagon \( \mathcal{B} \) can be dissected into isosceles right triangles, then the size of every inner angle of \( \mathcal{B} \) is \( \frac{3\pi }{4} \) (see Fig. 19). Let \( \left| {AH}\right| = a,\left| {BC}\right| = b,\left| {CD}\right| = m,\left| {DE}\right| = c,\left| {GF}\right| = d \) , \( \left| ... | Yes |
Theorem 0.3 Let \( \left( {X, h}\right) \) be a compact Hermitian manifold with real bisectional curvature bounded from above by a positive constant \( A \) . If \( X \) admits a Kähler metric \( {\omega }_{0} \), then the Kähler-Ricci flow running from \( {\omega }_{0} \) | \[ \frac{\mathrm{d}}{\mathrm{d}t}\omega = - \operatorname{Ric}\left( \omega \right) ,{\left. \;\omega \right| }_{t = 0} = {\omega }_{0} \] \( \left( {0.3}\right) \) has a smooth solution on \( X \times \lbrack 0,\frac{1}{aA}) \) where \( a = \mathop{\sup }\limits_{X}{\operatorname{tr}}_{{\omega }_{0}}h > 0 \) . | Yes |
Lemma 3.1 Let \( {H}_{i} \in \left( {0,1}\right) ,{K}_{i} \in (0,1\rbrack, i = 1,2 \) ,\n\n\[ \n{\lambda }_{t} : = \operatorname{Var}\left( {{S}_{t}^{{H}_{1},{K}_{1}} - {S}_{t}^{{H}_{2},{K}_{2}}}\right) = \left( {{2}^{{K}_{1}} - {2}^{2{H}_{1}{K}_{1} - 1}}\right) {t}^{2{H}_{1}{K}_{1}} + \left( {{2}^{{K}_{2}} - {2}^{2{H}... | Proof Since \( {S}^{{H}_{1},{K}_{1}} \) and \( {S}^{{H}_{2},{K}_{2}} \) are two independent sub-bifractional Brownian motions, for any \( \xi ,\eta \in \mathbb{R} \) ,\n\n\[ \n\operatorname{Var}\left\lbrack {\xi \left( {{S}_{t}^{{H}_{1},{K}_{1}} - {S}_{t}^{{H}_{2},{K}_{2}}}\right) + \eta \left( {{S}_{s}^{{H}_{1},{K}_{1... | Yes |
Lemma 3.2 For any \( x \in \lbrack - 1,1) \), we have\n\n\[ \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{\left( {{2n} - 1}\right) !!}{\left( {{2n} - 2}\right) !!}{x}^{n} = x{\left( 1 - x\right) }^{-\frac{3}{2}} \]\n\nwhere \( \left( {{2n} - 1}\right) !! = 1 \cdot 3 \cdot 5\cdots \left( {{2n} - 1}\right) \) and \( \le... | We have, for all \( t, s \geq 0 \) and \( s \neq t \),\n\n\[ \frac{{\rho }_{s, t}^{2}}{{\left( {\lambda }_{t}{\lambda }_{s} - {\rho }_{s, t}^{2}\right) }^{\frac{3}{2}}} = {\left( \frac{1}{{\lambda }_{t}{\lambda }_{s}}\right) }^{\frac{1}{2}}\frac{{\rho }_{s, t}^{2}}{{\lambda }_{t}{\lambda }_{s}}{\left( 1 - \frac{{\rho }... | Yes |
Lemma 3.2 For any \( 0 \leq t \leq \min \left\{ {{T}^{\varepsilon },1}\right\} \)\n\n\[ \n\parallel \left( {\sigma, u,\theta }\right) {\parallel }_{{L}^{2}}^{2} + \parallel \left( {u,\theta }\right) {\parallel }_{{L}_{t}^{2}\left( {H}^{1}\right) }^{2}\mathrm{\;d}s \leq {C}_{0}\left( {M}_{0}\right) \exp \left( {\sqrt{t}... | Proof Multiplying (0.8)-(0.10) by \( {R\sigma }, u \) and \( \theta \), respectively, and integrating over \( \Omega \times \left( {0, t}\right) \) , we obtain\n\n\[ \n\frac{1}{2}\parallel \left( {\sqrt{R}\sigma ,\sqrt{\rho }u,\sqrt{{C}_{v}\rho }\theta }\right) {\parallel }_{{L}^{2}}^{2} + {\int }_{0}^{t}\parallel \lef... | Yes |
Lemma 3.3 For any \( 0 \leq t \leq \min \left\{ {{T}^{\varepsilon },1}\right\} \) , \n\n\[ \n\parallel \left( {\nabla \sigma ,\mathrm{{div}}\left( u\right) ,\nabla \theta }\right) {\parallel }_{{L}^{2}}^{2} + \parallel \left( {\nabla \mathrm{{div}}\left( u\right) ,{\Delta \theta }}\right) {\parallel }_{{L}_{t}^{2}\left... | Proof We integrate the product of \( \nabla \operatorname{div}\left( u\right) \) and (0.9) over \( \Omega \times \left( {0, t}\right) \) to show \n\n\[ \n\frac{1}{2}\parallel \sqrt{\rho }\operatorname{div}\left( u\right) {\parallel }_{{L}^{2}}^{2} + \left( {\mu + \lambda }\right) \parallel \nabla \operatorname{div}\lef... | Yes |
Lemma 3.8 For any \( 0 \leq t \leq \min \left\{ {{T}^{\varepsilon },1}\right\} \), there exist positive constants \( {\mu }_{1} \) and \( {\kappa }_{1} \), such that \[ \frac{1}{2}{\int }_{\widetilde{\Omega }}J{\chi }^{2}\widetilde{\rho }{\left| {D}_{\zeta \tau }\widetilde{u}\right| }^{2}\mathrm{\;d}y + \frac{R}{2}{\in... | Proof Applying \( {D}_{\zeta \tau } \) to (3.21) and taking the product of the resulting equation and \( J{\chi }^{2}{D}_{\zeta \tau }{\widetilde{u}}_{i} \), we obtain \[ \frac{1}{2}{\int }_{\widetilde{\Omega }}J{\chi }^{2}\widetilde{\rho }{\left| {D}_{\zeta \tau }\widetilde{u}\right| }^{2}\mathrm{\;d}y + \frac{R}{\var... | Yes |
Lemma 3.12 For any \( 0 \leq t \leq \min \left\{ {{T}^{\varepsilon },1}\right\} \) | \[ \begin{aligned} {\begin{Vmatrix}{\chi }_{0}{\nabla }^{2}u\end{Vmatrix}}_{{L}^{2}}^{2} + {\int }_{\widetilde{\Omega }}J{\chi }^{2}\widetilde{\rho }{\left| {D}_{\zeta \tau }\widetilde{u}\right| }^{2}\mathrm{d}y & + {\begin{Vmatrix}\left( {\nabla }^{2}\sigma ,{\nabla }^{2}\theta \right) \end{Vmatrix}}_{{L}^{2}}^{2} \\ ... | Yes |
Lemma 1.2 \( {}^{\left\lbrack {16}\right\rbrack } \) Assume that \( A \) is an \( n \times n \) matrix. Let \( {s}_{\left( r,{1}^{n - r}\right) } \) be the Schur polynomial corresponding to the partition \( \left( {r,{1}^{n - r}}\right) \) . The identity for single-hook immanants | \[ {d}_{r + 1}\left( A\right) = \mathop{\sum }\limits_{{A}_{r}}{d}_{\left( r\right) }\left( {A}_{r}\right) {d}_{\left( {1}^{n - r}\right) }\left( {A}_{r}^{\prime }\right) - {d}_{r}\left( A\right) ,\] i.e., \[ {d}_{r + 1}\left( A\right) = \mathop{\sum }\limits_{{A}_{r}}\operatorname{per}\left( {A}_{r}\right) \det \left(... | Yes |
Lemma 1.4 \( \;\operatorname{per}L\left( {K}_{n}^{r}\right) = \left( {n - r}\right) \cdot \operatorname{per}L\left( {K}_{n}^{r - 1}\right) + n \cdot \left( {r - 1}\right) \cdot \operatorname{per}L\left( {K}_{n}^{r - 2}\right) . \) | Proof By Lemma 1.3, expanding per \( L\left( {K}_{n}^{r}\right) \) along the first row, we have\n\n\[ \operatorname{per}L\left( {K}_{n}^{r}\right) = \left( {n - 1}\right) \cdot \operatorname{per}L\left( {K}_{n}^{r - 1}\right) - \mathop{\sum }\limits_{{t = 2}}^{r}\operatorname{per}L\left( {K}_{n}^{r}\right) \left( {1 \m... | Yes |
