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Proposition 2.1. Any Euclidean vector bundle admits a Riemannian connection. | Proof. Any trivial Euclidean bundle \( \xi \) admits a Riemannian connection: If \( {X}_{1},\ldots ,{X}_{k} \) is an orthonormal parallelization of \( \xi \) and \( u \in E\left( \xi \right) \), define a section \( {X}^{u} \mathrel{\text{:=}} \sum \left\langle {u,{X}_{i}\left( {\pi \left( u\right) }\right) }\right\rang... | Yes |
Proposition 2.2. If \( R \) denotes the curvature tensor of a Riemannian connection \( \nabla \) on \( \xi \), then\n\n\[ \langle R\left( {U, V}\right) X, Y\rangle = - \langle R\left( {U, V}\right) Y, X\rangle ,\;U, V \in \mathfrak{X}M,\;X, Y \in {\Gamma \xi }.\] | Proof. Given \( u, v \in {M}_{p}, R\left( {u, v}\right) \) belongs to the Lie algebra of the holonomy group at \( p \) by Proposition 3.2 in Chapter 4. Since the connection is Riemannian, the holonomy group is a subgroup of the orthogonal group \( O\left( {E}_{p}\right) \) , so that \( R\left( {u, v}\right) \) is a ske... | Yes |
Proposition 2.3. Let \( M \) be a Riemannian manifold with Levi-Civita connection \( \nabla \) . If \( f : N \rightarrow M \) and \( U, V, W \in \mathfrak{X}N \), then\n\n\[ \left\langle {{\nabla }_{U}{f}_{ * }V,{f}_{ * }W}\right\rangle = \frac{1}{2}\left\{ {U\left\langle {{f}_{ * }V,{f}_{ * }W}\right\rangle + V\left\l... | Proof. By Lemma 2.1,\n\n\[ U\left\langle {{f}_{ * }V,{f}_{ * }W}\right\rangle = \left\langle {{\nabla }_{U}{f}_{ * }V,{f}_{ * }W}\right\rangle + \left\langle {{f}_{ * }V,{\nabla }_{U}{f}_{ * }W}\right\rangle \]\n\nThe result then follows from the proof of the uniqueness part in Theorem 2.2, once we establish that\n\n\[... | Yes |
Proposition 2.4. Let \( i : N \rightarrow M \) be an isometric immersion between Riemannian manifolds. If \( {\kappa }_{N},{\kappa }_{M} \) denote the connection maps of \( {\tau N},{\tau M} \) , and \( {\nabla }^{N},{\nabla }^{M} \) are the respective Levi-Civita connections, then\n\n(1) \( {\imath }_{ * }{\kappa }_{N... | Proof. Let \( X, Y, Z \in \mathfrak{X}N \) . By Theorem 2.2 and Proposition 2.3,\n\n\[ \n\left\langle {{\nabla }_{X}^{M}{\imath }_{ * }Y,{\imath }_{ * }Z}\right\rangle = \left\langle {{\nabla }_{X}^{N}Y, Z}\right\rangle \n\]\n\nbecause \( i \) is isometric. Furthermore, \( \left\langle {{\nabla }_{X}^{N}Y, Z}\right\ran... | Yes |
Proposition 3.1. Let \( R \) denote the curvature tensor of a Riemannian manifold \( M \) . The following identities hold for any vector fields \( X, Y, Z, U \) on \( M \) :\n\n(1) \( R\left( {X, Y}\right) Z = - R\left( {Y, X}\right) Z \) .\n\n(2) \( \langle R\left( {X, Y}\right) Z, U\rangle = - \langle R\left( {X, Y}\... | Proof. Statement (1) is true for any connection and follows from the definition of \( R \), whereas (2) holds for Riemannian connections and is the content of Proposition 2.2. Statement (3) is a consequence of the fact that the Levi-connection is torsion-free: We may assume that the vector fields involved have vanishin... | Yes |
Proposition 4.1 (The Gauss Equations). Consider an isometric immersion \( \imath : M \rightarrow \widetilde{M} \), where \( \dim \widetilde{M} \geq 2 \), and set \( m = \dim \widetilde{M} - \dim M \) . Given \( p \in M \), let \( x, y, z \in {M}_{p} \), and \( {n}_{1},\ldots ,{n}_{m} \) be an orthonormal basis of \( {\... | Proof. We only prove the first equation, since the second one is an immediate consequence of it. Extend \( x, y, z,{n}_{j} \) locally to \( X, Y, Z,{N}_{j} \) . Then\n\n\[ {\widetilde{\nabla }}_{Y}{\imath }_{ * }Z = {\left( {\widetilde{\nabla }}_{Y}{\imath }_{ * }Z\right) }^{T} + {\left( {\widetilde{\nabla }}_{Y}{\imat... | Yes |
