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Proposition 3.2.2. Let \( N, X = \left\lbrack {0,1}\right\rbrack, W,\sigma ,{\tau }_{0},\ldots ,{\tau }_{N - 1},\Omega = \{ 0,\ldots, N - 1{\} }^{\mathbb{N}} \) , and \( {P}_{x}^{\left( n\right) }\left( {\{ k\} }\right) \) be as described above. In particular, assume that\n\n\[ \mathop{\sum }\limits_{{k = 0}}^{{N - 1}}... | Proof. We claim that\n\n\[ \mathop{\sum }\limits_{{k = - {N}^{n}}}^{{\left( {N - 1}\right) {N}^{n} - 1}}{P}_{x}^{\left( n + 1\right) }\left( {\{ k\} }\right) = \mathop{\sum }\limits_{{k = - {N}^{n}}}^{{\left( {N - 1}\right) {N}^{n} - 1}}{P}_{x}^{\left( n + 1\right) }\left( \left\{ {k + {N}^{n}}\right\} \right) . \]\n\n... | Yes |
Theorem 3.3.1. Let \( N, X = \left\lbrack {0,1}\right\rbrack, W,\sigma ,{\tau }_{0},\ldots ,{\tau }_{N - 1} \), and \( \Omega \) be as in Proposition 3.2.2. In particular, we assume that \( W : \left\lbrack {0,1}\right\rbrack \rightarrow \left\lbrack {0,1}\right\rbrack \) is measurable and satisfies (3.2.7). For \( x \... | Proof. Let \( x \in \left\lbrack {0,1}\right\rbrack \) be given. Suppose the numbers \( {n}_{0} \) and \( b \) have been chosen such that (3.3.3) holds. Then set\n\n\[ {f}_{n}\left( k\right) = {\chi }_{\left\lbrack -{N}^{n},\left( N - 1\right) {N}^{n}\right) }\left( k\right) W\left( \frac{x + k}{N}\right) \cdots W\left... | Yes |
Lemma 4.2.1. Let the set \( {X}_{4} \) with associated measure \( \mu \) denote the Cantor construction of Figure 4.1(b) and (4.1.3)-(4.1.4). Specifically, \( \mu \) is the Hausdorff measure on the fractal \( {X}_{4} \) with Hausdorff dimension \( 1/2 \) . Let\n\n\[ \Lambda = {\Lambda }_{4} \mathrel{\text{:=}} \left\{ ... | Proof of Lemma 4.2.1. Setting\n\n\[ C\left( \xi \right) \mathrel{\text{:=}} {\int }_{{X}_{4}}{e}_{\xi }\left( x\right) {d\mu }\left( x\right) \]\n\n(4.2.1)\n\nwe get the scaling relation\n\n\[ C\left( \xi \right) = \frac{1}{2}\left( {1 + {e}^{i\pi \xi }}\right) C\left( \frac{\xi }{4}\right) ,\;\xi \in \mathbb{R}. \]\n\... | Yes |
To verify the ONB property, it is enough to verify that the function\n\n\\[ \n{h}_{\Lambda }\left( x\right) \mathrel{\text{:=}} \mathop{\sum }\limits_{{\lambda \in \Lambda }}{\left| C\left( x - \lambda \right) \right| }^{2} \n\\]\n\n(4.3.1)\n\nis constant and equal to \\( 1, x \in \\mathbb{R} \\) . | Setting \\( W\\left( x\\right) \\mathrel{\text{:=}} {\\cos }^{2}\\left( {2\\pi x}\\right) \\), see (4.1.1), we get\n\n\\[ \n{R}_{W}{h}_{\Lambda } = {h}_{\Lambda } \n\\]\n\n(4.3.2)\n\nwhere\n\n\\[ \n\\left( {{R}_{W}f}\\right) \\left( x\\right) \\mathrel{\text{:=}} W\\left( \\frac{x}{4}\\right) f\\left( \\frac{x}{4}\\rig... | No |
Proposition 4.4.1. [JoPe98] With the stated properties, we have \( {P}_{x}\left( {\mathbb{N}}_{0}\right) = 1 \) for all \( x \in X \) . | Proof. Let\n\n\[ \n{\chi }_{n}\left( \lambda \right) = \left\{ \begin{array}{ll} 1 & \text{ if }\lambda \in \Lambda \text{ and }\lambda < \frac{1}{3}\left( {{4}^{n} - 1}\right) , \\ 0 & \text{ otherwise,} \end{array}\right. \n\] \n\nand set \n\n\[ \n{F}_{x}^{\left( n\right) }\left( \lambda \right) \mathrel{\text{:=}} {... | Yes |
Lemma 5.2.1. Let \( \left( {X,\mathcal{B}}\right) \) be a measure space as described in Chapter 2. Let \( \sigma : X \rightarrow \) \( X \) and \( N \) be as described, and let \( {\tau }_{0},\ldots ,{\tau }_{N - 1} \) be a choice of branches of \( {\sigma }^{-1} \) . Let \( W : X \rightarrow \left\lbrack {0,1}\right\r... | Proof. As in Chapter 2, set \( \Omega \mathrel{\text{:=}} \{ 0,1,\ldots, N - 1{\} }^{\mathbb{N}} \), and let \( \mathbf{0} = \{ 0,0,0,\ldots \} \) be the point in \( \Omega \) which is given by an infinite string of zeroes. Let \( A\left( \underbrace{0,0,\ldots ,0}\right) \) be\nthe cylinder sets from (2.3.5), and let ... | Yes |
Proposition 5.3.2. Let \( X,\sigma ,{\tau }_{0},\ldots ,{\tau }_{N - 1}, W \), and \( {\left( {P}_{x}\right) }_{x \in X} \) be as specified in Lemma 5.2.1. Then the function\n\n\[ h\left( x\right) \mathrel{\text{:=}} {P}_{x}\left( {\mathbb{N}}_{0}\right) ,\;x \in X, \]\n\n(5.3.5)\n\nis harmonic for \( {R}_{W} \) . | Proof. Since \( {P}_{x} \) is a Radon probability measure on \( \Omega \) for each \( x \in X \), by Lemma 2.4.1, it follows that the following infinite-sum representation for \( h\left( x\right) \) is convergent:\n\n\[ h\left( x\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }\mathop{\sum }\limits_{\left( {i}_{1},\l... | Yes |
Theorem 5.4.1. Let \( X,\sigma ,{\tau }_{0},\ldots ,{\tau }_{N - 1}, W \), and \( {\left( {P}_{x}\right) }_{x \in X} \) be as above. Suppose for some \( x \in X \) there is an \( {n}_{0} \in \mathbb{N} \) and \( b \in {\mathbb{R}}_{ + } \) such that\n\n\[ \mathop{\prod }\limits_{{p \geq {n}_{0}}}W\left( {{\tau }_{{i}_{... | Proof of Theorem 5.4.1. Suppose condition (5.4.2) above holds. Let \( {n}_{0} \) and \( b \) be chosen as stated. Since \( W \leq 1 \), we get\n\n\[ \mathop{\prod }\limits_{{p \geq n}}W\left( {{\tau }_{{i}_{p}}\cdots {\tau }_{{i}_{1}}x}\right) \geq \mathop{\prod }\limits_{{p \geq {n}_{0}}}W\left( {{\tau }_{{i}_{p}}\cdo... | Yes |
Example 5.4.3. \( W\left( x\right) \mathrel{\text{:=}} {\cos }^{2}\left( {3\pi x}\right), X = \left\lbrack {0,1}\right\rbrack ,\sigma : x \rightarrow {2x}{\;\operatorname{mod}\;1},{\tau }_{i}\left( x\right) = \left( {x + i}\right) /2, i \in \{ 0,1\} \). See Figure 5.1. The condition (5.4.2) is not satisfied for any \( ... | To see this, recall that if \[ k = {i}_{1} + 2{i}_{2} + \cdots + {2}^{n - 1}{i}_{n} \] then \[ W\left( {{\tau }_{{i}_{n}}\cdots {\tau }_{{i}_{1}}x}\right) = {\cos }^{2}\left( \frac{{3\pi }\left( {x + k}\right) }{{2}^{n + 1}}\right) . \] To show that (5.4.2) does not hold, we check that for all \( b \in {\mathbb{R}}_{ +... | Yes |
Let \( X = \left\lbrack {0,1}\right\rbrack, N \in \mathbb{N}, N \geq 2 \), be given, and let \( \sigma ,{\tau }_{0},\ldots ,{\tau }_{N - 1} \) be the \( N \) -adic maps of (5.4.9). Let \( W : \left\lbrack {0,1}\right\rbrack \rightarrow \left\lbrack {0,1}\right\rbrack \) satisfy (2.4.1), i.e., \[ \mathop{\sum }\limits_{... | Following Remark 1.4.2, we extend the measure \( {P}_{x} \) on \( {\mathbb{N}}_{0} \) to \( \mathbb{Z} \), making use of two copies of \( \mathbf{\Omega } = \{ 0,1,\ldots, N - 1{\} }^{\mathbb{N}} \) . Identifying \( k \in {\mathbb{N}}_{0} \) with \( \omega \left( k\right) \), consider the case \( - {N}^{n} \leq k < 0 \... | Yes |
Proposition 5.5.2. There is a shift-invariant function \( v : \Omega \rightarrow \left\lbrack {0,1}\right\rbrack \) such that\n\n\[ V\left( {x,\omega }\right) = h\left( {v\left( \omega \right) }\right) ,\;\omega \in \Omega ,\]\n\n(5.5.2)\n\nis the cocycle corresponding to the harmonic function \( h \) in (5.4.11). More... | Proof. Since there is a 1-1 correspondence between harmonic functions and cocy-cles, see Theorem 2.7.1, the cocycle corresponding to \( h \) in (5.4.11) is unique. If we show that it has the form (5.4.11), it follows from Corollary 5.4.4 that the function \( v \) must be shift-invariant, i.e., that\n\n\[ v\left( {{\ome... | Yes |
Theorem 6.1.1. Let \( X,\mathcal{B}, N, W \) be as in Chapter 2, and \( \sigma : X \rightarrow X \) an endomorphism such that \( \# {\sigma }^{-1}\left( {\{ x\} }\right) = N, x \in X \), and assume in addition that \( X \) is a compact Hausdorff space. Suppose branches of \( {\sigma }^{-1} \) may be chosen such that, f... | Proof of Theorem 6.1.1. Let \( k \in {\mathbb{N}}_{0} \), and consider the \( N \) -adic representation \( k = \) \( {i}_{1} + {i}_{2}N + \cdots + {i}_{n}{N}^{n - 1} \) . Note that\n\n\[ \omega \left( k\right) = \left( {{i}_{1},\ldots ,{i}_{n},\underset{\infty \text{ string of zeroes }}{\underbrace{0,0,0,\ldots }}}\rig... | Yes |