Lemma 1. \( {6}^{\left\lbrack 9\right\rbrack } \) Let \( G \) be a connected graph on \( n \) vertices with Laplacian eigenvalues \( {\lambda }_{n} \geq \) \( {\lambda }_{n - 1} \geq \cdots \geq {\lambda }_{2} \geq {\lambda }_{1} = 0 \) . The number of spanning trees \( \tau \left( G\right) \) of \( G \) is given by | \[ \tau \left( G\right) = \frac{1}{n}\mathop{\sum }\limits_{{i = 2}}^{n}{\lambda }_{i} \] | Yes |
Theorem 1.1\n\n\[ \n{d}_{r}\left( {L\left( {K}_{n}\right) }\right) = \left\{ \begin{array}{ll} 0, & r = 1 \\ {n}^{n}\left( {n - 1}\right) , & r = 2 \\ \left( \begin{matrix} n \\ r - 1 \end{matrix}\right) \cdot {n}^{n - r} \cdot \left( {r - 1}\right) \cdot \left( {n - r - 1}\right) \cdot \operatorname{per}L\left( {K}_{n... | Proof When \( r = 1,2 \), the results hold from Lemmas 1.5-1.7.\n\nIn the sequel we shall consider the case that \( 3 \leq r \leq n \) . It is evident that all principle submatrices \( L\left( {K}_{n}^{r}\right) \) of \( L\left( {K}_{n}\right) \) have the same permanents while their complement submatrices \( L\left( {K... | Yes |
Theorem 1.2 \( {d}_{r + 1}\left( {A\left( {K}_{n}\right) }\right) = \mathop{\sum }\limits_{{i = 1}}^{r}\left( \begin{aligned} n \\ i \end{aligned}\right) \cdot {\left( -1\right) }^{n - i - 1}\left( {n - i - 1}\right) {\delta }_{i} + {\left( -1\right) }^{n}\left( {n - 1}\right) \), where \( {\delta }_{i} \) is the \( i ... | Proof Assume that \( {A}_{i} \) is the \( i \times i \) principal submatrix of \( A\left( {K}_{n}\right) \) and \( {A}_{i}^{\prime } \) is the \( \left( {n - i}\right) \times \left( {n - i}\right) \) complement principal submatrix of \( {A}_{i} \) . By Lemma 1.2, we have\n\n\[ \n{d}_{2}\left( A\right) = \mathop{\sum }\... | Yes |
Theorem 2.1 Let \( {C}_{n} \) be the cycle on \( n \) vertices. Then\n\n(a) \( \operatorname{per}\left( {L\left( {C}_{n}\right) }\right) = 2 \cdot \operatorname{per}\left( {A}_{n - 1}\right) + 2 \cdot \operatorname{per}\left( {A}_{n - 2}\right) + 2 \cdot {\left( -1\right) }^{n - 2} \), where\n\n\[ \n{A}_{n} = \left( \b... | Proof The items (a) and (c) can be easily verified from Lemma 1.3. To verify the explicit expression (b), it suffices to calculate \( \operatorname{per}\left( {A}_{n - 1}\right) \) and \( \operatorname{per}\left( {A}_{n - 2}\right) \) .\n\nBy Lemma 1.3, it follows that\n\n\[ \n\operatorname{per}\left( {A}_{k}\right) = ... | Yes |
Lemma 0.1 An optimal balanced \( \left( {v, W,1}\right) - \mathrm{{OOC}} \) is equivalent to an optimal balanced \( \left( {v, W,1}\right) \) -CDP. | In view of design theory, the following result is clear. | No |
Lemma 1.2 There exists a balanced \( h \) -perfect \( \left( {{vh},{gh},\{ 3,4\} ,1}\right) \) -RDF for \( \left( {v, g, h}\right) = \) \( \left( {{24},6,1}\right) ,\left( {{24},6,2}\right) ,\left( {{128},2,1}\right) ,\left( {{128},2,2}\right) \) . | Proof The desired balanced \( h \) -perfect \( \left( {{vh},{gh},\{ 3,4\} ,1}\right) \) -RDFs are displayed below:\n\n\( \left( {v, g, h}\right) = \left( {{24},6,1}\right) : \{ 0,1,3,{10}\} ,\{ 0,5,{11}\} .\n\n\( \left( {v, g, h}\right) = \left( {{24},6,2}\right) : \{ 0,1,6,{35}\} ,\{ 0,3,{26},{33}\} ,\{ 0,2,{11}\} ,\{... | Yes |
Lemma 2.1 If \( v \) is a positive integer whose prime factors are congruent to 1 (mod 6) and no less than 7, then there is a balanced \( \left( {{6v},6,\{ 3,4\} ,1}\right) \) -RDF. | Proof When \( v \equiv 7\left( {\;\operatorname{mod}\;{12}}\right) \), the conclusion comes from Theorem 6.2 of [39].\n\nWhen \( v \equiv 1\left( {\;\operatorname{mod}\;{12}}\right) \), let \( v = {p}_{1}^{{a}_{1}}{p}_{2}^{{a}_{2}}\cdots {p}_{l}^{{a}_{l}} \) be the factorization of \( v \) where each \( {p}_{i} \geq 7 ... | Yes |
Lemma 2.2 If \( v \) is a positive integer whose prime factors are congruent to 1 (mod 6) and no less than 7, then there exists a balanced \( \left( {3 \cdot {2}^{a}v,3 \cdot {2}^{a},\{ 3,4\} ,1}\right) \) -RDF, where \( a = 0,1,2 \) . | Proof When \( a = 1 \), the conclusion comes from Lemma 2.1. When \( a \neq 1 \), the conclusion comes from Theorem 6.4 of [39]. | No |
Lemma 2.3 There exists a balanced \( \left( {3 \cdot {2}^{a},{6g},\{ 3,4\} ,1}\right) \) -RDF for any integer \( a \geq 3 \), where\n\n\[ g = \left\{ \begin{array}{ll} 1, & \text{ if }a\text{ is odd; } \\ 2, & \text{ if }a\text{ is even. } \end{array}\right. \] | Proof Case 1: \( a = 3,4 \) . The conclusion is from Lemma 1.2.\n\nCase 2: \( a = 5 \) . A balanced \( \left( {{96},6,\{ 3,4\} ,1}\right) \) -RDF is displayed below:\n\n\( \{ 0,{28},{49}\} ,\{ 0,{38},{60}\} ,\{ 0,6,{15}\} ,\{ 0,8,{41},{67}\} ,\{ 0,2,{54}\} ,\{ 0,{12},{25},{65}\} ,\{ 0,4,5,{23}\} \) ,\n\n\( \{ 0,{10},{3... | Yes |
Lemma 3.2 Let \( p \equiv 1\left( {\;\operatorname{mod}\;{18}}\right) \) be a prime. If there exist two elements \( x, y \) of \( {\mathbb{Z}}_{p} \) satisfying one of the following conditions:\n\n(1) \( \{ x, x - 2\} \) are in different cosets among \( \left\{ {{C}_{0}^{9},{C}_{3}^{9}}\right\}, x - 1 \in {C}_{3}^{9}, ... | Proof First, we construct a balanced \( \left( {4,\{ 3,4\} ,{18}}\right) \) -DF as follows:\n\n\[ \{ 0,1,2,3\} ,\{ 0,1,2,3\} ,\{ 0,1,2,3\} ,\{ 0,1,2\} ,\{ 0,1,2\} ,\{ 0,1,2\} . \]\n\nSince \( p \equiv 1\left( {\;\operatorname{mod}\;{18}}\right) \) is a prime, we can identity \( {\mathbb{Z}}_{4} \times {\mathbb{Z}}_{p} ... | Yes |
Lemma 3.3 Let \( p \equiv 1\\left( {\\operatorname{mod}{18}}\\right) \) be a prime, then there exists a balanced \( \\left( {{4p},4,\\{ 3,4\\} ,1}\\right) \) - RDF. | Proof Applying Lemma 3.1 with \( m = 9 \) and \( s = 3 \), one can find two elements \( x, y \) of \( {\\mathbb{Z}}_{p} \) satisfying Condition (1) in Lemma 3.2 when \( p > {1478656} \). Then we have the conclusion. | No |
Lemma 3.4 If \( n \geq 0 \) is an integer and \( a \in \{ 1,2,3,4,5,6\} \), then there is a balanced \( \left( {{2}^{{6n} + a},{2}^{a},\{ 3,4\} ,1}\right) \) -RDF. | Proof We use induction on \( n \) .
Case 1: \( n = 0 \) . The conclusion is trivial.