Proposition 5.1. Let \( \pi : M \rightarrow B \) be a submersion, \( \mathcal{H} \) a distribution complementary to \( \ker {\pi }_{ * } \), and \( c : \left\lbrack {a, b}\right\rbrack \rightarrow B \) a regular curve. If \( M \) is compact, then for any \( p \in {\pi }^{-1}\left( {c\left( a\right) }\right) \), there e... | Proof. Let \( X \) be the vector field defined above, and consider the maximal integral curve \( \gamma \) of \( X \) with \( \gamma \left( a\right) = \left( {a, p}\right) \) . Now, \( {c}^{ * }M \) is compact because \( M \) is, and by Exercise 18 in Chapter \( 1,\gamma \) is defined on all of \( \left\lbrack {a, b}\r... | No |
Lemma 5.3. Suppose \( \widetilde{X},\widetilde{Y} \in \mathfrak{X}M \) are basic. Then \( {\left( {\nabla }_{\widetilde{X}}^{M}\widetilde{Y}\right) }^{h} \) is basic. In fact, if \( X, Y \in \mathfrak{X}B \) are \( \pi \) -related to \( \widetilde{X},\widetilde{Y} \), then \( {\nabla }_{X}^{B}Y \) is \( \pi \) -related... | Proof. This is an immediate consequence of (2.2) together with the fact that \( {\pi }_{ * }\left\lbrack {\widetilde{X},\widetilde{Y}}\right\rbrack = \left\lbrack {X, Y}\right\rbrack \circ \pi \) . | Yes |
Proposition 5.2. If \( x \in \mathcal{H} \), then \( {\dot{\gamma }}_{x}\left( t\right) \in \mathcal{H} \) for all \( t \), and \( \pi \circ {\gamma }_{x} = {\gamma }_{{\pi }_{ * }x} \) . | Proof. Let \( x \in {\mathcal{H}}_{p} \) . It clearly suffices to prove the statement in a neighborhood of \( p \) ; choosing this neighborhood to be compact, Proposition 5.1 guarantees the existence of a horizontal lift \( c \) of \( {\gamma }_{{\pi }_{ * }x} \) at \( p \) . Extend \( {\dot{\gamma }}_{{\pi }_{ * }x} \... | Yes |
Proposition 5.3. Let \( \widetilde{X},\widetilde{Y},\widetilde{Z} \in \mathfrak{X}M \) be basic, and denote by \( X, Y, Z \in \mathfrak{X}B \) the corresponding \( \pi \) -related vector fields on \( B \) . Then\n\n(1) \( {\pi }_{ * }{R}_{M}\left( {\widetilde{X},\widetilde{Y}}\right) \widetilde{Z} = {R}_{B}\left( {X, Y... | Proof. Statements (2) and (3) are direct consequences of (1). For (1), we have that\n\n\[ \n{R}_{M}\left( {\widetilde{X},\widetilde{Y}}\right) \widetilde{Z} = {\nabla }_{\widetilde{X}}^{M}{\nabla }_{\widetilde{Y}}^{M}\widetilde{Z} - {\nabla }_{\widetilde{Y}}^{M}{\nabla }_{\widetilde{X}}^{M}\widetilde{Z} - {\nabla }_{\l... | No |
Proposition 6.1. Let \( c : I \rightarrow M \) be a geodesic, \( {t}_{0} \in I \) . For any \( v, w \in \) \( {M}_{c\left( {t}_{0}\right) } \) there exists a unique Jacobi field \( Y \) along \( c \) with \( Y\left( {t}_{0}\right) = v \) and \( {Y}^{\prime }\left( {t}_{0}\right) = \) \( w \) . | Proof. Let \( {X}_{1},\ldots ,{X}_{n} \) be parallel fields along \( c \) such that \( {X}_{1}\left( {t}_{0}\right) ,\ldots ,{X}_{n - 1}\left( {t}_{0}\right) \) form an orthonormal basis of \( \dot{c}{\left( {t}_{0}\right) }^{ \bot } \), and \( {X}_{n} = \dot{c} \) . Any vector field \( Y \) along \( c \) can then be e... | Yes |
Let \( c : \left\lbrack {0, b}\right\rbrack \rightarrow M \) be a geodesic. If \( V \) is a variation of \( c \) through geodesics-meaning that \( t \mapsto V\left( {t, s}\right) \) is a geodesic for each \( s \), then the variational vector field \( t \mapsto {V}_{ * }{D}_{2}\left( {t,0}\right) \) is Jacobi along \( c... | Given a variation \( V \) of \( c \) through geodesics, define vector fields \( \widetilde{X} \) and \( \widetilde{Y} \) along \( V \) by \( \widetilde{X} = {V}_{ * }{D}_{1},\widetilde{Y} = {V}_{ * }{D}_{2} \) . By assumption, \( {\nabla }_{{D}_{1}}\widetilde{X} = 0 \), so that\n\n\[ R\left( {\widetilde{Y},\widetilde{X... | Yes |