The corresponding wavelet function \( \psi \) is given by\n\n\[ \psi \left( t\right) = \varphi \left( {2t}\right) - \varphi \left( {{2t} - 1}\right) ,\;t \in \mathbb{R}. \] | Even without Fourier transformation (1.3.3), it is immediate by inspection that the equations (6.2.1)-(6.2.2) are solved by the Haar father function,\n\n\[ \varphi \left( t\right) = \left\{ \begin{array}{ll} 1, & 0 \leq t < 1 \\ 0 & \text{ otherwise } \end{array}\right. \]\n\nand the Haar mother function,\n\n\[ \psi \l... | Yes |
Lemma 7.4.1. Let \( X,\sigma, N,{\tau }_{0},\ldots ,{\tau }_{N - 1},\mathcal{H} \), and \( W \) be as described above.\n\n(a) Then, for every \( x \in X \), there is a unique positive operator-valued Radon probability measure \( {P}_{x} \) on \( \Omega = \{ 0,1,\ldots, N - 1{\} }^{\mathbb{N}} \) such that\n\n\[ \n{P}_{... | Proof sketch. For each \( n \in \mathbb{N} \), consider functions \( f : \mathbf{\Omega } \rightarrow \mathcal{H} \) such that \( f\left( \omega \right) = \) \( f\left( {{\omega }_{1},\ldots ,{\omega }_{n}}\right) \), and define \n\n\[ \n{P}_{x}^{\left( n\right) }\left\lbrack f\right\rbrack = \mathop{\sum }\limits_{{\l... | Yes |
Lemma 7.4.3. Let \( X,\sigma, N,{\tau }_{0},\ldots ,{\tau }_{N - 1},\mathcal{H} \), and \( {\left( {W}_{i}\right) }_{0 \leq i < N} \) be given satisfying the conditions in Definitions 7.3.1, and (7.4.10). Then for every \( x \in X \), there is a unique positive operator-valued Radon probability measure \( {P}_{x} \) on... | Proof sketch. For each \( n \in \mathbb{N} \), consider functions \( f : {\mathbb{N}}_{0} \times \Omega \rightarrow \mathcal{H} \) such that \( f\left( {\omega ,\xi }\right) = f\left( {{\omega }_{1},\ldots ,{\omega }_{n};{\xi }_{1},\ldots ,{\xi }_{n}}\right) \) . Define \[ {P}_{x}^{\left( n\right) }\left\lbrack f\right... | Yes |
Proposition 7.5.2. Let the functions \( {\varphi }_{0},{\varphi }_{1},{\varphi }_{2},\ldots \) be defined by\n\n\[ \n{\varphi }_{2k}\left( t\right) = 2\mathop{\sum }\limits_{{j \in \mathbb{Z}}}{a}_{j}{\varphi }_{k}\left( {{2t} - j}\right) \n\]\n\n(7.5.5)\n\n\[ \n{\varphi }_{{2k} + 1}\left( t\right) = 2\mathop{\sum }\li... | Proof. The detailed steps are quite analogous to those given in Chapter 2, the main difference being that now Lemma 7.4.3 is used in place of Lemma 2.4.1 in the standard wavelet case. | No |
Lemma 7.6.1. Let the functions \( {m}_{0},{m}_{1} \) satisfy the unitarity condition (7.5.4) and let \( {S}_{i}, i = 0,1 \), be the corresponding operators. Then the operator relations (7.6.1) of Cuntz are satisfied. | Proof. We refer to [BrJo02b] for the detailed verification that (7.6.1) is satisfied when the unitarity property (7.5.4) is assumed. The essential step is the following identity:\n\n\[ \mathop{\sum }\limits_{{i = 0}}^{1}{\begin{Vmatrix}{S}_{i}^{ * }f\end{Vmatrix}}^{2} = \parallel f{\parallel }^{2},\;f \in {L}^{2}\left(... | Yes |
Lemma 7.6.2. For \( n \in \mathbb{N} \) and \( \omega \in \Omega \left( n\right) \), set\n\n\[ \n{S}_{\omega } \mathrel{\text{:=}} {S}_{\omega \left( 1\right) }\cdots {S}_{\omega \left( n\right) }\n\]\n\n(7.6.8)\n\nand\n\n\[ \n{E}_{\omega } \mathrel{\text{:=}} {S}_{\omega }{S}_{\omega }^{ * } = {S}_{\omega \left( 1\rig... | Proof. This is a direct verification; see also [BrJo02b, Jor05]. | No |
Theorem 7.6.3. There is a unique projection-valued measure \( P \) defined on the Borel subsets of \( \left\lbrack {0,1}\right\rbrack \) such that\n\n\[ P\left( \left\lbrack {\frac{\omega \left( 1\right) }{2} + \cdots + \frac{\omega \left( n\right) }{{2}^{n}},\frac{\omega \left( 1\right) }{2} + \cdots + \frac{\omega \l... | Proof. The result follows from Lemmas 2.5.1 (Kolmogorov's construction), 2.6.2, and 7.6.2 above; see also [Jor05, Jor04b]. | No |
Lemma 7.6.5. For the case \( n = 1 \) in Theorem 7.6.3, and the measure \( {\mu }_{0} \) of Remark 7.6.4, we have\n\n(a) \( {\mu }_{0}\left( \left\lbrack {0,\frac{1}{2}}\right) \right) = 2\mathop{\sum }\limits_{{j \in \mathbb{Z}}}{\left| {a}_{2j}\right| }^{2} \) ,\n\n(b) \( {\mu }_{0}\left( \left\lbrack {\frac{1}{2},1}... | Proof of Lemma 7.6.5. Part (a):\n\n\[ {\mu }_{0}\left( \left\lbrack {0,\frac{1}{2}}\right) \right) \underset{\text{by }\left( {7.6.12}\right) }{ = }{\begin{Vmatrix}{S}_{0}^{ * }{e}_{0}\end{Vmatrix}}^{2}\underset{\text{by }\left( {7.6.5}\right) }{ = }2{\int }_{0}^{1}{\left| \frac{1}{2}\mathop{\sum }\limits_{{r = 0}}^{1}... | No |
Theorem 7.6.7. Let \( {\left( {a}_{j}\right) }_{j \in \mathbb{Z}},{m}_{0} \), and \( {m}_{1} \) be as described above. Let \( n \in \mathbb{N} \), and \( \omega \in \Omega \left( n\right) \). Then\n\n\[ \n{\mu }_{0}\left( \left\lbrack {\frac{\omega \left( 1\right) }{2} + \cdots + \frac{\omega \left( n\right) }{{2}^{n}}... | Proof. For \( n \in \mathbb{N},\omega \in \Omega \left( n\right) \), set\n\n\[ \n{m}_{\omega }^{\left( n\right) }\left( x\right) = {m}_{\omega \left( 1\right) }\left( x\right) {m}_{\omega \left( 2\right) }\left( {2x}\right) \cdots {m}_{\omega \left( n\right) }\left( {{2}^{n - 1}x}\right) . \n\]\n\n(7.6.17)\n\nThen\n\n\... | Yes |
Theorem 7.7.2. Let \( A \) be a given operator satisfying (7.7.7), and let \( W = {W}_{A} \) be the corresponding matrix function (7.7.9) defined on \( \Omega \) . If \( n \in \mathbb{N} \), and \( \omega \in \Omega \left( n\right) \) , then \( W\left( \omega \right) \) is an \( n \times n \) complex matrix, and we set... | Proof. We know that the state \( {\rho }_{A} \) on \( \mathfrak{A} \) is well defined, and determined uniquely by (7.7.11). Let \( C\left( \Omega \right) \) be embedded in \( \mathfrak{A} \) as described, and let \( {\left. {\rho }_{A}\right| }_{C\left( \Omega \right) } \) be the restriction. Clearly the restriction is... | Yes |
Lemma 8.2.2. Let \( N \in \mathbb{N}, N \geq 2 \), and let \( \sigma : X \rightarrow X \) be an \( N \) -to-1 mapping which is onto \( X \) . Pick branches of the inverse,\n\n\[ \n{\tau }_{i} : X \rightarrow X,\;i = 0,1,\ldots, N - 1,\n\]\n\n(8.2.8)\n\nsuch that (2.1.2) holds.\n\nOn \( {\ell }^{2}\left( X\right) \), le... | Proof. The formulas (8.2.11) follow directly by an application of the expressions (8.2.9) and (8.2.10) for the operators \( {P}_{i} \) and their adjoints \( {P}_{i}^{ * } \) . The computation of the adjoint operator to \( {P}_{i} \) in (8.2.9) is straightforward. | No |
Lemma 8.2.3. (a) Let \( {\left( {S}_{i}\right) }_{i = 0}^{N - 1} \) be a representation of the Cuntz algebra \( {\mathcal{O}}_{N} \) on the Hilbert space \( \mathcal{K} \), and set\n\n\[ U \mathrel{\text{:=}} \mathop{\sum }\limits_{{i = 0}}^{{N - 1}}{P}_{i} \otimes {S}_{i}^{ * } \]\n\n(8.2.12)\n\nThen \( U : {\ell }^{2... | Proof. We will prove (a), but leave (b) to the reader.\n\nFirst,\n\n\[ U{U}^{ * }\; = \;\mathop{\sum }\limits_{i}\mathop{\sum }\limits_{j}{P}_{i}{P}_{j}^{ * } \otimes {S}_{i}^{ * }{S}_{j} \]\n\n\[ \underset{\text{by }\left( {8.2.13}\right) }{ = }\mathop{\sum }\limits_{i}\mathop{\sum }\limits_{j}{P}_{i}{P}_{i}^{ * } \ot... | No |