Case 2: \( n = 1 \) . When \( a = 1,2 \), the conclusion comes from Lemma 1.2. When \( a = 3 \), the conclusion comes from Appendix III. When \( a = 4,5,6 \), a balanced 1-perfect \( \left( {{2}^{7},2,\{ 3,4\} ,1}\right) \) -RDF and ... | Yes |
Lemma 3.5 If \( v \) is a positive integer whose prime factors are congruent to 1 (mod 18) and no less than 19, then there exists a balanced \( \left( {{2}^{a}v,{2}^{a},\{ 3,4\} ,1}\right) \) -RDF, where \( a = 0,1,2 \) . | Proof When \( a = 1 \), the conclusion is from Theorem 1.11 of [40].\n\nWhen \( a \neq 1 \) . Let \( v = {p}_{1}^{{a}_{1}}{p}_{2}^{{a}_{2}}\cdots {p}_{l}^{{a}_{l}} \) be the factorization of \( v \) where each \( {p}_{i} \geq {19} \) is a prime and each integer \( {a}_{i} \geq 1 \) . For each prime \( {p}_{i} \), there... | Yes |
Lemma 3.6 There exists an optimal balanced \( \left( {{2}^{a},\{ 3,4\} ,1}\right) \) -CDP for \( a = 1,2,3,4,5,6 \) . | Proof When \( a = 1,2,3,4 \), the conclusion is trivial. When \( a = 5,6 \), the desired optimal balanced \( \left( {{2}^{a},\{ 3,4\} ,1}\right) \) -RDFs are displayed below:\n\n\[ a = 5 : \{ 0,1,3,7\} ,\{ 0,5,{13}\} \text{.} \]\n\n\[ a = 6 : \{ 0,1,3,7\} ,\{ 0,5,{13},{22}\} ,\{ 0,{10},{21},{33}\} ,\{ 0,{14},{29}\} ,\{... | Yes |
Theorem 2.1 Let \( {C}_{6} \) be a cyclic group of order 6 generated by \( \sigma .{K}_{2}\left( {\mathbb{Z}\left\lbrack {C}_{6}\right\rbrack }\right) \) is an elementary abelian group of rank 3 with generators \( \{ \sigma , - 1\} ,\{ - \sigma , - 1\} ,\left\langle {\sigma \left( {{\sigma }^{3} - 1}\right) ,\sigma \le... | Proof According to [3, Theorem 5.4], the order of \( {K}_{2}\left( {\mathbb{Z}\left\lbrack {C}_{6}\right\rbrack }\right) \) is at least 8, then the result follows from [5, Proposition 2.10]. | No |
Corollary 2.1 Let \( {C}_{6} \) be a cyclic group of order 6 generated by \( \sigma .W{h}_{2}\left( {C}_{6}\right) \) is a cyclic of order 2 with generator \( \left\langle {\sigma \left( {{\sigma }^{3} - 1}\right) ,\sigma \left( {{\sigma }^{3} + 1}\right) }\right\rangle \) . | Proof This follows immediately from the definition of \( W{h}_{2}\left( G\right) \) . | No |
Corollary 2.2 Let \( {C}_{2} \) be a cyclic group of order 2 generated by \( \tau \) . Then \( {K}_{2}\left( {\mathbb{Z}\left\lbrack {\zeta }_{3}\right\rbrack \left\lbrack {C}_{2}\right\rbrack }\right) \) is a cyclic group of order 2 with generators \( \left\langle {{\zeta }_{3}\left( {\tau - 1}\right) ,{\zeta }_{3}\le... | Proof Let \( {C}_{3} \times {C}_{2} = \left\langle {\omega ,\tau \mid {\omega }^{3} = {\tau }^{2} = 1,{\omega \tau } = {\tau \omega }}\right\rangle \) . Then the square\n\n\n\n(2.1)\n\nis Cartesian, where the maps \(... | Yes |
Theorem 2.1 Suppose \( \Lambda = {k\Gamma }/I \) is a monomial algebra, and \( c \) is a cycle. Then the dimension tree \( \Delta \left( c\right) \) is infinite if one of the following is true:\n\n(1) \( c \) is a loop;\n\n(2) \( c \) is not a loop such that any cycle shifted from \( c \) lies outside the ideal \( I \)... | Proof (1) There is a positive integer, say \( n \geq 1 \), smallest with respect to \( {c}^{n + 1} = 0 \) . Now, it is easy to check \( {c}^{n} \in \mathcal{L}\left( c\right) \) and \( c \in \mathcal{L}\left( {c}^{n}\right) \) . Therefore, with Topdown, one sees the dimension tree \( \Delta \left( c\right) \) contains ... | Yes |
Theorem 1.1.2. Let \( {X}_{1},\ldots ,{X}_{m} \) be \( m \) vector fields on \( M \) such that \( {\left( {X}_{1}\right) }_{x} \) , \( \ldots ,{\left( {X}_{m}\right) }_{x} \) form a basis of \( {T}_{xc}\left( M\right) \) for each \( x \in M \) . For any multiindex \( \left( \alpha \right) = \) \( \left( {{\alpha }_{1},... | Proof. For any integer \( r \geq 0 \), let \( {\mathfrak{D}}_{r} \) denote the complex vector space of all differential operators on \( M \) of the form \( \mathop{\sum }\limits_{{\left| \alpha \right| \leq r}}{f}_{\left( \alpha \right) }{X}^{\left( \alpha \right) } \), the \( {f}_{\left( \alpha \right) } \) being \( {... | Yes |
Theorem 1.1.4. Let \( \pi \) be a regular imbedding of \( M \) into \( N \) . Then \( \pi \left\lbrack M\right\rbrack \) is locally closed in \( N \) . For each \( x \in M \), we can choose \( U, V,{x}_{1},\ldots ,{x}_{m},{y}_{1},\ldots \) , \( {y}_{n} \) such that, in addition to (1.1.24), we have\n\n\[ \pi \left\lbra... | Proof. Let \( {U}^{\prime },{V}^{\prime },{x}_{1},\ldots ,{x}_{m},{y}_{1},\ldots ,{y}_{n} \), and \( {a}^{\prime } > 0 \) be such that the relations (1.1.24) are satisfied (with \( {U}^{\prime },{V}^{\prime } \), and \( {a}^{\prime } \) replacing \( U, V \), and \( a \) , respectively). Since \( \pi \) is a homeomorphi... | Yes |
Theorem 1.1.5. Let \( N \) be a \( {C}^{\infty } \) manifold and let \( M \subseteq N \). In order that \( M \), equipped with the relative topology, be a (regular) submanifold of \( N \), it is necessary and sufficient that the following be satisfied. There exists an integer \( m \) with \( 1 \leq m \leq n \) such tha... | Proof. The only thing that needs to be proved is that if \( M \) satisfies the conditions described above, then it becomes a regular submanifold of \( N \) ; Theorem 1.1.4 implies the remaining assertions. Also if \( m = n,\left( {1.1.27}\right) \) reduces to the condition \( V \subseteq M \), so that in this case \( M... | Yes |
Lemma 1.3.1. Let \( M \) be an analytic manifold, \( X \) any real analytic vector field on \( M \), and \( x \in M \) a point such that \( {X}_{x} \neq 0 \) . Then there are analytic coordinates \( {x}_{1},\ldots ,{x}_{m} \) around \( x \) such that \( {X}_{y} = {\left( \partial /\partial {x}_{1}\right) }_{y} \) for a... | Proof. Select analytic coordinates \( {z}_{1},\ldots ,{z}_{m} \) around \( x \) such that \( {z}_{1}\left( x\right) \) \( = \cdots = {z}_{m}\left( x\right) = 0 \) and \( {X}_{x}\left( {z}_{1}\right) \neq 0 \) . Then there are real analytic functions \( {G}_{1},\ldots ,{G}_{m} \), defined on \( {I}_{a}^{m} \) (for some ... | Yes |
Lemma 1.3.2. Let \( M \) be an analytic manifold, \( x \in M \), and let \( {X}_{1},\ldots ,{X}_{p} \) be real analytic vector fields defined on an open set \( U \) containing \( x \) such that \( \left( i\right) \) \( {\left( {X}_{1}\right) }_{y},\ldots ,{\left( {X}_{p}\right) }_{y} \) are linearly independent for \( ... | Proof. We prove this by induction on \( p \) . For \( p = 1 \) this follows at once from Lemma 1.3.1. Let \( 1 < p \leq m \), and assume the result for \( {X}_{1},\ldots ,{X}_{p - 1} \) . Then we can choose a connected open set \( V \) with \( x \in V \subseteq U \) and coordinates \( {u}_{1},\ldots ,{u}_{m} \) on \( V... | Yes |
Lemma 1.3.5. Let \( A \) be a connected Hausdorff space which is locally connected. Suppose \( A = \mathop{\bigcup }\limits_{{n = 1}}^{\infty }{A}_{n} \) where each \( {A}_{n} \) is open in \( A \) and each connected component of \( {A}_{n} \) is second countable for each \( n \) . Then \( A \) is itself second countab... | Proof. Let \( {\mathcal{C}}_{n} \) be the class of (open) sets which are connected components of \( {A}_{n} \), and \( \mathcal{C} = \mathop{\bigcup }\limits_{{n = 1}}^{\infty }{\mathcal{C}}_{n} \) . Since there cannot exist an uncountable family of mutually disjoint nonempty open sets in a second countable space it fo... | Yes |
Theorem 1.3.6. (Global Frobenius Theorem) Let \( M \) be an analytic manifold \( \mathfrak{L}\left( {x \mapsto {\mathfrak{L}}_{x}}\right) \) an involutive analytic system of tangent spaces of rank \( p \) . Given any point of \( M \), there is one and exactly one maximal integral manifold of \( \mathfrak{L} \) containi... | Proof. Let \( \mathfrak{I} \) be the collection of all subsets of \( M \) which are unions of integral manifolds of \( \mathcal{L} \) . It follows from Lemma 1.3.4 that 3 is a topology for \( M \) finer than its original topology. It is clear that \( \left( {M,\mathfrak{J}}\right) \) is a Hausdorff locally connected sp... | Yes |