Lemma 7.3. If \( A \subset M \) is compact, then \( {\operatorname{inj}}_{A} > 0 \) . | Proof. By Theorem 4.1 in Chapter 4, there exists an open neighborhood \( U \) of the zero section \( \left\{ {{0}_{p} \mid p \in A}\right\} \) in \( {\left. TM\right| }_{A} \) on which \( \left( {\pi ,\exp }\right) \) is an imbedding. By compactness of \( A \), there exist \( {p}_{1},\ldots ,{p}_{k} \in A \) and \( {\e... | Yes |
Proposition 10.1. Let \( N \) be a compact submanifold of a Riemannian manifold \( M \) with normal bundle \( \nu \) in \( M \) . There exists \( \epsilon > 0 \) such that \( \exp \) : \( E\left( {\nu }^{\epsilon }\right) \rightarrow {B}_{\epsilon }\left( N\right) \) is a diffeomorphism of the total space \( E\left( {\... | Proof. Since exp has maximal rank on the zero section \( s\left( N\right) \) of \( \nu \) and is injective on \( s\left( N\right) \), there exists, by Lemma 1.1 in Chapter 3, a neighborhood \( U \) of \( s\left( N\right) \) in \( \nu \) such that \( \exp : U \rightarrow M \) is a diffeomorphism onto its image. By compa... | No |
Proposition 10.2. Given \( p \in M \), denote by \( H \) the isotropy group \( {G}_{p} \) at \( p, \) by \( {\nu }_{p} \) the normal bundle in \( M \) of the orbit \( G\left( p\right) = G/H \) of \( p \), and by \( {\nu }_{p}^{\epsilon } \) the corresponding disk bundle of radius \( \epsilon \) . There exists \( \epsil... | Proof. Choose \( \epsilon > 0 \) so that Proposition 10.1 holds for \( N = G\left( p\right) \), and define \( F : G{ \times }_{H}U \rightarrow {B}_{\epsilon }\left( {G\left( p\right) }\right) \) by \( F\left\lbrack {g, u}\right\rbrack = g \circ \exp u \) . \( G \) is assumed to act by isometries, so that \( h \circ \ex... | Yes |
Proposition 0.3. The element \( w \in {H}^{2}\left( M\right) \) represented by \( \operatorname{Tr} \circ R \) is independent of the choice of connection. | Proof. Consider connections \( {\mathcal{H}}_{i} \) on \( \xi \) with curvature \( {R}_{i}, i = 1,2 \) . Let \( I = \left\lbrack {0,1}\right\rbrack \), and denote by \( p : M \times I \rightarrow M \) and \( t : M \times I \rightarrow I \) the respective projections. The bundle \( {p}^{ * }\xi \) then admits connection... | Yes |
Proposition 1.1. Let \( T \) denote a symmetric \( \left( {0, k}\right) \) tensor on \( \mathfrak{{gl}}\left( n\right) \), and \( \xi \) a rank \( n \) bundle over \( M \) with total space \( E \) and connection \( \nabla \) . If \( T \) is invariant, then it induces a parallel section \( \bar{T} \) of \( {\operatornam... | Proof. Given \( p \in M \), choose an isomorphism \( b : {\mathbb{R}}^{n} \rightarrow {E}_{p} \), and define\n\n\[ \bar{T}\left( p\right) \left( {{L}_{1} \otimes \cdots \otimes {L}_{k}}\right) = T\left( {{b}^{-1} \circ {L}_{1} \circ b,\ldots ,{b}^{-1} \circ {L}_{k} \circ b}\right) ,\;{L}_{i} \in \mathfrak{{gl}}\left( {... | Yes |
Proposition 1.2. Let \( \\xi \) denote a vector bundle over \( M, f : N \\rightarrow M \) . If \( w,\\widetilde{w} \) denote the Weil homomorphisms associated to \( \\xi ,{f}^{ * }\\xi \) respectively, then \( \\widetilde{w} = {f}^{ * } \\circ w. \) | Proof. Let \( R \) denote the curvature tensor of some connection on \( \\xi \) . By Cartan’s structure equation, the induced connection on \( {f}^{ * }\\xi \) is \( {f}^{ * }R \), so that for \( T \\in {S}_{{GL}\\left( n\\right) } \) of type \( \\left( {0, k}\\right) \), \n\n\[ \n\\widetilde{w}\\left( T\\right) = \\le... | Yes |