Lemma 8.2.4. Let \( {\left( {\varphi }_{n}\right) }_{n \in {\mathbb{N}}_{0}} \) be the sequence in \( {L}^{2}\left( \mathbb{R}\right) \) from Proposition 7.5.2, and let \( \mathcal{K} \) be the Hilbert space \( {L}^{2}\left( \mathbb{T}\right) \) with basis \[ {e}_{j}\left( z\right) \mathrel{\text{:=}} {z}^{j},\;z \in \... | Proof. The conclusion (7.5.8) from Proposition 7.5.2 is the assertion that the double-indexed family \[ \left\{ {{\varphi }_{n}\left( {\cdot - j}\right) \mid n \in {\mathbb{N}}_{0}, j \in \mathbb{Z}}\right\} \] forms a tight frame (also called a PARSEVAL frame) for the Hilbert space \( {L}^{2}\left( \mathbb{R}\right) \... | Yes |
Theorem 8.2.6. Let \( {P}_{i}, i = 0,1 \), be the representation of \( {\mathcal{O}}_{2} \) which is described in Lemma 8.2.2 and (8.2.1)-(8.2.2), and let \( {S}_{i}, i = 0,1 \), be one of the representations of \( {\mathcal{O}}_{2} \) described in Lemma 7.6.1. Let\n\n\[ W : {\ell }^{2}\left( {\mathbb{N}}_{0}\right) \o... | Proof. First\n\n\[ W\mathop{\sum }\limits_{{i = 0}}^{1}{P}_{i} \otimes {S}_{i}^{ * }\left( {|n\rangle \otimes {e}_{j}}\right) = W\left( {\mathop{\sum }\limits_{{i = 0}}^{1}|{2n} + i\rangle \otimes {S}_{i}^{ * }{e}_{j}}\right) \]\n\n(8.2.19)\n\n\[ = \mathop{\sum }\limits_{{i = 0}}^{1}\mathop{\sum }\limits_{{k \in \mathb... | Yes |
Corollary 8.2.7. Let the two representations of \( {\mathcal{O}}_{2} \) be as described in the theorem, and assume in addition that the functions in (8.2.16) form an orthonormal basis (ONB) in \( {L}^{2}\left( \mathbb{R}\right) \). Then the unitary scaling operator \( {U}_{2} \) in \( {L}^{2}\left( \mathbb{R}\right) \)... | Proof. Immediate from the theorem, and Remark 8.2.5. | No |
Proposition 8.2.8. (a) Let the two representations of \( {\mathcal{O}}_{2} \) be as described in Theorem 8.2.6, and assume the functions in (8.2.16) are orthonormal.\n\nThen\n\n\[\n\begin{aligned} \left\langle {{\varphi }_{m}\left( {\cdot - k}\right) \mid {U}_{2}{\varphi }_{n}\left( {\cdot - j}\right) }\right\rangle = ... | Proof. It follows from sesquilinearity that the second conclusion is implied by the first. To verify (8.2.22), we carry out a computation and use the results in Section 7.5.\n\nBefore starting the computation of the left-hand side in (8.2.22), write \( m \in {\mathbb{N}}_{0} \) in the form \( m = p + {2l}, p \in \{ 0,1... | Yes |
Corollary 8.2.9. ([CoWi93],[Wic93]) Let the two representations of \( {\mathcal{O}}_{2} \) be as described in Theorem 8.2.6, and let \( {\varphi }_{0},{\varphi }_{1},\ldots \) be the functions generated by the algorithm (7.5.7). Consider a subset\n\n\[ \n\mathcal{A} \subset {\mathbb{N}}_{0} \times {\mathbb{N}}_{0} \n\]... | Proof. We give a proof which relies on Theorem 8.2.6 above. Having the representation\n\n\[ \n{U}_{2} = \mathop{\sum }\limits_{{i = 0}}^{1}{P}_{i} \otimes {S}_{i}^{ * } \n\]\n\n(8.2.30)\n\nmakes it clear that\n\n\[ \n{U}_{2}^{p} = \mathop{\sum }\limits_{I}{P}_{I} \otimes {S}_{I}^{ * } \n\]\n\n(8.2.31)\n\n(where the sum... | Yes |
Proposition 8.3.2. Let \( {\left( {S}_{i}\right) }_{i = 0}^{1} \) be some representation of \( {\mathcal{O}}_{2} \) on \( {L}^{2}\left( \mathbb{T}\right) \) which corresponds to a quadrature-mirror filter \( \left( {{m}_{0},{m}_{1}}\right) \) as described in Lemma 7.6.1. Let \( p, n \in {\mathbb{N}}_{0} \), and \( j \i... | Proof. The result follows from (8.2.34) in Corollary 8.2.9 above. This is the formula which gives the expansion of \( {U}_{2}^{p}{\varphi }_{n}\left( {\cdot - j}\right) \) in the ONB (8.3.8). Hence, to prove the proposition, we only need to compute the \( {\ell }^{2} \) -norm of the expansion coefficients from (8.2.34)... | Yes |
Theorem 8.4.3. Let \( \\left( {U,\\mathcal{H},{\\mathcal{H}}_{0}}\\right) \) and \( \\left( {{U}^{\\prime },{\\mathcal{H}}^{\\prime },{\\mathcal{H}}_{0}^{\\prime }}\\right) \) be two given systems as stated in Definition 8.4.1, and assume that all three conditions (i)–(iii) hold for both systems, i.e., that the two mul... | Proof. This is a standard result in operator theory, and we refer the reader to [BrJo02b, Chapter 2]. We further add that the underlying ideas date back to Kolmogorov [Kol77] in the 1930s. | No |
Theorem 8.4.4. Let \( N \in \mathbb{N}, N \geq 2 \), and let \( \left( {X,\sigma ,{\tau }_{i}}\right) \) satisfy the conditions in Lemma 8.2.2. Let \( {\left( {P}_{i}\right) }_{i = 0}^{N - 1} \) be the corresponding representation of \( {\mathcal{O}}_{N} \) in \( {\ell }^{2}\left( X\right) \). Let \( {\left( {S}_{i}\ri... | Proof. To prove that (i) in Definition 8.4.1 is satisfied note that for every subset \( E \subset X \), we have \[ E \subset {\sigma }^{-1}\left( {\sigma \left( E\right) }\right) \] (8.4.11) Substituting (a) from the theorem, we get \[ E \subset {\sigma }^{-1}\left( E\right) = \mathop{\bigcup }\limits_{{i = 0}}^{{N - 1... | Yes |
Lemma 8.4.7. Let \( N \in \mathbb{N}, N \geq 2 \), and let \( \left( {X,\sigma ,{\tau }_{i}}\right) \) satisfy the conditions in Lemma 8.2.2. Suppose some subset \( E \subset X \) satisfies conditions (a) and (b) in Theorem 8.4.4. Then there are subsets \( {E}_{i} \subset E \) such that\n\n\[ \sigma \left( {E}_{i}\righ... | Proof. Zorn's lemma. | No |
Proposition 8.4.9. Let \( N \in \mathbb{N}, N \geq 2 \), and let \( \left( {X,\sigma ,{\tau }_{i}}\right) \) satisfy the conditions in Lemma 8.2.2. Let \( \left( {E,{\left( {E}_{i}\right) }_{i \in I}}\right) \) be a system of subsets in \( X \) which satisfies the conditions in Lemma 8.4.7, where 1 is a chosen index se... | Proof. We leave the easy details to the reader. | No |
Lemma 9.3.2. Let \( {\mathcal{H}}_{i}, i = 1,2 \), be complex Hilbert spaces, and let \( n \in \mathbb{N}, n \geq 2 \) , be given. Let \( \left( {S}_{i}\right) \in \operatorname{Rep}\left( {{\mathcal{O}}_{n},{\mathcal{H}}_{1}}\right) \), and let \( {V}_{1},\ldots ,{V}_{n} \) be a system of operators in the Hilbert spac... | Proof. Consider arbitrary pairs of vectors \( {x}_{i},{y}_{i} \in {\mathcal{H}}_{i}, i = 1,2 \) . Using the relations (9.3.1) for the \( S \) -system, we find that the unitarity property (i) is equivalent to the following two identities:\n\n\[ \left\langle {{x}_{1} \mid {y}_{1}}\right\rangle \left\langle {{x}_{2}\left|... | Yes |
Lemma 9.3.5. [BrJ099a] Up to unitary equivalence of representations every per-mutative representation \( \left( {S}_{i}\right) \) of \( {\mathcal{O}}_{n} \) in a Hilbert space \( \mathcal{H} \) has the following form for some set \( X \) and some \( n \) -fold branch mapping \( R : X \rightarrow X \). | Proof. Let \( \left( {S}_{i}\right) \in \operatorname{Rep}\left( {{\mathcal{O}}_{n},\mathcal{H}}\right) \) be a permutative representation. Let \( X \) be an index set for some ONB which is permuted by the respective isometries \( {S}_{i} \) . As a result, there are maps \( {\sigma }_{i} : X \rightarrow X \) such that\... | Yes |
Lemma 9.5.2. The Fourier transform\n\n\\[ \n\\widehat{f}\\left( x\\right) = {\\int }_{-\\infty }^{\\infty }{e}^{-{i2\\pi xt}}f\\left( t\\right) {dt} \n\\]\n\n(9.5.7)\n\nrealizes a unitary isomorphism\n\n\\[ \nW : {L}^{2}\\left( \\mathbb{R}\\right) \\cong {\\ell }^{2}\\left( \\mathbb{Z}\\right) \\otimes {L}^{2}\\left( \... | Proof. Initially, \\( {Wf}\\left( \\cdot \\right) \\) is just a function from \\( \\mathbb{R} \\) mapping into \\( {\\ell }^{2}\\left( \\mathbb{Z}\\right) \\) . But an application of Parseval's identity and Fubini's theorem shows that\n\n\\[ \n{\\int }_{0}^{1}\\mathop{\\sum }\\limits_{{k \\in \\mathbb{Z}}}{\\left| \\wi... | Yes |