Theorem 1.4.1. Let \( a > 0 \) and let \( {G}_{1},\ldots ,{G}_{m} \) be real functions defined and analytic on \( {I}_{a}^{m} \) . Then\n\n(a) if \( {u}_{j},{v}_{j}\left( {1 \leq j \leq m}\right) \) are analytic functions defined on an open interval \( \Delta \) containing 0 such that \( \left( {{u}_{1},\ldots ,{u}_{m}... | Proof. (a) If \( \left( {{\varphi }_{1},\ldots ,{\varphi }_{m}}\right) \) is solution of (1.4.1), we have, for \( 1 \leq j \leq m \) ,\n\n\[ {\varphi }_{j}^{\prime }\left( 0\right) = {G}_{j}\left( {{\varphi }_{1}\left( 0\right) ,\ldots ,{\varphi }_{m}\left( 0\right) }\right) \]\n\n\[ {\varphi }_{j}^{\prime \prime }\lef... | Yes |
Theorem 1.4.2. Let \( N \) be an analytic manifold, \( a > 0 \), and let the real functions \( {G}_{j} \) be defined and analytic on \( {I}_{a}^{m} \times N \) . Fix \( x \in N \) . Then we can find \( b \) with \( 0 < b < a \), an open subset \( {N}_{x} \) of \( N \) containing \( x \), and real analytic functions \( ... | Proof. We may assume that for some \( d > 0, N = {I}_{d}^{n}, x = \\left( {0,\\ldots ,0}\\right) \), and that the \( {G}_{j} \) are the restrictions to \( {I}_{a}^{m} \\times {I}_{d}^{n} \) of functions (denoted again by \( {G}_{j} \) ) defined and holomorphic on \( {J}_{a}^{m} \\times {J}_{d}^{n} \) . let \( 0 < c < a... | No |
Theorem 2.1.1 Let \( G \) be a Lie group, real or complex. Suppose \( H \) is a subgroup which is at the same time a quasi-regularly imbedded submanifold of G. Then \( H \), together with this analytic structure, is a Lie subgroup of \( G \) . If \( H \) is a regularly imbedded submanifold of \( G \), then \( H \) is c... | Proof. The map \( \left( {x, y}\right) \mapsto x{y}^{-1} \) of \( G \times G \) into \( G \) is analytic. Hence, by restriction, \( \varphi : \left( {x, y}\right) \mapsto x{y}^{-1} \) is an analytic map of \( H \times H \) into \( G \) . Since \( \varphi \left\lbrack {H \times H}\right\rbrack \subseteq H \) and \( H \)... | Yes |
Theorem 2.1.2. Let \( G \) be a real (resp. complex) algebraic group. Then \( G \) is a closed real (resp. complex) Lie subgroup of \( {GL}\left( {n,\mathbf{R}}\right) \) (resp. \( {GL}\left( {n,\mathbf{C}}\right) \) ). | Proof. We discuss only the real case; the complex case is treated along the same lines. Let \( \varphi \) be the algebra of all polynomials in the entries \( {a}_{ij} \) with real coefficients. Write \( U = {GL}\left( {n,\mathbf{R}}\right) \), and let \( G \subseteq U \) be an algebraic group. In view of Theorem 2.1.1,... | Yes |
Theorem 2.4.1. Let \( G \) be a Lie group, \( \mathfrak{g} \) its Lie algebra, \( \mathfrak{G} \) its enveloping algebra. Suppose \( \left\{ {{X}_{1},\ldots ,{X}_{n}}\right\} \) is any basis for \( \mathfrak{g} \) . For any \( n \) -tuple \( \left( {{r}_{1},\ldots ,{r}_{n}}\right) \) of integers \( \geq 0 \) let us def... | Proof. We work with real Lie groups; the complex case can be treated similarly. Now, \( {X}_{1},\ldots ,{X}_{n} \) are real analytic vector fields on \( G \) with the property that \( {\left( {X}_{1}\right) }_{a},\ldots ,{\left( {X}_{n}\right) }_{a} \) form a basis for \( {T}_{a}\left( G\right) \) for any \( a \in G \)... | Yes |
Lemma 2.5.1. Let \( G \) be a Lie group, \( \mathfrak{g} \) its Lie algebra, and \( \mathfrak{h} \) a subalgebra of \( \mathfrak{g} \). For any \( x \in G \), let \( {\mathfrak{L}}_{x}^{\mathfrak{h}} \) be the subspace of \( {T}_{x}\left( G\right) \) consisting of the set of all tangent vectors of the form \( {X}_{x}, ... | Proof. We work with real Lie groups; the complex case is analogous. Put \( \mathcal{L} = {\mathcal{L}}^{\mathfrak{h}} \), let \( p = \dim \mathfrak{h} \), and let \( {X}_{1},\ldots ,{X}_{n} \) be a basis for \( \mathfrak{g} \) such that\n\n\( {X}_{1},\ldots ,{X}_{p} \) span \( \mathfrak{h} \). Since \( \mathfrak{h} \) ... | Yes |
Theorem 2.5.2. The correspondence, which assigns to any analytic subgroup of \( G \) the subalgebra of \( \mathfrak{g} \) defined by it, is a bijection of the set of all analytic subgroups of \( G \) onto the set of all subalgebras of \( \mathfrak{g} \) . If \( \mathfrak{h} \subseteq \mathfrak{g} \) is a subalgebra, th... | Proof. Let \( \mathfrak{L} \) be as in Lemma 2.5.1. We can apply the Chevalley-Fro-benius theory of involutive systems (cf. §1.3) to it. Let \( H \) be the maximal integral manifold of \( \mathfrak{L} \) containing 1 .\n\nFrom the definition of \( \mathcal{L} \) it is clear that\n\n(2.5.6)\n\n\[{\left( d{l}_{x}\right) ... | Yes |
Lemma 2.5.3. Let \( A \) and \( B \) be locally compact second countable groups, and let \( \varphi \) be a continuous homomorphism of \( A \) onto \( B \) . Then \( \varphi \) is open. If \( \varphi \) is one-to-one, it is a homeomorphism. | Proof. Let \( {1}_{A} \) and \( {1}_{B} \) be the respective identities of \( A \) and \( B \) . Let \( V \) be an open set containing \( {1}_{A} \) . Select a compact neighborhood \( {V}_{1} \) of \( {1}_{A} \) such that \( {V}_{1}{V}_{1}^{-1} \subseteq V \) . Since \( A \) is second countable, we can find a sequence ... | Yes |
Theorem 2.5.4. Let \( G \) be a Lie group and \( H \) a Lie subgroup of \( G \). Then \( H \) is quasi-regularly imbedded in \( G \). Moreover, the following conditions on \( H \) are equivalent:\n\n(i) \( H \) is a topological subgroup of \( G \)\n\n(ii) \( H \) is regularly imbedded in \( G \)\n\n(iii) \( H \) is a c... | Proof. Let \( {H}^{0} \) be the component of the identity of \( H.{H}^{0} \) is quasi-regularly imbedded in \( G \), as we observed in the proof of Theorem 2.5.2. From this the quasi-regularity of \( H \) follows easily. For the rest of the theorem (i) \( \Rightarrow \) (ii) by definition, while (ii) \( \Rightarrow \) ... | Yes |
Corollary 2.6.2. Let \( {G}_{1} \) and \( {G}_{2} \) be admissible groups satisfying the second axiom of countability. Let\n\n\[ \pi : {G}_{1} \rightarrow {G}_{2} \]\n\nbe a continuous homomorphism whose kernel \( D \) is a discrete subgroup of \( {G}_{1} \) and which maps \( {G}_{1} \) onto \( {G}_{2} \) . Suppose \( ... | Proof. For definiteness, let \( {G}_{1} \) be an analytic group. Since \( D \) is discrete, we can select a compact neighborhood \( {K}_{1} \) of the identity \( {1}_{1} \) of \( {G}_{1} \) such that \( {K}_{1}{K}_{1}^{-1} \cap D = \left\{ {1}_{1}\right\} \) . Let \( {V}_{1} \) be the interior of \( {K}_{1} \) and \( V... | Yes |
Lemma 2.7.1. Let \( M, N \) be analytic manifolds and let \( \pi \) be a submersive analytic map of \( M \) onto \( N \) . Then \( \mathcal{L} : x \mapsto \operatorname{kernel}{\left( d\pi \right) }_{x}\left( {x \in M}\right) \) is an involutive analytic system of tangent spaces of rank equal to \( \dim \left( M\right)... | Proof. Let \( m = \dim \left( M\right), n = \dim \left( N\right) \) . Clearly, \( m = n \) if and only if \( {\left( d\pi \right) }_{x} \) is bijective for all \( x \in M \) . In this case \( \pi \) is a local diffeomorphism. If \( x \in M, y = \pi \left( x\right) \), and \( U \) is an open subset of \( M \) containing... | Yes |
Theorem 2.7.3. Let \( {G}_{j} \) be a Lie group with Lie algebra \( {\mathfrak{g}}_{j}\left( {j = 1,2}\right) \), and let \( \pi \) be an analytic homomorphism of \( {G}_{1} \) into \( {G}_{2} \) . Then for any \( X \in {\mathfrak{g}}_{1} \) there exists exactly one \( {X}^{\prime } \in {\mathfrak{g}}_{2} \) such that ... | Proof. Let \( {1}_{j} \) be the identity of \( {G}_{j}\left( {j = 1,2}\right) \) . It is obvious that given \( X \in {\mathfrak{g}}_{1} \), there is exactly one \( {X}^{\prime } \in {\mathfrak{g}}_{2} \) such that \( {X}_{{1}_{2}}^{\prime } = {\left( d\pi \right) }_{{1}_{1}}\left( {X}_{{1}_{1}}\right) .\pi \) being a h... | Yes |