Lemma 3.1. For \( A \in \mathfrak{o}\left( {2k}\right) \), \[ \operatorname{Pf}\left( A\right) = \mathop{\sum }\limits_{{\left\{ {\left( {{i}_{1},{j}_{1}}\right) ,\ldots ,\left( {{i}_{k},{j}_{k}}\right) }\right\} \in P}}{\epsilon }^{{i}_{1}{j}_{1}\ldots {i}_{k}{j}_{k}}{a}_{{i}_{1}{j}_{1}}\cdots {a}_{{i}_{k}{j}_{k}}, \]... | Proof. Notice that the expression \( \left( {\operatorname{sgn}\sigma }\right) {a}_{\sigma \left( 1\right) \sigma \left( 2\right) }\cdots {a}_{\sigma \left( {{2k} - 1}\right) \sigma \left( {2k}\right) } \) remains unchanged when two pairs \( \left( {\sigma \left( {{2l} - 1}\right) ,\sigma \left( {2l}\right) }\right) \)... | Yes |
Proposition 3.1. For \( n = {2k} \) and \( A, B \in \mathfrak{{gl}}\left( n\right) ,\operatorname{Pf}\left( {{B}^{t}{AB}}\right) = \left( {\det B}\right) \operatorname{Pf}\left( A\right) \) . In particular, if \( B \in {SO}\left( n\right) \), then \( \operatorname{Pf}\left( {{B}^{-1}{AB}}\right) = \operatorname{Pf}\lef... | Proof.\n\n\[ \n{2}^{k}k!\operatorname{Pf}\left( {{B}^{t}{AB}}\right) = \mathop{\sum }\limits_{{\sigma \in {P}_{n}}}\left( {\operatorname{sgn}\sigma }\right) )\left( {\mathop{\sum }\limits_{{{i}_{1},{i}_{2} = 1}}^{n}{b}_{{i}_{1}\sigma \left( 1\right) }{a}_{{i}_{1}{i}_{2}}{b}_{{i}_{2}\sigma \left( 2\right) }}\right) \n\]... | Yes |
Corollary 3.1. For \( A \in \mathfrak{o}\left( {2k}\right) \) , \( \det A = \mathop{\operatorname{Pf}}\limits^{2}\left( A\right) \) . | Proof. Choose \( B \in O\left( n\right) \) such that \( {BA}{B}^{-1} = \left( {{\lambda }_{1}\ldots {\lambda }_{k}}\right) \) . It follows from Lemma 3.1 that \( \det \left( {{\lambda }_{1}\ldots {\lambda }_{k}}\right) = {\lambda }_{1}\cdots {\lambda }_{k} \) . By Proposition 3.1,\n\n\[ \operatorname{Pf}\left( A\right)... | Yes |
Lemma 4.1. \( \operatorname{Pf}\left( {A \circledast B}\right) = \operatorname{Pf}\left( A\right) \operatorname{Pf}\left( B\right) \) . | Proof. The statement is clear if \( n \) or \( m \) is odd, since\n\n\[ \n{\operatorname{Pf}}^{2}\left( {A \circledast B}\right) = \det \left( {A \circledast B}\right) = \det A \cdot \det B = {\operatorname{Pf}}^{2}\left( A\right) \cdot {\operatorname{Pf}}^{2}\left( B\right) ,\n\]\n\nand both sides vanish. If \( n = {2... | Yes |
Proposition 5.1. Let \( \xi \) be a Euclidean bundle over \( {M}^{4} \) with curvature tensor \( R \) . The first Pontrjagin form of \( \xi \) is given by \[ {p}_{1} = \frac{1}{{\left( 2\pi \right) }^{2}}\left( {{\left| {R}^{ + }\right| }^{2} - {\left| {R}^{ - }\right| }^{2}}\right) \omega . \] | Proof. Let \( {R}^{ij} \) denote as before the local 2-form on \( M \) given by \( {R}^{ij}\left( p\right) \left( {x, y}\right) = \) \( \left\langle {R\left( {x, y}\right) {U}_{j}\left( p\right) ,{U}_{i}\left( p\right) }\right\rangle \), where \( \left\{ {U}_{i}\right\} \) is a local orthonormal basis of sections of th... | Yes |
Proposition 5.3. The subbundles \( {\Lambda }_{2}^{ \pm }\left( \xi \right) \) are parallel under the induced connection; i.e., \( \widetilde{R} = {\widetilde{R}}_{ + } + {\widetilde{R}}_{ - } \), with \( {\widetilde{R}}_{ \pm } \in {A}_{2}\left( {M,\operatorname{End}{\Lambda }_{2}^{ \pm }\left( \xi \right) }\right) \)... | Proof. By Exercise 157, \( \widetilde{R}\left( {x, y}\right) \alpha = \left\lbrack {R\left( {x, y}\right) ,\alpha }\right\rbrack \) . Since \( {\Lambda }_{2}\left( {E}_{p}\right) \) is a direct sum of the ideals \( {\Lambda }_{2}^{ \pm }\left( {E}_{p}\right) \), the first statement is clear. The inner product on \( \ma... | No |