Theorem 9.5.5. Let \( {m}_{0},{m}_{1},\ldots ,{m}_{N - 1} \) be functions in \( {L}^{\infty }\left( \mathbb{T}\right) \) which satisfy the conditions in Lemma 9.5.3(b), and let \( {\varphi }_{0},{\varphi }_{1},{\varphi }_{2},\ldots \) be the corresponding sequence in \( {L}^{2}\left( \mathbb{R}\right) \) . Then the dou... | Proof. The fact that (9.5.14) is an ONB in \( {L}^{2}\left( \mathbb{R}\right) \) follows from Lemma 9.5.3(b). When the operators \( {V}_{i}, i = 0,1,\ldots, N - 1 \), are defined from the filter functions \( {m}_{i} \) , as in (9.5.6), it follows from Lemma 9.5.1 that \( \left( {V}_{i}\right) \in \operatorname{Rep}\lef... | Yes |
Then the unitary scaling operator\n\n\[ Uf\left( t\right) \mathrel{\text{:=}} \sqrt{N}f\left( {Nt}\right) \;\text{ for }f \in {L}^{2}\left( \mathbb{R}\right), t \in \mathbb{R} \]\n\nhas the tensor factorization\n\n\[ U = \mathop{\sum }\limits_{{i = 0}}^{{N - 1}}{S}_{i} \otimes {V}_{i}^{ * } \] | Proof. Once we have identified the isomorphism (9.5.18), the result follows from (9.5.16) in Theorem 9.5.5. Specifically,\n\n\[ U\left( {\left| n\right\rangle \otimes \left| k\right\rangle }\right) = \sqrt{N}\;{\varphi }_{n}\left( {{Nt} - k}\right) \]\n\n\[ = \mathop{\sum }\limits_{{i = 0}}^{{N - 1}}\mathop{\sum }\limi... | Yes |
Theorem 9.6.2. (a) There is an an orthonormal system \( {\varphi }_{0},{\varphi }_{1},{\varphi }_{2},\ldots \) in\n\n\( {L}^{2}\left( {{\mathcal{R}}_{4},{\left( dt\right) }^{1/2}}\right) \) which solves the system of equations\n\n\[ \begin{cases} {\varphi }_{4n}\left( t\right) & = {\varphi }_{n}\left( {4t}\right) + {\v... | Proof. We have presented the details in Sections 9.4 and 9.5 so that the proofs of the two theorems follow as an easy application. The main point is the observation that the two recursive systems (9.6.17) and (9.6.20) admit solutions which satisfy the respective orthogonality conditions. But the two Hilbert spaces \( {... | No |
Theorem 9.7.3. Let \( N = 4 \), and let \( \left\{ {{\varphi }_{n} \mid n \in {\mathbb{N}}_{0}}\right\} \) be the orthonormal sequence in \( {L}^{2}\left( {{\mathcal{R}}_{4},{\left( dt\right) }^{1/2}}\right) \) constructed in Theorem 9.6.2.\n\nThen the family\n\n\[ \left\{ {{e}_{\lambda }\left( t\right) {\varphi }_{j}\... | Proof. The proof follows from a combination of (i) Lemma 9.7.1, (ii) the discussion of the affine fractals in Section 9.6, and (iii) some basic observations of scaling and the Hausdorff measure; see also [DuJo06b] and the following list.\n\nSeveral observations are needed:\n\n(1) \( {\Lambda }_{4} \subset {\mathcal{R}}... | No |
Theorem 1.10. If \( G \) and \( {G}^{\prime } \) are Lie groups and \( \varphi : G \rightarrow {G}^{\prime } \) is a homomorphism of Lie groups, then \( \varphi \) has constant rank and \( \ker \varphi \) is a (closed) regular Lie subgroup of \( G \) of dimension \( \dim G - \operatorname{rk}\varphi \) where \( \operat... | Proof. It is well known (see [8]) that if a smooth map \( \varphi \) has constant rank, then \( {\varphi }^{-1}\{ e\} \) is a closed regular submanifold of \( G \) of dimension \( \dim G - \operatorname{rk}\varphi \) . Since \( \ker \varphi \) is a subgroup, it suffices to show that \( \varphi \) has constant rank. Wri... | Yes |
Theorem 1.15. Let \( G \) be a connected Lie group and \( U \) a neighborhood of \( e \) . Then \( U \) generates \( G \), i.e., \( G = { \cup }_{n = 1}^{\infty }{U}^{n} \) where \( {U}^{n} \) consists of all \( n \) -fold products of elements of \( U \) . | Proof. We may assume \( U \) is open without loss of generality. Let \( V = U \cap {U}^{-1} \subseteq U \) where \( {U}^{-1} \) is the set of all inverses of elements in \( U \) . This is an open set since the inverse map is continuous. Let \( H = { \cup }_{n = 1}^{\infty }{V}^{n} \) . By construction, \( H \) is an op... | Yes |
Lemma 1.17. Let \( G \) be a Lie group. The connected component \( {G}^{0} \) is a regular Lie subgroup of \( G \) . If \( {G}^{1} \) is any connected component of \( G \) with \( {g}_{1} \in {G}^{1} \), then \( {G}^{1} = \) \( {g}_{1}{G}^{0} \) . | Proof. We prove the second statement of the lemma first. Since left multiplication by \( {g}_{1} \) is a homeomorphism, it follows easily that \( {g}_{1}{G}^{0} \) is a connected component of \( G \) . But since \( e \in {G}^{0} \), this means that \( {g}_{1} \in {g}_{1}{G}^{0} \) so \( {g}_{1}{G}^{0} \cap {G}^{1} \neq... | Yes |
Theorem 1.18. If \( G \) is a Lie group and \( H \) a connected Lie subgroup so that \( G/H \) is also connected, then \( G \) is connected. | Proof. Since \( H \) is connected and contains \( e, H \subseteq {G}^{0} \), so there is a continuous map \( \pi : G/H \rightarrow G/{G}^{0} \) defined by \( \pi \left( {gH}\right) = g{G}^{0} \) . It is trivial that \( G/{G}^{0} \) has the discrete topology with respect to the quotient topology. The assumption that \( ... | Yes |
Theorem 1.20. The compact classical groups, \( {SO}\left( n\right) ,{SU}\left( n\right) \), and \( {Sp}\left( n\right) \), are connected. | Proof. Start with \( {SO}\left( n\right) \) and proceed by induction on \( n \) . As \( {SO}\left( 1\right) = \{ 1\} \), the case \( n = 1 \) is trivial. Next, observe that \( {SO}\left( n\right) \) has a transitive action on \( {S}^{n - 1} \) in \( {\mathbb{R}}^{n} \) by matrix multiplication. For \( n \geq 2 \), the ... | Yes |
Lemma 1.21. If \( H \) is a discrete normal subgroup of a connected Lie group \( G \), then \( H \) is contained in the center of \( G \) . | Proof. For each \( h \in H \), consider \( {C}_{h} = \left\{ {{gh}{g}^{-1} \mid g \in G}\right\} \) . Since \( {C}_{h} \) is the continuous image of the connected set \( G,{C}_{h} \) is connected. Normality of \( H \) implies \( {C}_{h} \subseteq H \) . Discreteness of \( H \) and connectedness of \( {C}_{h} \) imply t... | Yes |
Theorem 1.22. Let \( G \) be a connected Lie group.\n\n(1) The connected simply connected cover \( \widetilde{G} \) is a Lie group.\n\n(2) If \( \pi \) is the covering map and \( \widetilde{Z} = \ker \pi \), then \( \widetilde{Z} \) is a discrete central subgroup of \( \widetilde{G} \).\n\n(3) \( \pi \) induces a diffe... | Proof. Because coverings satisfy the lifting property (e.g., for any smooth map \( f \) of a connected simply connected manifold \( M \) to \( G \) with \( {m}_{0} \in M \) and \( {g}_{0} \in {\pi }^{-1}\left( {f\left( {m}_{0}\right) }\right) \) , there exists a unique smooth map \( \widetilde{f} : M \rightarrow \widet... | No |
Theorem 1.24. (1) \( {\pi }_{1}\left( {{SO}\left( 2\right) }\right) \cong \mathbb{Z} \) and \( {\pi }_{1}\left( {{SO}\left( n\right) }\right) \cong \mathbb{Z}/2\mathbb{Z} \) for \( n \geq 3 \) . | Proof. Start with \( {SO}\left( n\right) \) . As \( {SO}\left( 2\right) \cong {S}^{1},{\pi }_{1}\left( {{SO}\left( 2\right) }\right) \cong \mathbb{Z} \) . Recall from the proof of Theorem 1.20 that \( {SO}\left( n\right) \) has a transitive action on \( {S}^{n - 1} \) with stabilizer isomorphic to \( {SO}\left( {n - 1}... | Yes |
Theorem 1.40. (I) \( {\operatorname{Pin}}_{n}\left( \mathbb{R}\right) \) has two connected \( \left( {n \geq 2}\right) \) components with \( {\operatorname{Spin}}_{n}\left( \mathbb{R}\right) = \) \( {\operatorname{Pin}}_{n}{\left( \mathbb{R}\right) }^{0} \) . | Proof. For \( n \geq 2 \), consider the path \( t \rightarrow \gamma \left( t\right) = \cos t + {e}_{1}{e}_{2}\sin t \) . Since \( \gamma \left( t\right) = {e}_{1}\left( {-{e}_{1}\cos t + {e}_{2}\sin t}\right) \), it follows that \( \gamma \left( t\right) \in {\operatorname{Spin}}_{n}\left( \mathbb{R}\right) \) and so ... | Yes |