Corollary 2.7.4. Let \( \pi \) be as above. Then \( {d\pi } \) is surjective if and only if \( \pi \left\lbrack {G}_{1}^{0}\right\rbrack = {G}_{2}^{0} \), and it is injective if and only if the kernel of \( \pi \) is discrete. | \[ \left( {d\pi }\right) {\left( X\right) }_{1}, = {\left( \frac{d}{dt}\left( \pi \circ x\right) \right) }_{t = 0} \] | No |
Corollary 2.7.6. Let \( G \) be a simply connected analytic group, \( \mathfrak{g} \) its Lie algebra. Suppose \( \xi \left( {X \mapsto {X}^{\xi }}\right) \) is an automorphism of \( \mathfrak{g} \) . Then there exists exactly one automorphism \( \eta \left( {x \mapsto {x}^{\eta }}\right) \) of \( G \) such that \( \xi... | Proof. By Theorem 2.7.5 there is a unique homomorphism \( \eta \) of \( G \) into itself such that \( \xi = {d\eta } \) . It remains to check that \( \eta \) is an automorphism. By Corollary 2.7.4, \( \eta \) is surjective and its kernel is discrete. \( G \) is therefore a covering group of itself, \( \eta \) being the... | Yes |
Lemma 2.8.1. Let \( {G}_{1} \) and \( {G}_{2} \) be two analytic groups. Then they are locally analytically isomorphic if and only if the following condition is satisfied: there exists a simply connected analytic group \( G \) and homomorphisms \( {\omega }_{j} \) of \( G \) into \( {G}_{j} \) such that \( {\omega }_{j... | Proof. Let \( {G}_{1} \) and \( {G}_{2} \) be locally analytically isomorphic and suppose that \( {U}_{1},{U}_{2} \), and \( \omega \) are as in the definition. Let \( G \) be a simply connected admissible covering group of \( {G}_{1} \) with covering homomorphism \( {\omega }_{1} \) . In view of the discussion in \( §... | Yes |
Theorem 2.8.2. Let \( {G}_{j} \) be Lie groups and \( {\mathfrak{g}}_{j} \) the corresponding Lie algebras, \( j = 1,2 \) . Then \( {\mathfrak{g}}_{1} \) and \( {\mathfrak{g}}_{2} \) are isomorphic if and only if \( {G}_{1} \) and \( {G}_{2} \) are locally analytically isomorphic. | Proof. We may assume that \( {G}_{1} \) and \( {G}_{2} \) are analytic groups. Suppose \( {G}_{1} \) and \( {G}_{2} \) are locally analytically isomorphic; then there is a simply connected analytic group \( G \) and analytic homomorphisms \( {\omega }_{j} \) of \( G \) onto \( {G}_{j} \) with discrete kernels. Let \( \... | Yes |
Let \( G \) be a Lie group acting transitively and analytically on an analytic manifold \( M \) and let \( {x}_{0} \in M \) . Then the stability subgroup \( {G}_{{x}_{0}} \) is a closed Lie subgroup of \( G \) . If \( G \) is transitive, the map \[ \gamma : g \mapsto g \cdot {g}_{0}\;\left( {g \in G}\right) \] is an an... | Proof. We have \[ \gamma \circ {l}_{g} = {t}_{g} \circ \gamma \;\left( {g \in G}\right) , \] \( {l}_{g} \) as usual denoting the left translation by \( g \) . If we write \( {V}_{g} = {\left( d\gamma \right) }_{g}\left\lbrack {{T}_{g}\left( G\right) }\right\rbrack \) , it follows from (2.9.10) that \( {V}_{g} = {\left(... | Yes |
Lemma 2.9.3. Let \( G \) be a Lie group with Lie algebra \( \mathfrak{g} \), and \( H \) a closed Lie subgroup which defines the subalgebra \( \mathfrak{h} \) of \( \mathfrak{g} \). We denote by \( \mathfrak{L}\left( {x \mapsto {\mathfrak{L}}_{x}}\right) \) the involutive analytic system \( {}^{1} \) of tangent spaces ... | Proof. Choose coordinates \( {x}_{1},\ldots ,{x}_{m} \) on an open subset \( V \) of \( G \) containing 1 such that \( {\left( \partial /\partial {x}_{1}\right) }_{y},\ldots ,{\left( \partial /\partial {x}_{p}\right) }_{y} \) span \( {\mathfrak{L}}_{y} \) for all \( y \in V \); we may (and do) assume that \( y \mapsto ... | Yes |
Theorem 2.9.5. Let \( G, H \) be as above and let \( N = G/H \) . Suppose that \( x \in N \) and that \( y \in G \) is such that \( \beta \left( y\right) = x \) . Then there is a section \( c \) defined in an open neighborhood of \( x \) such that \( c\left( x\right) = y \) . | Proof. Let \( A \) and \( {U}_{0} \) be as in the proof of Theorem 2.9.4. Then \( \beta \) is an analytic diffeomorphism of \( {U}_{0} \) onto \( A \) . As before, let \( \gamma \) be the map of \( A \) onto \( {U}_{0} \) which inverts the restriction of \( \beta \) to \( {U}_{0} \) . Then \( \gamma \) is a section for... | Yes |
Theorem 2.9.6. Let \( G \) be a Lie group, \( H \) a closed normal Lie subgroup. Then the analytic structure induced on the topological group \( N = G/H \) from \( G \) converts \( N \) into a Lie group. The natural map \( \beta \) is then an analytic homomorphism of \( G \) onto \( N \) . | Proof. Let \( \eta \) be the map \( \left( {x, y}\right) \mapsto x{y}^{-1} \) of \( N \times N \) into \( N \) ; \( \bar{\eta } \) the map \( \left( {g,{g}^{\prime }}\right) \mapsto g{g}^{\prime - 1} \) of \( G \times G \) into \( G \) ; and \( \pi \) the map \( \left( {g,{g}^{\prime }}\right) \mapsto \left( {\beta \le... | Yes |
Theorem 2.9.7. \( G \cdot x \) is a submanifold of \( M \) of dimension \( = \dim \left( G\right) - \) \( \dim \left( {G}_{x}\right) \) . In order that \( G \cdot x \) be a regular submanifold of \( M \) it is necessary and sufficient that it be locally closed in \( M \) . | Proof. Let \( i \) be the inclusion map of \( N = G \cdot x \) into \( M \) . As in Lemma 2.9.2, \( {\left( di\right) }_{y}\left\lbrack {{T}_{y}\left( N\right) }\right\rbrack \) has the same dimension for all \( y \in N \) . Since \( i \) is one-to-one and one-to-one submersions are imbeddings, we can use Lemma 2.7.2 t... | Yes |
Lemma 2.9.9. The quotient topology on \( X \) is Hausdorff if and only if \( \Gamma \) is closed in \( M \times M \) . In this case all the orbits in \( M \) are closed regular submanifolds of \( M \) . | Proof. Let \( \Gamma \) be closed, and let \( x, y \in M \) be such that \( \pi \left( x\right) \neq \pi \left( y\right) \) . Then \( \left( {x, y}\right) \notin \Gamma \), so we can find open subsets \( U, V \) of \( M \) such that \( \left( {x, y}\right) \in \) \( U \times V \subseteq \left( {M \times M}\right) \smal... | Yes |
Theorem 2.9.10. Let \( G \) act freely on \( M \) . Then the following are equivalent:\n\n(1) \( \Gamma \) is closed in \( M \times M \), and \( \gamma \) is a homeomorphism of \( M \times G \) onto \( \Gamma \) .\n\n(2) \( \Gamma \) is closed in \( M \times M \) ; moreover, given any \( x \in M \) we can find a regula... | Proof. (1) \( \Rightarrow \) (2). Let \( x \in M \) . Choose a regular submanifold \( {N}_{1} \) of \( M \) passing through \( x \) such that \( {T}_{x}\left( M\right) \) is the direct sum of \( {T}_{x}\left( {G \cdot x}\right) \) and \( {T}_{x}\left( {N}_{1}\right) \) . Let \( \psi \) be the map \( \left( {g, y}\right... | Yes |
Corollary 2.9.11. Let \( G \) be a compact Lie group acting freely and analytically on \( M \) . Then \( M/G \) admits a unique analytic structure for which \( \pi \) is a submersion. | Proof. In this case it is easily seen that (1) of Theorem 2.9.10. is satisfied. | No |
Corollary 2.9.12. Let \( G \) be a discrete group acting freely and analytically on \( M \), and let \( \Gamma \) be closed in \( M \times M \) . Then, in order that \( X = M/G \) admit an analytic structure such that \( \pi \) is a submersion, it is necessary and sufficient that the following condition be satisfied: f... | Proof. We shall verify that this condition is equivalent to the fact that \( \gamma \) is a homeomorphism. Suppose \( \gamma \) is a homeomorphism and \( x \in M \) . If an open neighborhood of \( x \) with the required properties does not exist, we can find sequences \( \left\{ {x}_{n}\right\} \) and \( \left\{ {y}_{n... | Yes |