Proposition 5.4. Let \( {p}_{ \pm },{p}_{1} \) denote the first Pontrjagin forms of \( {\Lambda }_{2}^{ \pm }\left( \xi \right) \) , \( \xi \), and \( e \) the Euler form of \( \xi \) . Then\n\n(1) \( {p}_{ + } = \frac{2}{{\left( 2\pi \right) }^{2}}\left( {{\left| {R}_{ + }^{ + }\right| }^{2} - {\left| {R}_{ + }^{ - }\... | Proof. Consider a positively oriented orthonormal basis \( {u}_{1},\ldots ,{u}_{4} \) of \( {E}_{q} \) , and denote by \( {I}^{ \pm },{J}^{ \pm },{K}^{ \pm } \) the induced orthonormal bases of \( {\Lambda }_{2}^{ \pm }\left( {E}_{q}\right) \) ; i.e.,\n\n\( {I}^{ \pm } = \left( {1/\sqrt{2}}\right) \left( {{u}_{1} \land... | Yes |
Let \( V \) be a complex vector space, \( J \) the induced complex structure on the underlying real space \( {V}_{\mathbb{R}} \) . Given any inner product \( \langle \) , \( \rangle \) on \( {V}_{\mathbb{R}} \) for which \( J \) is skew-adjoint, the formula\n\n\[ \langle v, w{\rangle }_{\mathbb{C}} \mathrel{\text{:=}} ... | Proof. Given a real inner product on \( {V}_{\mathbb{R}} \) ,(8.2) defines a complex-valued function on \( V \times V \) that is clearly additive in the first variable. Given \( \alpha = \) \( a + {ib} \in \mathbb{C}, \)\n\n\[ \langle {\alpha v}, w{\rangle }_{\mathbb{C}} = \langle \left( {a + {ib}}\right) v, w\rangle +... | Yes |
Proposition 8.2. For any symplectic vector space \( \left( {V,\sigma }\right) \), there exists a basis \( {\alpha }_{1},\ldots ,{\alpha }_{2n} \) of the dual \( {V}^{ * } \) such that \( \sigma = \mathop{\sum }\limits_{{k = 1}}^{n}{\alpha }_{k} \land {\alpha }_{n + k} \) . | Proof. Let \( {v}_{1} \) be any nonzero vector in \( V \) . Since \( \sigma \) is nondegenerate, there exists some \( w \in V \) with \( \sigma \left( {{v}_{1}, w}\right) = 1 \) . Set \( {v}_{n + 1} \mathrel{\text{:=}} w, W \mathrel{\text{:=}} \operatorname{span}\left\{ {{v}_{1},{v}_{n + 1}}\right\} \), and \( Z \mathr... | Yes |
Proposition 8.3. If a real vector space has a complex structure \( J \), then it admits a compatible symplectic form \( \sigma \) . Conversely, any symplectic form \( \sigma \) on \( V \) induces a compatible complex structure \( J \) . In each case, there exists an inner product on \( V \) such that \( \sigma \left( {... | Proof. Given a complex structure \( J \) on \( V \), choose an inner product for which \( J \) is skew-adjoint, and hence also isometric. Then \( \sigma \), where \( \sigma \left( {v, w}\right) \mathrel{\text{:=}} \) \( \langle {Jv}, w\rangle \) is symplectic. Furthermore,\n\n\[ \sigma \left( {{Jv},{Jw}}\right) = \left... | Yes |
Proposition 8.4. If \( V \) is a complex vector space, then its realification \( {V}_{\mathbb{R}} \) inherits a canonical orientation. | Proof. An arbitrary basis \( \left\{ {{v}_{1},\ldots ,{v}_{n}}\right\} \) of \( V \) induces an element\n\n\[ \n{v}_{1} \land \cdots \land {v}_{n} \land J{v}_{1} \land \cdots \land J{v}_{n} \in \left( {{\Lambda }_{2n}{V}_{\mathbb{R}}}\right) \smallsetminus \{ 0\} .\n\]\n\nThe component of \( \left( {{\Lambda }_{2n}{V}_... | Yes |
For \( M \in {M}_{n, n}\left( \mathbb{C}\right) \) , \( \det h\left( M\right) = {\left| \det M\right| }^{2} \) . | The claim is easily seen to be true for diagonalizable matrices. But the latter are dense in \( {M}_{n, n}\left( \mathbb{C}\right) \) ; in fact, we may assume that \( M \in {M}_{n, n}\left( \mathbb{C}\right) \) is in Jordan canonical form. If not all the diagonal terms are distinct, then modifying them appropriately yi... | No |