Lemma 1.44. (1) Up to multiplication by a nonzero scalar, there is a unique left invariant volume form on \( G \) . | Proof. Since \( \dim \mathop{\bigwedge }\limits_{n}^{ * }{\left( G\right) }_{e} = 1 \), up to multiplication by a nonzero scalar, there is a unique choice of \( {\omega }_{e} \in \mathop{\bigwedge }\limits_{n}^{ * }{\left( G\right) }_{e} \) . This choice uniquely extends to a left invariant \( n \) - form, \( \omega \)... | Yes |
Theorem 1.46. Let \( G \) be compact. The measure \( {dg} \) is left invariant, right invariant, and invariant under inversion, i.e.,\n\n\[{\int }_{G}f\left( {hg}\right) {dg} = {\int }_{G}f\left( {gh}\right) {dg} = {\int }_{G}f\left( {g}^{-1}\right) {dg} = {\int }_{G}f\left( g\right) {dg}\]\n\nfor \( h \in G \) and \( ... | Proof. It suffices to work with continuous \( f \) . Left invariance follows from the left invariance of \( {\omega }_{G} \) and the change of variables formula in Equation 1.41 \( \left( {l}_{h}\right. \) is clearly orientation preserving):\n\n\[{\int }_{G}f\left( {hg}\right) {dg} = {\int }_{G}\left( {f \circ {l}_{h}}... | No |
Theorem 1.47. For compact \( G \), the measure \( {dg} \) is the unique left invariant Borel measure on \( G \) normalized so \( G \) has measure 1. | Proof. Suppose \( {dh} \) is a left invariant Borel measure on \( G \) normalized so \( G \) has measure 1. Then for nonnegative measurable \( f \), definitions and the Fubini-Tonelli Theorem show that\n\n\[ \n{\int }_{G}f\left( g\right) {dg} = {\int }_{G}{\int }_{G}f\left( g\right) {dgdh} = {\int }_{G}{\int }_{G}f\lef... | Yes |
Theorem 2.8. As algebras, \[ {\mathcal{C}}_{n}\left( \mathbb{C}\right) \cong \left\{ \begin{matrix} \text{ End }\bigwedge W & n\text{ even } \\ \left( {\text{ End }\bigwedge W}\right) \bigoplus \left( {\text{ End }\bigwedge W}\right) & n\text{ odd. } \end{matrix}\right. \] | Proof. \( n \) even: For \( z = w + {w}^{\prime } \in {\mathbb{C}}^{n} \), define \( \widetilde{\Phi } : {\mathbb{C}}^{n} \rightarrow \) End \( / \smallsetminus W \) by \[ \widetilde{\Phi }\left( z\right) = \epsilon \left( w\right) - {2\iota }\left( {w}^{\prime }\right) \] As an algebra map, extend \( \widetilde{\Phi }... | No |
Theorem 2.9. As algebras,\n\n\\[ \n{\\mathcal{C}}_{n}^{ + }\\left( \\mathbb{C}\\right) \\cong \\left\\{ \\begin{matrix} \\left( {\\text{End}{\\bigwedge }^{ + }W}\\right) \\bigoplus \\left( {\\text{End}{\\bigwedge }^{ - }W}\\right) & n\\text{ even} \\\\ \\left( {\\text{End}\\bigwedge W}\\right) & n\\text{ odd.} \\end{ma... | Proof. \\( n \\) even: From the definition of \\( \\widetilde{\\Phi } \\) in the proof of Theorem 2.8, it is clear that the operators in \\( \\widetilde{\\Phi }\\left( {{\\mathcal{C}}_{n}^{ + }\\left( \\mathbb{C}\\right) }\\right) \\) preserve \\( \\mathop{\\bigwedge }\\limits^{ \\pm }W \\) . Thus restricted to \\( {\\... | No |
Theorem 2.12 (Schur’s Lemma). Let \( V \) and \( W \) be finite-dimensional representations of a Lie group \( G \) . If \( V \) and \( W \) are irreducible, then\n\n\[ \n\dim {\operatorname{Hom}}_{G}\left( {V, W}\right) = \left\{ \begin{array}{ll} 1 & \text{ if }V \cong W \\ 0 & \text{ if }V ≆ W. \end{array}\right. \n\... | Proof. If nonzero \( T \in {\operatorname{Hom}}_{G}\left( {V, W}\right) \), then \( \ker T \) is not all of \( V \) and \( G \) -invariant so irreducibility implies \( T \) is injective. Similarly, the image of \( T \) is nonzero and \( G \) -invariant, so irreducibility implies \( T \) is surjective and therefore a bi... | Yes |
Theorem 2.15. Every representation of a compact Lie group is unitary. | Proof. Begin with any inner product \( \langle \cdot , \cdot \rangle \) on \( V \) and define\n\n\[ \left( {v,{v}^{\prime }}\right) = {\int }_{G}\left\langle {{gv}, g{v}^{\prime }}\right\rangle {dg}. \]\n\nThis is well defined since \( G \) is compact and \( g \rightarrow \left\langle {{gv}, g{v}^{\prime }}\right\rangl... | Yes |
Corollary 2.17. Finite-dimensional representations of compact Lie groups are completely reducible. | Proof. Suppose \( V \) is a representation of a compact Lie group \( G \) that is reducible. Let \( \left( {\cdot , \cdot }\right) \) be a \( G \) -invariant inner product. If \( W \subseteq V \) is a proper \( G \) -invariant subspace, then \( V = W \oplus {W}^{ \bot } \) . Moreover, \( {W}^{ \bot } \) is also a prope... | Yes |
Corollary 2.19. If \( V \) is a finite-dimensional representation of a compact Lie group \( G, V \) is irreducible if and only if \( \dim {\operatorname{Hom}}_{G}\left( {V, V}\right) = 1 \) . | Proof. If \( V \) is irreducible, then Schur’s Lemma (Theorem 2.12) implies that \( \dim {\operatorname{Hom}}_{G}\left( {V, V}\right) = 1 \) . On the other hand, if \( V \) is reducible, then \( V = W \oplus {W}^{\prime } \) for proper submodules \( W,{W}^{\prime } \) of \( V \) . In particular, this shows that \( \dim... | Yes |
Corollary 2.20. (1) If \( V \) is a finite-dimensional representation of a compact Lie group \( G \), then \( \bar{V} \cong {V}^{ * } \). | Proof. For part (1), let \( \left( {\cdot , \cdot }\right) \) be a \( G \) -invariant inner product on \( V \) . Define the bijective linear map \( T : \bar{V} \rightarrow {V}^{ * } \) by \( {Tv} = \left( {\cdot, v}\right) \) for \( v \in V \) . To see that it is a \( G \) -map, calculate that \( g\left( {Tv}\right) = ... | Yes |
Corollary 2.21. Let \( V \) be a finite-dimensional representation of a compact Lie group \( G \) with a \( G \) -invariant inner product \( \left( {\cdot , \cdot }\right) \) . If \( {V}_{1},{V}_{2} \) are inequivalent irreducible submodules of \( V \), then \( {V}_{1} \bot {V}_{2} \), i.e., \( \left( {{V}_{1},{V}_{2}}... | Proof. Consider \( W = \left\{ {{v}_{1} \in {V}_{1} \mid \left( {{v}_{1},{V}_{2}}\right) = 0}\right\} \) . Since \( \left( {\cdot , \cdot }\right) \) is \( G \) -invariant, \( W \) is a submodule of \( {V}_{1} \) . If \( \left( {{V}_{1},{V}_{2}}\right) \neq 0 \), i.e., \( W \neq {V}_{1} \), then irreducibility implies ... | Yes |
Lemma 2.23. If \( {V}_{1},{V}_{2} \) are direct sums of irreducible submodules isomorphic to \( {E}_{\pi } \) , then so is \( {V}_{1} + {V}_{2} \) . | Proof. By finite dimensionality, it suffices to check the following: if \( \left\{ {W}_{i}\right\} \) are \( G \) - submodules of a representation and \( {W}_{1} \) is irreducible satisfying \( {W}_{1} \nsubseteq {W}_{2} \oplus \cdots \oplus {W}_{n} \) , then \( {W}_{1} \cap \left( {{W}_{2} \oplus \cdots \oplus {W}_{n}... | No |
Theorem 2.24 (Canonical Decomposition). Let \( V \) be a finite-dimensional representation of a compact Lie group \( G \) .\n\n(1) There is a \( G \) -intertwining isomorphism \( {\iota }_{\pi } \)\n\n\[ \n{\operatorname{Hom}}_{G}\left( {{E}_{\pi }, V}\right) \otimes {E}_{\pi }\overset{ \cong }{ \rightarrow }{V}_{\left... | Proof. For part (1), let \( T \in {\operatorname{Hom}}_{G}\left( {{E}_{\pi }, V}\right) \) be nonzero. Then \( \ker T = \{ 0\} \) by the irreducibility of \( {E}_{\pi } \) . Thus \( T \) is an equivalence of \( {E}_{\pi } \) with \( T\left( {E}_{\pi }\right) \), and so \( T\left( {E}_{\pi }\right) \subseteq \) \( {V}_{... | Yes |