Lemma 2.10.2. Let \( V \) be as above and let \( \varepsilon \) be the algebra of endomorphisms of \( V \) . For \( A \in \mathcal{E} \) let\n\n\[ \n{e}^{A} = 1 + A + \frac{{A}^{2}}{2!} + \cdots + \frac{{A}^{n}}{n!} + \cdots \n\]\n\nThen \( {e}^{A} \) is well defined and \( A \mapsto {e}^{A} \) is an analytic map of \(... | Proof. It is enough to prove everything when \( V \) is a complex vector space.\n\nSuppose \( \left\{ {{v}_{1},\ldots ,{v}_{m}}\right\} \) is a basis for \( V \) and \( c > 0 \) is a constant such that all the matrix entries of an \( A \in \mathcal{E} \) relative to this basis are \( \leq c \) . Then an easy induction ... | Yes |
Theorem 2.10.3. (1) Let \( {G}_{i} \) be a real (resp. complex) Lie group with Lie algebra \( {\mathfrak{g}}_{i}\left( {i = 1,2}\right) \), and let \( \pi \left( {{G}_{1} \rightarrow {G}_{2}}\right) \) be an analytic homomorphism. Then\n\n\[ \pi \left( {\exp X}\right) = \exp \left( {d\pi }\right) \left( X\right) \;\lef... | Proof. We prove both assertions in the real case. Let \( {1}_{i} \) be the identity of \( {G}_{i}\left( {i = 1,2}\right) \).\n\nTo prove (1), let \( X \in {\mathfrak{g}}_{1} \) and \( {\eta }_{X}\left( t\right) = \pi \left( {\exp {tX}}\right) ,\left( {t \in \mathbf{R}}\right) \). Then \( {\dot{\eta }}_{X}\left( 0\right... | Yes |
Let \( {G}_{1},{G}_{2} \) be real Lie groups and \( \pi \) a homomorphism of \( {G}_{1} \) into \( {G}_{2} \). In order that \( \pi \) be analytic it is necessary and sufficient that for every continuous homomorphism \( \alpha \) of \( \mathbf{R} \) into \( {G}_{1},\pi \circ \alpha \) is continuous. In particular, if \... | The necessity of the conditions is obvious. We now prove their sufficiency. Let \( {\mathfrak{g}}_{i} \) be the Lie algebra of \( {G}_{i}, i = 1,2 \). Let \( {X}_{1},\ldots ,{X}_{m} \) be a basis for \( {\mathfrak{g}}_{1} \). By our assumption, for \( 1 \leq i \leq m,{\eta }_{i} : t \mapsto \pi \left( {\exp t{X}_{i}}\r... | Yes |
Theorem 2.11.3. Let \( {G}_{1},{G}_{2} \) be real Lie groups and \( \varphi \) a continuous one-one homomorphism of \( {G}_{1} \) onto \( {G}_{2} \) . Then \( \varphi \) is an analytic isomorphism of \( {G}_{1} \) onto \( {G}_{2} \) . In particular, a (second countable) topological group can admit at most one real anal... | Proof. By Lemma 2.5.3, \( \varphi \) is a homeomorphism. So both \( \varphi \) and \( {\varphi }^{-1} \) are analytic by the previous theorem. For the second assertion, let \( G \) be a real Lie group and \( {G}^{ * } \) another real Lie group having the same underlying topological group as \( G \) . If we apply the fi... | Yes |
Lemma 2.12.1. Let \( x \in G, X \in \mathfrak{g} \). Then for any integer \( k \geq 0 \) and any function \( f \) defined and \( {C}^{\infty } \) around \( x \), \[ \left( {{X}^{k}f}\right) \left( x\right) = f\left( {x;{X}^{k}}\right) = {\left( \frac{{d}^{k}}{d{t}^{k}}f\left( x\exp tX\right) \right) }_{t = 0}. \] Iff i... | Proof. We shall prove by induction on \( k \) the more general formula \[ f\left( {x\exp {tX};{X}^{k}}\right) = \frac{{d}^{k}}{d{t}^{k}}f\left( {x\exp {tX}}\right) \] for all \( t \in \mathbf{R} \) and all \( f \) defined and \( {C}^{\infty } \) around \( x \). For \( k = 0 \) this is obvious. Assume (2.12.4) for some ... | Yes |
Lemma 2.12.2. Let \( x \in G,{X}_{1},\ldots ,{X}_{s} \in \mathfrak{g} \) . If \( f \) is a function defined and \( {C}^{\infty } \) around \( x \), then\n\n\[ \left\{ {\;\begin{aligned} \left( {{X}_{1}\cdots {X}_{s}f}\right) \left( x\right) & = f\left( {x;{X}_{1}\cdots {X}_{s}}\right) \\ = {\left( \frac{{\partial }^{s}... | Proof. Let \( F \) be the function \( \left( {{t}_{1},\ldots ,{t}_{s}}\right) \mapsto f\left( {x\exp {t}_{1}{X}_{1}\cdots \exp {t}_{s}{X}_{s}}\right) \) , defined in a neighborhood of the origin in \( {\mathbf{R}}^{s} \) . Then for all sufficiently small \( \left| {t}_{1}\right| ,\ldots ,\left| {t}_{s - 1}\right| \)\n\... | Yes |
Theorem 2.12.3. Let \( x \in G \) and let \( f \) be a function defined and analytic around \( x \) . Let \( {X}_{1},\ldots ,{X}_{s} \in \mathfrak{g} \) . Then there is an \( a > 0 \) such that\n\n(2.12.7)\n\n\[ f\left( {x\exp \left( {{t}_{1}{X}_{1} + \cdots + {t}_{s}{X}_{s}}\right) }\right) = \mathop{\sum }\limits_{{{... | Proof. Let \( F \) be the function \( \left( {{t}_{1},\ldots ,{t}_{s}}\right) \mapsto f\left( {x\exp \left( {{t}_{1}{X}_{1} + \cdots + {t}_{s}{X}_{s}}\right) }\right. \) , defined and analytic around the origin in \( {\mathbf{R}}^{s} \) . Write \( {D}_{j} = \partial /\partial {t}_{j},1 \leq j \leq s \) . Then for some ... | Yes |
Theorem 2.12.4. Let \( s \geq 1 \) and \( {X}_{1},\ldots ,{X}_{s} \in \mathfrak{g} \) . Then\n\n\[ \exp t{X}_{1}\cdots \exp t{X}_{s} = \exp \left\{ {t\mathop{\sum }\limits_{{1 \leq i \leq s}}{X}_{i} + \frac{{t}^{2}}{2}\mathop{\sum }\limits_{{1 \leq i < j \leq s}}\left\lbrack {{X}_{i},{X}_{j}}\right\rbrack + O\left( {t}... | Proof. Fix \( s \geq 1,{X}_{1},\ldots ,{X}_{s} \in \mathfrak{g} \) . Let \( f \) be a function defined and analytic around 1 and let \( F \) be the function \( \left( {{t}_{1},\ldots ,{t}_{s}}\right) \mapsto f\left( {\exp {t}_{1}{X}_{1}\cdots \exp {t}_{s}{X}_{s}}\right) \) . Then for sufficiently small \( \left| t\righ... | Yes |
Corollary 2.12.5. Let \( X, Y \in \mathfrak{g} \) and let \( \left\{ {X}_{n}\right\} ,\left\{ {Y}_{n}\right\} \) be sequences in \( \mathfrak{g} \) such that \( {X}_{n} \rightarrow X \) and \( {Y}_{n} \rightarrow Y \) as \( n \rightarrow \infty \) . Then\n\n(2.12.14)\n\n\[ \left\{ \begin{array}{l} \exp \left( {X + Y}\r... | This corollary would follow if we showed that the \( O\left( {t}^{3}\right) \) estimates in (2.12.10) are uniform when \( X \) and \( Y \) vary over compact subsets of \( \mathfrak{g} \) ; and for this it would be enough to verify that the \( O\left( {t}^{3}\right) \) estimates in (2.12.11) are uniform when \( {X}_{1},... | No |
Lemma 2.13.1. Let \( G \) be a Lie group acting analytically on an analytic manifold \( M \). Let \( {x}_{0} \in M \), and let \( {G}_{0} \) be the stability subgroup of \( G \) at \( {x}_{0} \). For each \( g \in G \), let \( {t}_{g} \) denote the diffeomorphism \( x \mapsto g \cdot x \) of \( M \). Then for each \( g... | Proof. By Lemma 2.9.2, \( {G}_{0} \) is a closed Lie subgroup of \( G \). If \( g \in {G}_{0} \) , \( {t}_{g} \cdot {x}_{0} = {t}_{{g}^{-1}} \cdot {x}_{0} = {x}_{0} \), so the linear map \( {\left( d{t}_{g}\right) }_{{x}_{0}} \) is a well-defined endomorphism of the tangent space \( {T}_{{x}_{0}}\left( M\right) \) havi... | Yes |
Let \( G \) be a Lie group with Lie algebra \( \mathfrak{g} \). Then the differential of the adjoint representation of \( G \) is the adjoint representation of \( \mathfrak{g} \). In particular, its kernel is the centralizer of \( {G}^{0} \) in \( G \), and the subalgebra of \( \mathfrak{g} \) defined by this kernel is... | Proof. Let \( \lambda \) denote the differential of the adjoint representation of \( G \). Then by Theorem 2.10.3 and the fact the that exponential map of \( \mathfrak{{gl}}\left( g\right) \) is the usual matrix exponential, we have \( \operatorname{Ad}\left( {\exp {tX}}\right) = {e}^{{t\lambda }\left( X\right) } \) fo... | Yes |