Lemma 8.2. \( h\left( {U\left( n\right) }\right) = h\left( {{GL}\left( {n,\mathbb{C}}\right) }\right) \cap {SO}\left( {2n}\right) \) | Proof. By Lemma 8.1, it suffices to show that \( M \in U\left( n\right) \) iff \( h\left( M\right) \in \) \( O\left( {2n}\right) \) . But since \( h\left( {\bar{M}}^{t}\right) = h{\left( M\right) }^{t} \), we have that \( A \in U\left( n\right) \) iff \( A{\bar{A}}^{t} = {I}_{n} \) iff \( h\left( A\right) h{\left( A\ri... | Yes |
Proposition 8.5. For any \( M \in \mathfrak{u}\left( n\right) \), there exists \( A \in U\left( n\right) \) such that\n\n\[ \n{AM}{A}^{-1} = \left( \begin{array}{lll} i{\lambda }_{1} & & \\ & \ddots & \\ & & i{\lambda }_{n} \end{array}\right) ,\;{\lambda }_{i} \in \mathbb{R} \]\n\nequivalently,\n\n\[ \nh\left( A\right)... | Proof. Recall that an endomorphism \( L \) of \( {\mathbb{C}}^{n} \) is normal if \( L{L}^{ * } = {L}^{ * }L \) . The spectral theorem asserts that a normal endomorphism of \( {\mathbb{C}}^{n} \) has \( n \) orthonormal eigenvectors. The endomorphism \( v \mapsto {Lv} \mathrel{\text{:=}} M \cdot v \) is skew-adjoint, h... | Yes |
Proposition 9.1. If \( \xi \) is a complex bundle, then the total Chern class of its conjugate is given by\n\n\[ c\left( \bar{\xi }\right) = 1 - {c}_{1}\left( \xi \right) + {c}_{2}\left( \xi \right) - {c}_{3}\left( \xi \right) + \cdots \] | Proof. A Hermitian inner product \( \langle \) , \( \rangle {on\xi } \) induces a Hermitian inner product \( \langle \) , \( \rangle {on}\bar{\xi } \) given by\n\n\[ \langle U, V\rangle \mathrel{\text{:=}} \overline{\langle U, V\rangle } = \langle V, U\rangle ,\;U, V \in {\Gamma \xi }.\]\n\nA Hermitian connection \( \n... | Yes |
Proposition 1.1. The function \( \delta : G \rightarrow {\mathbb{R}}_{ + }^{ \times } \) is a quasicharacter. The measure \( \delta \left( h\right) {\mu }_{L}\left( h\right) \) is right-invariant. | Proof. Conjugation by first \( {g}_{1} \) and then \( {g}_{2} \) is the same as conjugation by \( {g}_{1}{g}_{2} \) in one step. Thus \( \delta \left( {{g}_{1}{g}_{2}}\right) = \delta \left( {g}_{1}\right) \delta \left( {g}_{2}\right) \), so \( \delta \) is a quasicharacter. Using (1.1), \[ \delta \left( g\right) {\int... | Yes |
Proposition 1.2. If \( G \) is compact, then \( G \) is unimodular and \( {\mu }_{L}\left( G\right) < \infty \) . | Proof. Since \( \delta \) is a homomorphism, the image of \( \delta \) is a subgroup of \( {\mathbb{R}}_{ + }^{ \times } \) . Since \( G \) is compact, \( \delta \left( G\right) \) is also compact, and the only compact subgroup of \( {\mathbb{R}}_{ + }^{ \times } \) is just \( \{ 1\} \) . Thus \( \delta \) is trivial, ... | Yes |
Proposition 1.3. If \( G \) is unimodular, then the map \( g \rightarrow {g}^{-1} \) is an isometry. | Proof. It is easy to see that \( g \rightarrow {g}^{-1} \) turns a left Haar measure into a right Haar measure. If left and right Haar measures agree, then \( g \rightarrow {g}^{-1} \) multiplies the left Haar measure by a positive constant, which must be 1 since the map has order 2. | Yes |
Proposition 2.1. If \( G \) is compact and \( \left( {\pi, V}\right) \) is any finite-dimensional complex representation, then \( V \) admits a \( G \) -equivariant inner product. | Proof. Start with an arbitrary inner product \( \langle \langle \) , \( \rangle \rangle \) . Averaging it gives another inner product,\n\n\[ \langle v, w\rangle = {\int }_{G}\langle \langle \pi \left( g\right) v,\pi \left( g\right) w\rangle \rangle \mathrm{d}g \]\n\nfor it is easy to see that this inner product is Herm... | Yes |
Proposition 2.2. If \( G \) is compact, then each finite-dimensional representation is the direct sum of irreducible representations. | Proof. Let \( \left( {\pi, V}\right) \) be given. Let \( {V}_{1} \) be a nonzero invariant subspace of minimal dimension. It is clearly irreducible. Let \( {V}_{1}^{ \bot } \) be the orthogonal complement of \( {V}_{1} \) with respect to a \( G \) -invariant inner product. It is easily checked to be invariant and is of... | Yes |