Lemma 2.27. With respect to the inner product \( \langle \cdot , \cdot \rangle \) on \( {V}_{m}\left( {\mathbb{R}}^{n}\right) ,{\mathcal{H}}_{m}{\left( {\mathbb{R}}^{n}\right) }^{ \bot } = \) \( {\left| x\right| }^{2}{V}_{m - 2}\left( {\mathbb{R}}^{n}\right) \) where \( {\left| x\right| }^{2} = \mathop{\sum }\limits_{{... | Proof. Let \( p \in {V}_{m}\left( {\mathbb{R}}^{n}\right) \) and \( q \in {V}_{m - 2}\left( {\mathbb{R}}^{n}\right) \) . Then \( \left\langle {p,{\left| x\right| }^{2}q}\right\rangle = {\partial }_{{\left| x\right| }^{2}}{}_{\bar{q}}p = {\partial }_{\bar{q}}{\Delta p} = \) \( \langle {\Delta p}, q\rangle \) . Thus \( {... | Yes |
Lemma 2.29. If \( G \) is a compact Lie group with finite-dimensional representations \( U, V, W \) satisfying \( U \oplus V \cong U \oplus W \), then \( V \cong W \) . | Proof. Using Equation 2.18, decompose \( U \cong {\bigoplus }_{\left\lbrack \pi \right\rbrack \in \widehat{G}}{m}_{\pi }{E}_{\pi }, V \cong {\bigoplus }_{\left\lbrack \pi \right\rbrack \in \widehat{G}}{m}_{\pi }^{\prime }{E}_{\pi } \) , and \( W \cong {\bigoplus }_{\left\lbrack \pi \right\rbrack \in \widehat{G}}{m}_{\p... | Yes |
Lemma 2.31. \[ {\left. {\mathcal{H}}_{m}\left( {\mathbb{R}}^{n}\right) \right| }_{O\left( {n - 1}\right) } \cong {\mathcal{H}}_{m}\left( {\mathbb{R}}^{n - 1}\right) \oplus {\mathcal{H}}_{m - 1}\left( {\mathbb{R}}^{n - 1}\right) \oplus \cdots \oplus {\mathcal{H}}_{0}\left( {\mathbb{R}}^{n - 1}\right) . \] | Proof. Any \( p \in {V}_{m}\left( {\mathbb{R}}^{n}\right) \) may be uniquely written as \( p = \mathop{\sum }\limits_{{k = 0}}^{m}{x}_{1}^{k}{p}_{k} \) with \( {p}_{k} \in \) \( {V}_{m - k}\left( {\mathbb{R}}^{n - 1}\right) \) where \( {\mathbb{R}}^{n} \) is viewed as \( \mathbb{R} \times {\mathbb{R}}^{n - 1} \) . Sinc... | Yes |
Theorem 2.33. \( {\mathcal{H}}_{m}\left( {\mathbb{R}}^{n}\right) \) is an irreducible \( O\left( n\right) \) -module and, in fact, is irreducible under \( \operatorname{SO}\left( n\right) \) for \( n \geq 3 \) . | Proof. See Exercise 2.31 for the case of \( n = 2 \) . In this proof assume \( n \geq 3 \) .\n\n\( {\left. {\mathcal{H}}_{m}\left( {\mathbb{R}}^{n}\right) \right| }_{{SO}\left( {n - 1}\right) } \) contains, up to scalar multiplication, a unique \( {SO}\left( {n - 1}\right) \) - invariant function: If \( f \in {\mathcal... | No |
Lemma 3.2. \( {MC}\left( G\right) \) is a subalgebra of the set of smooth functions on \( G \) and contains the constant functions. If \( {\left\{ {v}_{i}^{\pi }\right\} }_{i = 1}^{{n}_{\pi }} \) is a basis for \( {E}_{\pi },\left\lbrack \pi \right\rbrack \in \widehat{G} \), then \( \left\{ {{f}_{{v}_{i}^{\pi },{v}_{j}... | Proof. By definition, a matrix coefficient is clearly a smooth function on \( G \) . If \( V,{V}^{\prime } \) are unitary representations of \( G \) with \( G \) -invariant inner products \( {\left( \cdot , \cdot \right) }_{V} \) and \( {\left( \cdot , \cdot \right) }_{{V}^{\prime }} \) , then \( U \oplus V \) is unita... | Yes |
Theorem 3.5. Let \( V,{V}_{i} \) be finite-dimensional representations of a compact Lie group \( G \) .\n\n(1) \( {\chi }_{V} \in {MC}\left( G\right) \) .\n\n(2) \( {\chi }_{V}\left( e\right) = \dim V \) .\n\n(3) If \( {V}_{1} \cong {V}_{2} \), then \( {\chi }_{{V}_{1}} = {\chi }_{{V}_{2}} \) .\n\n(4) \( {\chi }_{V}\le... | Proof. Each statement of the theorem is straightforward to prove. We prove parts (1), (4), (5), and (7) and leave the rest as an exercise (Exercise 3.3). For part (1), let \( \left\{ {v}_{i}\right\} \) be an orthonormal basis for \( V \) with respect to a \( G \) -invariant inner product \( \left( {\cdot , \cdot }\righ... | No |
Theorem 3.7. (1) Let \( V, W \) be finite-dimensional representations of a compact Lie group \( G \) . Then\n\n\[ \n{\int }_{G}{\chi }_{V}\left( g\right) \overline{{\chi }_{W}\left( g\right) }{dg} = \dim {\operatorname{Hom}}_{G}\left( {V, W}\right) .\n\]\n\nIn particular, \( {\int }_{G}{\chi }_{V}\left( g\right) {dg} =... | Proof. Begin with the assumption that \( V, W \) are irreducible. Let \( \left\{ {v}_{i}\right\} \) and \( \left\{ {w}_{j}\right\} \) be an orthonormal bases for \( V \) and \( W \) with respect to the \( G \) -invariant inner products \( {\left( \cdot , \cdot \right) }_{V} \) and \( {\left( \cdot , \cdot \right) }_{W}... | Yes |
Theorem 3.9. For compact Lie groups \( {G}_{i} \), a finite-dimensional representation \( W \) of \( {G}_{1} \times {G}_{2} \) is irreducible if and only if \( W \cong {V}_{1} \otimes {V}_{2} \) for finite-dimensional irreducible representations \( {V}_{i} \) of \( {G}_{i} \) . | Proof. If \( {V}_{i} \) are irreducible representations of \( {G}_{i} \), then \( {\int }_{{G}_{i}}{\left| {\chi }_{{V}_{i}}\left( g\right) \right| }^{2}{dg} = 1 \) . Since \( {\chi }_{{V}_{1} \otimes {V}_{2}}\left( {{g}_{1},{g}_{2}}\right) = {\chi }_{{V}_{1}}\left( {g}_{1}\right) {\chi }_{{V}_{2}}\left( {g}_{2}\right)... | Yes |
Lemma 3.13. Let \( \left( {\pi, V}\right) \) be a unitary representation of a compact Lie group \( G \) on a Hilbert space. There exists a nonzero finite-dimensional \( G \) -invariant (closed) subspace of \( V \) . | Proof. Begin with any self-adjoint positive compact operator \( {T}_{0} \in \operatorname{Hom}\left( {V, V}\right) \) , e.g., any nonzero finite rank projection will work. Using vector-valued integration in \( \operatorname{Hom}\left( {V, V}\right) \), define\n\n\[ T = {\int }_{G}\pi \left( g\right) \circ {T}_{0} \circ... | Yes |
Corollary 3.15. Let \( \left( {\pi, V}\right) \) be a unitary representation of a compact Lie group \( G \) on a Hilbert space. There exists finite-dimensional irreducible \( G \) -submodules \( {V}_{\alpha } \subseteq V \) so that\n\n\[ V = {\bigoplus }_{\alpha }{V}_{\alpha } \]\n\nIn particular, the irreducible unita... | Proof. Zorn's Lemma says that any partially ordered set has a maximal element if every linearly ordered subset has an upper bound. With this in mind, consider the collection of all sets \( \left\{ {{V}_{\alpha } \mid \alpha \in \mathcal{A}}\right\} \) satisfying the properties: (1) each \( {V}_{\alpha } \) is finite-di... | Yes |
Lemma 3.17. Let \( V \) be a unitary representation of a compact Lie group \( G \) on a Hilbert space with invariant inner product \( {\left( \cdot , \cdot \right) }_{V} \) and let \( {E}_{\pi },\left\lbrack \pi \right\rbrack \in \widehat{G} \), be an irreducible representation of \( G \) with invariant inner product \... | Proof. The adjoint of \( {T}_{2},{T}_{2}^{ * } \in \operatorname{Hom}\left( {V,{E}_{\pi }}\right) \), is still a \( G \) -map since\n\n\[{\left( {T}_{2}^{ * }\left( gv\right), x\right) }_{{E}_{\pi }} = {\left( gv,{T}_{2}x\right) }_{V} = {\left( v,{T}_{2}\left( {g}^{-1}x\right) \right) }_{V} = {\left( {T}_{2}^{ * }v,{g}... | Yes |
Theorem 3.19 (Canonical Decomposition). Let \( V \) be a unitary representation of a compact Lie group \( G \) on a Hilbert space.\n\n(1) There is a \( G \) -intertwining unitary isomorphism \( {\iota }_{\pi } \)\n\n\[ {\operatorname{Hom}}_{G}\left( {{E}_{\pi }, V}\right) \widehat{ \otimes }{E}_{\pi }\overset{ \cong }{... | Proof. As in the proof of Theorem 2.24, \( {\iota }_{\pi } \) is a well-defined \( G \) -map from \( {\operatorname{Hom}}_{G}\left( {{E}_{\pi }, V}\right) \otimes {E}_{\pi } \) to \( {V}_{\left\lbrack \pi \right\rbrack } \) with dense range (since \( {V}_{\left\lbrack \pi \right\rbrack } \) is a Hilbert space direct su... | Yes |