Corollary 2.13.3. If \( X, Y \in \mathfrak{g} \) and \( \left\lbrack {X, Y}\right\rbrack = 0 \), then\n\n\[ \exp \left( {X + Y}\right) = \exp X\exp Y. \] | Proof. Let \( Z \) be the component of the identity of the kernel of \( \mathrm{{Ad}} \), and \( 3 \) the corresponding subalgebra of \( g \) . By the theorem just proved \( 3 \) is the center of \( \mathfrak{g} \) . Since \( \mathfrak{g} \) is abelian if and only if \( \mathfrak{z} = \mathfrak{g} \), it follows that \... | Yes |
Theorem 2.13.4. Let \( G \) be an analytic group with Lie algebra \( \mathfrak{g}, H \) an analytic subgroup of \( G \), and \( \mathfrak{h} \) the corresponding subalgebra of \( \mathfrak{g} \) . Then \( H \) is normal in \( G \) if and only if \( \mathfrak{h} \) is an ideal in \( \mathfrak{g} \) ; in this case \( {\m... | Proof. Suppose \( H \) is normal in \( G \) and let \( X \in \mathfrak{h}, y \in G \) . Then \( \exp t{X}^{y} = \) \( y\exp {tX}{y}^{-1} \in H \) for all \( t \in \mathbf{R} \) . By (2) of Theorem 2.10.3, \( {X}^{y} \in \mathfrak{h} \) . Thus \( {\mathfrak{h}}^{y} = \mathfrak{h} \) . In particular, if \( X \in \mathfra... | Yes |
Lemma 2.14.1. Let \( l \) , \( n \) be integers with \( 0 \leq l \leq n \) . Then\n\n(2.14.3)\n\n\[ \mathop{\sum }\limits_{{0 \leq k \leq l}}{\left( -1\right) }^{k}\left( \begin{matrix} n + 1 \\ k \end{matrix}\right) = {\left( -1\right) }^{l}\left( \begin{array}{l} n \\ l \end{array}\right) \]\n\n(here, for integers \(... | Proof. Follows trivially by induction on \( l \) . | No |
Lemma 2.14.2. Let \( \\mathfrak{A} \) be an associative algebra over a field \( k \) of characteristic zero. For any \( a \\in \\mathfrak{A} \) let \( {d}_{a} \) be the endomorphism \( b \\mapsto {ab} - {ba} \) of \( \\mathfrak{A} \) . Then for any integer \( n \\geq 0 \) ,\n\n(2.14.4)\n\n\[ \n{d}_{a}^{n}\\left( b\\rig... | Proof. Let \( {l}_{a} \) (resp. \( {r}_{a} \) ) denote the endomorphism \( b \\mapsto {ab} \) (resp. \( b \\mapsto {ba} \) ) of \( \\mathfrak{A} \) . It is obvious that \( {l}_{a} \) commutes with \( {r}_{a} \) and that \( {d}_{a} = {l}_{a} - {r}_{a} \) . So for any integer \( n \\geq 0 \) ,\n\n\[ \n{d}_{a}^{n} = {\\le... | Yes |
Lemma 2.14.5. \( \mathfrak{v} \) is an open connected subset of \( \mathfrak{g} \) which is invariant under \( \operatorname{Ad}\left\lbrack G\right\rbrack \) . If \( \mathfrak{z} \) is the center of \( \mathfrak{g} \), \[ \mathfrak{v} + \mathfrak{z} = \mathfrak{v}. \] | Proof. For \( X \in \mathfrak{g} \) and \( y \in G \), the eigenvalues of ad \( X \) and ad \( {X}^{y} \) are the same, by (2.13.16). So \( \mathfrak{v} \) is invariant under Ad. If \( X \in \mathfrak{v}, Z \in \mathfrak{z} \), then \( \operatorname{ad}\left( {X + Z}\right) = \operatorname{ad}X \), because ad \( Z = 0 ... | Yes |
Theorem 2.14.6. Let notation be as above. Then exp \( \mathfrak{v} = U \) is a connected open neighborhood of 1 in \( G \) which is invariant under the inner automorphisms of \( G \) . The exponential map has bijective differential at all points of \( \mathfrak{v} \), and for \( X,{X}^{\prime } \in \mathfrak{v} \) , \(... | Proof. \( U \) is obviously connected, and (2.13.7) implies that it is invariant under the inner automorphisms of \( G \) . It is immediate from Theorem 2.14.3 that \( {\left( d\exp \right) }_{X} \) is bijective for all \( X \in \mathfrak{v} \) . This enables us to conclude that \( U \) is open in \( G \) . Suppose now... | Yes |
Lemma 2.15.1. Let \( V \) be a finite-dimensional vector space (over \( \mathbf{R} \) or \( \mathbf{C} \) ) equipped with a norm \( \left| \cdot \right| \) . Let \( E \) be the algebra of all endomorphisms on \( V \) and let \( \left| \cdot \right| \) denote the standard \( {}^{5} \) operator norm in \( E \) . Let \( a... | Proof. Elementary. | No |
Lemma 2.15.2. Let \( X, Y \in \mathfrak{g} \) and \( F \) be as in (2.15.3). Let \( a > 0 \) be such that \( a\left| X\right| < \eta, a\left| Y\right| < \eta \) . Then \( F \) is a solution to the equation\n\n(2.15.11)\n\n\[ \frac{dF}{dt} = f\left( {\operatorname{ad}F}\right) \left( {X + Y}\right) + \frac{1}{2}\left\lb... | Proof. For brevity we denote \( Z\left( {u : v : X : Y}\right) \) by \( Z\left( {u : v}\right) \) and \( F\left( {t : X : Y}\right) \) by \( F\left( t\right) \) . We shall also make the usual identification of the tangent spaces to \( G \) and \( g \) at each of their points with \( g \) . We have\n\n\[ \exp {uX}\exp {... | Yes |
Lemma 2.15.3. Let \( {c}_{n}\left( {X : Y}\right) \) be defined (2.15.4) for \( X, Y \in \mathfrak{g} \) . Then they are uniquely determined by the recursion formula\n\n\[ \left( {n + 1}\right) {c}_{n + 1}\left( {X : Y}\right) = \frac{1}{2}\left\lbrack {X - Y,{c}_{n}\left( {X : Y}\right) }\right\rbrack \]\n\n(2.15.15)\... | Proof. The relations (2.15.15) obviously determine all the \( {c}_{n} \) uniquely if \( {c}_{1} \) is known. We now prove (2.15.15). Fix \( X, Y \in \mathfrak{g} \) and write \( {c}_{n} \) for \( {c}_{n}\left( {X : Y}\right) \) . In what follows, if \( k \) is any integer \( \geq 1 \), denote by \( O\left( {t}^{k}\righ... | Yes |
For any \( a \in \mathcal{B},{\tau }_{\varphi }^{\infty }\left( a\right) \left( {x \mapsto {\tau }_{\varphi }^{\infty }{\left( a\right) }_{x}}\right) \) is an analytic differential operator on \( M \), and the map \( {\tau }_{\varphi }^{\infty }\left( {a \mapsto {\tau }_{\varphi }^{\infty }\left( a\right) }\right) \) i... | The first assertion is immediate from (2.16.4) on using local coordinates. To prove the second it is enough to show that \( {\tau }_{\varphi }^{\infty }\left( {ab}\right) = {\tau }_{\varphi }^{\infty }\left( a\right) {\tau }_{\varphi }^{\infty }\left( b\right) \) when \( a = {X}_{1}\cdots {X}_{r} \) and \( b = {Y}_{1}\... | Yes |
Lemma 2.16.3. Let notation be as above. Then\n\n(1) if \( {\lambda }_{h} \) denotes, for any \( h \in G \), the analytic diffeomorphism \( \left( {g, x}\right) \mapsto \) \( \left( {{hg}, x}\right) \) of \( G \times M \) onto itself, then\n\n\[{\lambda }_{h}\left\lbrack {S}_{\left( g, x\right) }\right\rbrack = {S}_{\le... | Proof. The proof of (1) is an immediate consequence of the fact that \( {\mathfrak{L}}^{\tau } \) is invariant under the \( {\lambda }_{h}\left( {h \in G}\right) \) . | No |
Corollary 2.16.4. Let \( \tau \) be a homomorphism of \( \mathfrak{g} \) into \( {\mathfrak{J}}_{a}\left( M\right) \) . If \( {\varphi }_{1} \) and \( {\varphi }_{2} \) are global \( G \) -transformation groups on \( M \) with \( {\tau }_{{\varphi }_{1}} = {\tau }_{{\varphi }_{2}} = \tau \), then \( {\varphi }_{1} = {\... | Proof. First assume that \( {\varphi }_{1} \) and \( {\varphi }_{2} \) are global \( G \) -transformation groups on \( M \) with \( {\tau }_{{\varphi }_{1}} = {\tau }_{{\varphi }_{2}} = \tau \) . Let \( {p}_{G} \) (resp. \( {p}_{M} \) ) be the projection of \( G \times M \) onto \( G \) (resp. onto \( M \) ), and let \... | Yes |
Lemma 2.16.7. Let \( \\left\\{ {{U}_{i} : i \\in I}\\right\\} \) be a locally finite open covering of \( M \) . For each \( \\left( {i, j}\\right) \\in I \\times I \) and any \( x \\in {U}_{i} \\cap {U}_{j} \), let \( {U}_{ijx} \) be a neighborhood of \( x \) contained in \( {U}_{i} \\cap {U}_{j} \) . Then for each \( ... | Proof. Let \( \\left\\{ {{W}_{i} : i \\in I}\\right\\} \) be a locally finite open covering of \( M \) with \( {Cl}\\left( {V}_{i}\\right) \\subseteq {U}_{i}\\left( {i \\in I}\\right) \) . For \( x \\in M \), denote by \( {U}_{x}^{\\prime } \) the intersection of all the \( {U}_{i},{V}_{j},{U}_{pqx} \) that contain \( ... | Yes |
Theorem 2.16.8. Let \( \tau \) be an infinitesimal \( G \) -transformation group on \( M \) . Then there is a local \( G \) -transformation group \( \varphi \) on \( M \) such that \( {\tau }_{\varphi } = \tau \) . | Proof. By Theorem 2.16.6 we can select a locally finite open covering \( \left\{ {{U}_{i} : i \in I}\right\} \) of \( M \) and local \( G \) -transformation groups \( {\varphi }_{i} \) on \( {U}_{i}\left( {i \in I}\right) \), such that \( {\tau }_{{\varphi }_{i}}\left( X\right) = \tau \left( X\right) \mid {U}_{i}\left(... | Yes |