Proposition 2.3. The matrix coefficients of \( G \) are continuous functions. The pointwise sum or product of two matrix coefficients is a matrix coefficient, so they form a ring. | Proof. If \( v \in V \), then \( g \rightarrow \pi \left( g\right) v \) is continuous since by definition a representation \( \pi : G \rightarrow \mathrm{{GL}}\left( V\right) \) is continuous and so a matrix coefficient \( L\left( {\pi \left( g\right) v}\right) \) is continuous.\n\nIf \( \left( {{\pi }_{1},{V}_{1}}\rig... | Yes |
Proposition 2.4. If \( f \) is a matrix coefficient of \( \left( {\pi, V}\right) \), then \( \check{f}\left( g\right) = f\left( {g}^{-1}\right) \) is a matrix coefficient of \( \left( {\widehat{\pi },{V}^{ * }}\right) \) . | Proof. This is clear from (2.3), regarding \( v \) as a linear functional on \( {V}^{ * } \) . | No |
Theorem 2.1. Let \( f \) be a function on \( G \) . The following are equivalent.\n\n(i) The functions \( \lambda \left( g\right) f \) span a finite-dimensional vector space.\n\n(ii) The functions \( \rho \left( g\right) f \) span a finite-dimensional vector space.\n\n(iii) The function \( f \) is a matrix coefficient ... | Proof. It is easy to check that if \( f \) is a matrix coefficient of a particular representation \( V \), then so are \( \lambda \left( g\right) f \) and \( \rho \left( g\right) f \) for any \( g \in G \) . Since \( V \) is finite-dimensional, its matrix coefficients span a finite-dimensional vector space; in fact, a ... | Yes |
Lemma 2.1. Suppose that \( \left( {{\pi }_{1},{V}_{1}}\right) \) and \( \left( {{\pi }_{2},{V}_{2}}\right) \) are complex representations of the compact group \( G \) . Let \( \langle \) , \( \rangle {be} \) any inner product on \( {V}_{1} \) . If \( {v}_{i},{w}_{i} \in {V}_{i} \) , then the map \( T : {V}_{1} \rightar... | Proof. We have\n\n\[ T\left( {{\pi }_{1}\left( h\right) w}\right) = {\int }_{G}\left\langle {{\pi }_{1}\left( {gh}\right) w,{v}_{1}}\right\rangle {\pi }_{2}\left( {g}^{-1}\right) {v}_{2}\mathrm{\;d}g. \]\n\nThe variable change \( g \rightarrow g{h}^{-1} \) shows that this equals \( {\pi }_{2}\left( h\right) T\left( w\r... | Yes |
Theorem 2.3 (Schur orthogonality). Suppose that \( \left( {{\pi }_{1},{V}_{1}}\right) \) and \( \left( {{\pi }_{2},{V}_{2}}\right) \) are irreducible representations of the compact group \( G \) . Either every matrix coefficient of \( {\pi }_{1} \) is orthogonal in \( {L}^{2}\left( G\right) \) to every matrix coefficie... | Proof. We must show that if there exist matrix coefficients \( {f}_{i} : G \rightarrow \mathbb{C} \) of \( {\pi }_{i} \) that are not orthogonal, then there is an isomorphism \( T : {V}_{1} \rightarrow {V}_{2} \) . We may assume that the \( {f}_{i} \) have the form \( {f}_{i}\left( g\right) = \left\langle {{\pi }_{i}\l... | Yes |
Theorem 2.4 (Schur orthogonality). Let \( \left( {\pi, V}\right) \) be an irreducible representation of the compact group \( G \), with invariant inner product \( \langle \) , \( \rangle \) . Then there exists a constant \( d > 0 \) such that\n\n\[{\int }_{G}\left\langle {\pi \left( g\right) {w}_{1},{v}_{1}}\right\rang... | Proof. We will show that if \( {v}_{1} \) and \( {v}_{2} \) are fixed, there exists a constant \( c\left( {{v}_{1},{v}_{2}}\right) \) such that\n\n\[{\int }_{G}\left\langle {\pi \left( g\right) {w}_{1},{v}_{1}}\right\rangle \overline{\left\langle \pi \left( g\right) {w}_{2},{v}_{2}\right\rangle }\mathrm{d}g = c\left( {... | Yes |