Lemma 3.20. The left and right actions of a compact Lie group \( G \) on \( C\left( G\right) \) and \( {L}^{2}\left( G\right) \) are representations and norm preserving. | Proof. The only statement from Definition 3.11 that still requires checking is continuity of the map \( \left( {g, f}\right) \rightarrow {l}_{g}f \) (since \( {r}_{g} \) is handled similarly). Working in \( C\left( G\right) \) first, calculate\n\n\[ \left| {{f}_{1}\left( {{g}_{1}^{-1}h}\right) - {f}_{2}\left( {{g}_{2}^... | Yes |
Lemma 3.23. With respect to the left action of a compact Lie group \( G \) on \( C{\left( G\right) }_{G\text{-fin }} \) and the action on \( {\operatorname{Hom}}_{G}\left( {{E}_{\pi }, C{\left( G\right) }_{G\text{-fin }}}\right) \) given by Equation 3.22,\n\n\[ \n{\operatorname{Hom}}_{G}\left( {{E}_{\pi }, C\left( G\ri... | Proof. Let \( T \in {\operatorname{Hom}}_{G}\left( {{E}_{\pi }, C{\left( G\right) }_{G\text{-fin }}}\right) \) and define \( {\lambda }_{T} \) as in the statement of the lemma. This is a \( G \) -map since\n\n\[ \n\left( {g{\lambda }_{T}}\right) \left( x\right) = {\lambda }_{T}\left( {{g}^{-1}x}\right) = \left( {T\left... | Yes |
Theorem 3.24. Let \( G \) be a compact Lie group. As a \( G \times G \) -module with \( \left( {{g}_{1},{g}_{2}}\right) \in \) \( G \times G \) acting as \( {r}_{{g}_{1}} \circ {l}_{{g}_{2}} = {l}_{{g}_{2}} \circ {r}_{{g}_{1}} \) on \( C{\left( G\right) }_{G\text{-fin }} \) , \[ C{\left( G\right) }_{G\text{-fin }} \con... | The intertwining isomorphism is induced by mapping \( \lambda \otimes x \in {E}_{\pi }^{ * } \otimes {E}_{\pi } \) to \( {f}_{\lambda \otimes x} \in \) \( C{\left( G\right) }_{G\text{-fin }} \) where \( {f}_{\lambda \otimes x}\left( g\right) = \lambda \left( {{g}^{-1}x}\right) \) for \( g \in G \) . Proof. The proof of... | Yes |
Theorem 3.25 (Peter-Weyl). Let \( G \) be a compact Lie group. \( C{\left( G\right) }_{G\text{-fin }} \) is dense in \( C\left( G\right) \) and in \( {L}^{2}\left( G\right) \) . | Proof. Since \( C\left( G\right) \) is dense in \( {L}^{2}\left( G\right) \), it suffices to prove the first statement. For this, recall that \( C{\left( G\right) }_{G\text{-fin }} \) is an algebra that is closed under complex conjugation and contains 1. By the Stone-Weierstrass Theorem, it only remains to show that \(... | Yes |
Corollary 3.27. Let \( G \) be a compact Lie group. If \( {\left\{ {v}_{i}^{\pi }\right\} }_{i = 1}^{{n}_{\pi }} \) is an orthonormal basis for \( {E}_{\pi },\left\lbrack \pi \right\rbrack \in \widehat{G} \), then \( \left\{ {{\left( \dim {E}_{\pi }\right) }^{\frac{1}{2}}{f}_{{v}_{i}^{\pi },{v}_{j}^{\pi }}^{{E}_{\pi }}... | Proof. This follows immediately from Lemma 3.2, Theorem 3.21, the Schur orthogonality relations, and the Peter-Weyl Theorem. | No |
Theorem 3.28. A compact Lie group \( G \) possesses a faithful representation, i.e., there exists a (finite-dimensional representation) \( \left( {\pi, V}\right) \) of \( G \) for which \( \pi \) is injective. | Proof. By the proof of the Peter-Weyl Theorem, for \( {g}_{1} \in {G}^{0},{g}_{1} \neq e \), there exists a finite-dimensional representation \( \left( {{\pi }_{1},{V}_{1}}\right) \) of \( G \), so that \( {\pi }_{1}\left( {g}_{1}\right) \) is not the identity operator. Thus \( \ker {\pi }_{1} \) is a closed proper Lie... | Yes |
Theorem 3.30. Let \( G \) be a compact Lie group and let \( \chi \) be the set of irreducible characters, i.e., \( \chi = \left\{ {{\chi }_{{E}_{\pi }} \mid \left\lbrack \pi \right\rbrack \in \widehat{G}}\right\} \). (1) The span of \( \chi \) equals the set of continuous class functions in \( C{\left( G\right) }_{G\te... | Proof. For part (1), recall from Theorem 3.24 that \( C{\left( G\right) }_{G\text{-fin }} \cong {\bigoplus }_{\left\lbrack \pi \right\rbrack \in \widehat{G}}{E}_{\pi }^{ * } \otimes {E}_{\pi } \) as a \( G \times G \) -module. View \( C{\left( G\right) }_{G\text{-fin }} \) and \( {E}_{\pi }^{ * } \otimes {E}_{\pi } \) ... | Yes |
Theorem 3.32. The map \( n \rightarrow {V}_{n}\left( {\mathbb{C}}^{2}\right) \) establishes an isomorphism \( \mathbb{N} \cong \widehat{{SU}\left( 2\right) } \) . | Proof. Viewing \( {S}^{1} \) as a subgroup of \( {SU}\left( 2\right) \) via the inclusion \( {e}^{i\theta } \rightarrow \operatorname{diag}\left( {{e}^{i\theta },{e}^{-{i\theta }}}\right) \) , Equation 2.25 calculates the character of \( {V}_{n}\left( {\mathbb{C}}^{2}\right) \) restricted to \( {S}^{1} \) to be\n\n(3.3... | No |
Corollary 3.40. Let \( G \) be a compact Lie group and \( f,{f}_{i} \in {L}^{2}\left( G\right) \). (1) Then the Parseval-Plancherel formula holds:\n\n\[ \parallel f{\parallel }_{{L}^{2}\left( G\right) }^{2} = \mathop{\sum }\limits_{{\left\lbrack \pi \right\rbrack \in \widehat{G}}}\dim {E}_{\pi }\parallel \pi \left( f\r... | Proof. Part (1) follows immediately from the Plancherel Theorem. | No |
Theorem 3.42 (Scalar Fourier Inversion). Let \( G \) be a compact Lie group and \( f \in \) \( \operatorname{span}\left( {{L}^{2}\left( G\right) * {L}^{2}\left( G\right) }\right) \subseteq C\left( G\right) \) . Then\n\n\[ f\left( e\right) = \mathop{\sum }\limits_{{\left\lbrack \pi \right\rbrack \in \widehat{G}}}\left( ... | Proof. If \( f = {f}_{1} * {f}_{2} \) for \( {f}_{i} \in {L}^{2}\left( G\right) \), then by Corollary 3.40,\n\n\[ f\left( e\right) = {\int }_{G}{f}_{1}\left( {g}^{-1}\right) {f}_{2}\left( g\right) {dg} = {\int }_{G}{f}_{2}\left( g\right) \overline{{\widetilde{f}}_{1}\left( g\right) }{dg} \]\n\n\[ = {\left( {f}_{2},{\wi... | Yes |
Theorem 3.43. Let \( G \) be a compact Lie group. The map \( f \rightarrow {\left( \widehat{f}\left( \pi \right) \right) }_{\left\lbrack \pi \right\rbrack \in \widehat{G}} \) establishes a unitary isomorphism\n\n\[ \n\left\{ {{L}^{2}\left( G\right) \text{ class functions }}\right\} \cong {l}^{2}\left( \widehat{G}\right... | Proof. This result is implicitly embedded in the proof of Theorem 3.30. However it is trivial to check directly. Observe that\n\n\[ \n\widehat{{\chi }_{{E}_{\gamma }}}\left( \pi \right) = {\int }_{G}{\chi }_{{E}_{\gamma }}\left( g\right) {\chi }_{{E}_{\pi }}\left( g\right) {dg}\n\]\n\nso that Theorem 3.7 implies \( {\c... | Yes |
Theorem 3.45. Let \( \\left( {\\gamma, V}\\right) \) be a representation of a compact Lie group \( G \) on a complete Hausdorff locally convex topological space.\n\n(1) For \( \\left\\lbrack \\pi \\right\\rbrack ,\\left\\lbrack {\\pi }^{\\prime }\\right\\rbrack \\in \\widehat{G} \), the operator \( \\left( {\\dim {E}_{... | Proof. For part (1), let \( g \\in G \) and \( v \\in V \), and observe that\n\n\[ \n\\left( {{g\\gamma }\\left( {\\chi }_{E\\pi }\\right) {g}^{-1}}\\right) v = {\\int }_{G}{\\chi }_{E\\pi }\\left( h\\right) {gh}{g}^{-1}{vdh} = {\\int }_{G}{\\chi }_{E\\pi }\\left( {{g}^{-1}{hg}}\\right) {hvdh} \n\]\n\n\[ \n= {\\int }_{... | Yes |
Theorem 3.46. Let \( \\left( {\\gamma, V}\\right) \) be a representation of a compact Lie group \( G \) on a Hausdorff complete locally convex topological space.\n\n(1) \( {V}_{G\\text{-fin }} = {\\bigoplus }_{\\left\\lbrack \\pi \\right\\rbrack \\in \\widehat{G}}{V}_{\\left\\lbrack \\pi \\right\\rbrack }^{0} \) . | Proof. For part (1), the definitions and Corollary 2.17 imply \( {V}_{G\\text{-fin }} = \\mathop{\\sum }\\limits_{{\\left\\lbrack \\pi \\right\\rbrack \\in \\widehat{G}}}{V}_{\\left\\lbrack \\pi \\right\\rbrack }^{0} \) , so it only remains to see the sum is direct. However, the existence of the projections in Theorem ... | Yes |