Theorem 2.16.9. Let \( M \) be compact and \( G \) simply connected. Then for any infinitesimal G-transformation group \( \tau \) on \( M \) there is a unique global G-transformation group \( \varphi \) on \( M \) such that \( {\tau }_{\varphi } = \tau \) . | Proof. The proof depends on the following elementary result whose verification is left to the reader: if \( Z \) is any Hausdorff space, \( z \in Z \), and \( A \) is a neighborhood of \( \{ z\} \times M \), there is an open neighborhood \( B \) of \( z \) such that \( B \times M \subseteq A \) . This said, let \( \psi... | No |
Lemma 2.16.10. Let \( \zeta \) be an one-parameter group of analytic diffeomor-phisms of \( M \), and \( Z \) the infinitesimal generator of \( \zeta \) . If \( X \in \mathfrak{g} \), then \( \tau \left( X\right) = Z \) and only if for each \( x \in M,\left( {\exp \left( {tX}\right) ,\zeta \left( {-t \cdot x}\right) }\... | Proof. Write \( f\left( t\right) = \left( {\exp \left( {tX}\right) ,\zeta \left( {-t}\right) \cdot x}\right) \left( {t \in \mathbf{R}}\right), x \in M \) being fixed. Suppose \( f\left( t\right) \in {S}_{\left( 1, x\right) } \) for all \( t \) . Then \( f \) is an analytic map of \( \mathbf{R} \) into \( {S}_{\left( 1,... | Yes |
Lemma 2.16.11. Suppose \( {X}_{1},\ldots ,{X}_{p} \in \mathfrak{g} \) are such that \( \tau \left( {X}_{i}\right) \) is global for all \( i = 1,\ldots, p \) . Let \( {\zeta }_{i} \) be the one-parameter group of analytic diffeomorphisms of \( M \) generated by \( \tau \left( {X}_{i}\right) \left( {1 \leq i \leq p}\righ... | Proof. The analyticity of \( \Phi \) is obvious. We prove the second assertion by induction on \( p \) . The case \( p = 1 \) is precisely the preceding lemma. Suppose \( p > 1 \), and assume the result when the number of elements considered from \( \mathrm{g} \) is \( p - 1 \) . Then\n\n\[ \left( {\exp \left( {{t}_{2}... | Yes |
Theorem 2.16.13. Let \( \tau \) be an infinitesimal \( G \) -transformation group on \( M \) . Suppose that \( G \) is simply connected and that \( \tau \left( X\right) \) is a global vector field on \( M \) for all \( X \in \mathfrak{g} \) . Then there is a unique global \( G \) -transformation group \( \varphi \) on ... | Proof. By the preceding lemma and the simple connectedness of \( G \), we obtain the result that \( {p}_{G, x} \) is a homeomorphism of \( {S}_{\left( 1, x\right) } \) onto \( G \) for each \( x \in M \) . For \( \left( {g, x}\right) \in G \times M \), define \( \varphi \left( {{g}^{-1} : x}\right) \) as the unique ele... | Yes |
Theorem 3.1.3. There are ideals \( {J}_{1},\ldots ,{J}_{n} \) of \( k\left\lbrack T\right\rbrack \) such that (i) \( 0 \neq {J}_{1} \subseteq \) \( {J}_{2}\cdots \subseteq {J}_{n} \neq k\left\lbrack T\right\rbrack \), and (ii) for suitable vectors \( {v}_{1},\ldots ,{v}_{n} \) of \( V, V \) is the direct sum of the sub... | It is obvious that \( 1 \leq n \leq m \) . Let \( {p}_{i}\left( {1 \leq i \leq n}\right) \) be monic polynomials generating the ideal \( {J}_{i} \) . Then \( {p}_{1},\ldots ,{p}_{n} \) form a complete set of invariants for the action of \( L \) on \( V \) . | No |
Lemma 3.1.10. Let notation be as above. Then for any endomorphism \( L \) of \( V,\zeta \) intertwines the actions of \( {L}_{1,1} \) and ad \( L \) ; i.e., \[ \zeta \circ {L}_{1,1} \circ {\zeta }^{-1} = \operatorname{ad}L. \] | Proof. Let \( v, w \in V,{v}^{ * } \in {V}^{ * } \) . Then \( \left\lbrack {L,\zeta \left( {v \otimes {v}^{ * }}\right) }\right\rbrack \left( w\right) = {v}^{ * }\left( w\right) {Lv} - \) \( {v}^{ * }\left( {Lw}\right) v \), so (ad \( L)\left( {\zeta \left( {v \otimes {v}^{ * }}\right) }\right) = \left( {\zeta \circ {L... | Yes |
Lemma 3.1.11. Let \( L \) be semisimple (resp. nilpotent). Then \( {L}_{r, s} \) is semisimple (resp. nilpotent) for all \( r, s \geq 0 \) . In particular, ad \( L \) is semisimple (resp. nilpotent). | Proof. If \( {M}_{1},{M}_{2} \) are nilpotent endomorphisms on a vector space of dimension \( n \), and \( \left\lbrack {{M}_{1},{M}_{2}}\right\rbrack = 0 \), then \( {\left( {M}_{1} + {M}_{2}\right) }^{2n} = 0 \) . In particular, if \( {N}_{i} \) are nilpotent endomorphism of a vector space \( {W}_{i}, i = 1,2 \), the... | Yes |
Corollary 3.1.12. Let \( L = S + N \) be the Jordan decomposition of \( L \), and let \( r, s \geq 0 \) . Then \( {L}_{r, s} = {S}_{r, s} + {N}_{r, s} \) is the Jordan decomposition of \( {L}_{r, s} \) . | Proof. By the lemma, \( {S}_{r, s} \) is semisimple and \( {N}_{r, s} \) is nilpotent. Since \( \left\lbrack {S, N}\right\rbrack = 0 \), we have \( \left\lbrack {{S}_{r, s},{N}_{r, s}}\right\rbrack = 0 \) . Hence, \( {L}_{r, s} = {S}_{r, s} + {N}_{r, s} \) is the Jordan decomposition of \( {L}_{r, s} \) . | Yes |
Theorem 3.1.13. Let notation be as above. Then:\n\n(i) For any \( L \in \mathfrak{{gl}}\left( V\right) ,\mathfrak{g}\left( L\right) \) is an abelian Lie algebra whose elements are of the form \( p\left( L\right) \) for \( p \in k\left\lbrack T\right\rbrack \) with zero constant term. | Proof. Since \( {\left\lbrack M, N\right\rbrack }_{r, s} = \left\lbrack {{M}_{r, s},{N}_{r, s}}\right\rbrack \), it is clear that \( \mathfrak{g}\left( L\right) \) is a Lie subalgebra of \( \mathfrak{{gl}}\left( V\right) \) . Fix \( M \in \mathfrak{g}\left( L\right) \) . By (3.1.14), \( M \in \{ L{\} }^{\prime \prime }... | Yes |
Corollary 3.1.14. Let \( A \) be an algebra over \( k \) . Assume that \( \dim A < \infty \) . If \( L \) is a derivation of \( A \), then any replica of \( L \) is also a derivation of \( A \) . In particular, the semisimple and nilpotent parts of \( A \) are also derivations of \( A \) . | Proof. Let \( L = S + N \) be the Jordan decomposition of \( L \) . By (vii) above, \( S \) and \( N \) are replicas of \( L \) . The present corollary is then an immediate consequence of Lemma 3.1.10 (cf. especially (3.1.15)). | No |
Theorem 3.1.15. Let \( N \) be a nilpotent endomorphism of \( V \) . Then\n\n\[ \mathfrak{g}\left( N\right) = \{ {cN} : c \in k\} . \] | Proof. We prove that \( \mathrm{g}\left( N\right) = k \cdot N \) by induction on \( \dim V \) . For \( \dim V = 1, N = 0 \), so \( g\left( N\right) = 0 \) . Let \( \dim V \geq 2 \) . Let \( s\left( {1 \leq s \leq \dim V}\right) \) be the integer such that \( {T}^{s} \) is the minimal polynomial of \( N \) . We may assu... | Yes |
Theorem 3.1.16. Let \( L \) be an endomorphism of \( V \) . Then \( L \) is nilpotent if and only if \[ \operatorname{tr}\left( {L{L}^{\prime }}\right) = 0\;\forall {L}^{\prime } \in \mathfrak{g}\left( L\right) \] | Proof. If \( L \) is nilpotent, then (3.1.25) is obvious. Conversely, let \( L \) be arbitrary but satisfying (3.1.25). Let \( L = S + N \) be the Jordan decomposition of \( L \) . We must prove that \( S = 0 \) . We may (and shall) assume that \( k \) is algebraically closed. Since \( \mathfrak{g}\left( S\right) \subs... | Yes |
Corollary 3.1.17. Let \( A \) be a finite-dimensional algebra over \( k \), and let \( L \) be a derivation of \( A \) . Then \( L \) is nilpotent if and only if \( \operatorname{tr}\left( {L{L}^{\prime }}\right) = 0 \) for all \( {L}^{\prime } \in \) \( k\left\lbrack L\right\rbrack \) that are derivations of \( A \) . | For by Corollary 3.1.14, any replica of \( L \) is a derivation of \( A \) and belongs to \( k\left\lbrack L\right\rbrack \) . | No |
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