Corollary 3.47. Let \( G \) be a compact Lie group. Suppose \( \mathcal{S} \subseteq C\left( G\right) \) is a subspace equipped with a topology so that:\n\n(a) \( \mathcal{S} \) is dense in \( C\left( G\right) \),\n\n(b) \( \mathcal{S} \) is a Hausdorff complete locally convex topological space,\n\n(c) the topology on ... | Proof. Clearly \( {\mathcal{S}}_{\left\lbrack \pi \right\rbrack } \subseteq C{\left( G\right) }_{\left\lbrack \pi \right\rbrack } \) for \( \left\lbrack \pi \right\rbrack \in \widehat{G} \) . Note \( C{\left( G\right) }_{\left\lbrack \pi \right\rbrack } \cong {E}_{\pi }^{ * } \otimes {E}_{\pi } \) by Theorem 3.24. Argu... | Yes |
Theorem 4.2. Let \( G \) be a Lie subgroup of \( {GL}\left( {n,\mathbb{C}}\right) \). (a) Then \( \mathfrak{g} \) is a real vector space. (b) The Lie bracket is linear in each variable, skew symmetric, i.e., \( \left\lbrack {X, Y}\right\rbrack = \) \( - \left\lbrack {Y, X}\right\rbrack \), and satisfies the Jacobi iden... | Proof. Let \( {X}_{i} = {\gamma }_{i}^{\prime }\left( 0\right) \in \mathfrak{g} \). For \( r \in \mathbb{R} \), consider the smooth curve \( \gamma \) that maps a neighborhood of \( 0 \in \mathbb{R} \) to \( G \) defined by \( \gamma \left( t\right) = {\gamma }_{1}\left( {rt}\right) {\gamma }_{2}\left( t\right) \). The... | No |
Theorem 4.5. Let \( G \) be a Lie subgroup of \( {GL}\left( {n,\mathbb{C}}\right) \) and \( X \in \mathfrak{g} \). (a) Then \[ {\gamma }_{X}\left( t\right) = \exp \left( {tX}\right) = {e}^{tX} = \mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{{t}^{n}}{n!}{X}^{n}. \] (b) Moreover \( {\gamma }_{X} \) is a homomorphism and... | Proof. It is a familiar fact that the map \( t \rightarrow {e}^{tX} \) is a well-defined smooth homomorphism of \( \mathbb{R} \) into \( {GL}\left( {n,\mathbb{C}}\right) \) (Exercise 4.3). Hence, first extend \( \widetilde{X} \) to a vector field on \( {GL}\left( {n,\mathbb{C}}\right) \) by \( {\widetilde{X}}_{g} = {gX... | No |
Theorem 4.6. Let \( G \) be a Lie subgroup of \( {GL}\left( {n,\mathbb{C}}\right) \). (a) \( \mathfrak{g} = \left\{ {X \in \mathfrak{{gl}}\left( {n,\mathbb{C}}\right) \mid {e}^{tX} \in G}\right. \) for \( \left. {t \in \mathbb{R}}\right\} \). (b) The map \( \exp : \mathfrak{g} \rightarrow G \) is a local diffeomorphism... | Proof. To see \( \mathfrak{g} \) is contained in \( \left\{ {X \in \mathfrak{{gl}}\left( {n,\mathbb{C}}\right) \mid {e}^{tX} \in G\text{for}t \in \mathbb{R}}\right\} \), use Theorem 4.5. Conversely, if \( {e}^{tX} \in G \) for \( t \in \mathbb{R} \) for all \( X \in \mathfrak{{gl}}\left( {n,\mathbb{C}}\right) \), apply... | Yes |
Theorem 4.8. Suppose \( \varphi ,{\varphi }_{i} : H \rightarrow G \) are homomorphisms of Lie subgroups of general linear groups.\n\n(a) The following diagram is commutative:\n\n\[ \exp \begin{matrix} \mathfrak{h} & \overset{d\varphi }{ \rightarrow } & \mathfrak{g} \\ \downarrow & & \downarrow \\ H & \overset{\varphi }... | Proof. For part (a), observe that since \( \varphi \) is a homomorphism that\n\n\[ \frac{d}{dt}\varphi \left( {e}^{tX}\right) = {\left. \frac{d}{ds}\varphi \left( {e}^{\left( {t + s}\right) X}\right) \right| }_{s = 0} = {\left. \varphi \left( {e}^{tX}\right) \frac{d}{ds}\varphi \left( {e}^{sX}\right) \right| }_{s = 0} ... | Yes |
Corollary 4.9. Let \( G \) be a Lie subgroup of \( {GL}\left( {n,\mathbb{C}}\right) \) and let \( \gamma : \mathbb{R} \rightarrow G \) be a smooth homomorphism, i.e., \( \gamma \left( {s + t}\right) = \gamma \left( s\right) \gamma \left( t\right) \) for \( s, t \in \mathbb{R} \) . If \( {\gamma }^{\prime }\left( 0\righ... | Proof. View the multiplicative group \( {\mathbb{R}}^{ + } \) as a Lie subgroup of \( {GL}\left( {1,\mathbb{C}}\right) \) . Let \( \widetilde{\gamma },\sigma \) : \( {\mathbb{R}}^{ + } \rightarrow G \) be the two homomorphisms defined by \( \widetilde{\gamma } = \gamma \circ \ln \) and \( \sigma \left( x\right) = {e}^{... | Yes |
Lemma 4.13. For \( A, B \in {M}_{n}\left( \mathbb{C}\right) ,\left\lbrack {{\partial }_{A},{\partial }_{B}}\right\rbrack = {\partial }_{\left\lbrack A, B\right\rbrack } \) . | Proof. For the sake of clarity of exposition, we will verify this lemma for \( {M}_{n}\left( \mathbb{R}\right) \) and leave the general case of \( {M}_{n}\left( \mathbb{C}\right) \) to the reader. In this setting and with \( A \in {M}_{n}\left( \mathbb{R}\right) ,{\partial }_{A} \) is simply \( \mathop{\sum }\limits_{{... | No |
Theorem 4.14. Let \( G \) be a Lie subgroup of \( {GL}\left( {n,\mathbb{C}}\right) \). There is a bijection between the set of connected Lie subgroups of \( G \) and the set of subalgebras of \( \mathfrak{g} \). If \( H \) is a connected Lie subgroup of \( G \), the correspondence maps \( H \) to its Lie algebra \( \ma... | Proof. Suppose \( \mathfrak{h} \) is a subalgebra of \( \mathfrak{g} \). Let \( H \) be the unique maximal connected submanifold of \( G \) so that \( I \in H \), and so the tangent space of \( H \) at \( h \) corresponds to \( h\mathfrak{h} \) for \( h \in H \). Now the connected submanifold \( {h}_{0}^{-1}H,{h}_{0} \... | Yes |
Theorem 4.15. Let \( H \) and \( G \) be connected Lie subgroups of general linear groups and \( \varphi : H \rightarrow G \) a homomorphism of Lie groups. Then \( \varphi \) is a covering map if and only if \( {d\varphi } \) is an isomorphism. | Proof. If \( \varphi \) is a covering, then there is a neighborhood \( U \) of \( I \) in \( H \) and a neighborhood \( V \) of \( I \) in \( G \), so that \( \varphi \) restricts to a diffeomorphism \( \varphi : U \rightarrow V \) . Thus the differential at \( I,{d\varphi } \), is an isomorphism.\n\nSuppose now that \... | Yes |
Theorem 4.16. Let \( H \) and \( G \) be connected Lie subgroups of general linear groups with \( H \) simply connected. If \( \psi : \mathfrak{h} \rightarrow \mathfrak{g} \) is a homomorphism of Lie algebras, then there exists a unique homomorphism of Lie groups \( \varphi : H \rightarrow G \) so that \( {d\varphi } =... | Proof. Uniqueness follows from Theorem 4.8. For existence, suppose \( H \) is a Lie subgroup of \( {GL}\left( {n,\mathbb{C}}\right) \) and \( G \) is a subgroup of \( {GL}\left( {m,\mathbb{C}}\right) \) . Then we may view \( H \times G \) as a block diagonal Lie subgroup of \( {GL}\left( {n + m,\mathbb{C}}\right) \) . ... | Yes |
Theorem 5.1. Let \( G \) be a Lie subgroup of \( {GL}\left( {n,\mathbb{C}}\right) \). (a) For \( X, Y \in \mathfrak{g},\left\lbrack {X, Y}\right\rbrack = 0 \) if and only if \( {e}^{tX} \) and \( {e}^{sY} \) commute for \( s, t \in \mathbb{R} \). In this case, \( {e}^{X + Y} = {e}^{X}{e}^{Y} \). (b) If \( A \) is a con... | Proof. Since part (b) follows from part (a) and Theorems 1.15 and 4.6, it suffices to prove part (a). It is a familiar fact (Exercise 4.3) that when \( X \) and \( Y \) commute, i.e., \( \left\lbrack {X, Y}\right\rbrack = 0 \), that \( {e}^{{tX} + {sY}} = {e}^{tX}{e}^{sY} \). Since \( {e}^{{tX} + {sY}} = {e}^{{sY} + {t... | No |
Theorem 5.2. (b) If \( G \) is a compact Abelian Lie group, then \( \exp \) is a surjective map to \( {G}^{0} \), the connected component of \( G \) . | Proof. Let \( G \) be a compact Abelian group. By Theorem 5.1, \( \exp : \mathfrak{g} \rightarrow {G}^{0} \) is a homomorphism. By Theorems 1.15 and 4.6 it follows that exp is surjective, so \( {G}^{0} \cong \) \( \mathfrak{g}/\ker \left( \exp \right) \) . Since \( \mathfrak{g} \cong {\mathbb{R}}^{\dim \mathfrak{g}} \)... | Yes |
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