Proposition 2.5. The character \( \chi \) of a representation \( \left( {\pi, V}\right) \) is a matrix coefficient of \( V \) . | Proof. If \( {v}_{1},\ldots ,{v}_{n} \) is a matrix of \( V \), and \( {L}_{1},\ldots ,{L}_{n} \) is the dual basis of \( {V}^{ * } \) , then \( \chi \left( g\right) = \mathop{\sum }\limits_{{i = 1}}^{n}{L}_{i}\left( {\pi \left( g\right) {v}_{i}}\right) \) . | Yes |
Proposition 2.6. Suppose that \( \left( {\pi, V}\right) \) is a representation of \( G \) . Let \( \chi \) be the character of \( \pi \) .\n\n(i) If \( g \in V \) then \( \chi \left( {g}^{-1}\right) = \overline{\chi \left( g\right) } \) .\n\n(ii) Let \( \left( {\widehat{\pi },{V}^{ * }}\right) \) be the contragredient ... | Proof. Since \( \pi \left( g\right) \) is unitary with respect to an invariant inner product \( \langle \) , \( \rangle , \) its eigenvalues \( {t}_{1},\ldots ,{t}_{n} \) all have absolute value 1, and so\n\n\[ \operatorname{tr}\pi {\left( g\right) }^{-1} = \mathop{\sum }\limits_{i}{t}_{i}^{-1} = \mathop{\sum }\limits_... | Yes |
Proposition 2.7. If \( \\left( {\\pi, V}\\right) \) is an irreducible representation and \( \\chi \) its character, then\n\n\[ \n{\\int }_{G}\\chi \\left( g\\right) \\mathrm{d}g = \\left\\{ \\begin{array}{l} 1\\text{ if }\\pi \\text{ is the trivial representation; } \\ \\ 0\\text{ otherwise. } \\end{array}\\right.\n\] | Proof. The character of the trivial representation is just the constant function 1, and since we normalized the Haar measure so that \( G \) has volume 1, this integral is 1 if \( \\pi \) is trivial. In general, we may regard \( {\\int }_{G}\\chi \\left( g\\right) \\mathrm{d}g \) as the inner product of \( \\chi \) wit... | Yes |
Proposition 2.8. If \( \left( {\pi, V}\right) \) is a representation of \( G \) and \( \chi \) its character, then\n\n\[ \n{\int }_{G}\chi \left( g\right) \mathrm{d}g = \dim \left( {V}^{G}\right) \n\] | Proof. Decompose \( V = { \oplus }_{i}{V}_{i} \) into a direct sum of irreducible invariant subspaces, and let \( {\chi }_{i} \) be the character of the restriction \( {\pi }_{i} \) of \( \pi \) to \( {V}_{i} \) . By Proposition \( {2.7},{\int }_{G}{\chi }_{i}\left( g\right) \mathrm{d}g = 1 \) if and only if \( {\pi }_... | Yes |
Lemma 2.2. Define a representation \( \Psi : \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \times \mathrm{{GL}}\left( {m,\mathbb{C}}\right) \rightarrow \mathrm{{GL}}\left( \Omega \right) \) where \( \Omega = {\operatorname{Mat}}_{n \times m}\left( \mathbb{C}\right) \) by \( \Psi \left( {{g}_{1},{g}_{2}}\right) : X \rightar... | Proof. Both \( \operatorname{tr}\Psi \left( {{g}_{1},{g}_{2}}\right) \) and \( \operatorname{tr}\left( {g}_{1}^{-1}\right) \operatorname{tr}\left( {g}_{2}\right) \) are continuous, and since diagonalizable matrices are dense in \( \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \) we may assume that both \( {g}_{1} \) and \(... | Yes |
Theorem 2.5 (Schur orthogonality). Let \( \left( {{\pi }_{1},{V}_{1}}\right) \) and \( \left( {{\pi }_{2},{V}_{2}}\right) \) be representations of \( G \) with characters \( {\chi }_{1} \) and \( {\chi }_{2} \) . Then\n\n\[ \n{\int }_{G}{\chi }_{1}\left( g\right) \overline{{\chi }_{2}\left( g\right) }\mathrm{d}g = \dim... | Proof. Define a representation \( \Pi \) of \( G \) on the space \( \Omega = {\operatorname{Hom}}_{\mathbb{C}}\left( {{V}_{1},{V}_{2}}\right) \) of all linear transformations \( T : {V}_{1} \rightarrow {V}_{2} \) by \n\n\[ \n\Pi \left( g\right) T = {\pi }_{2}\left( g\right) \circ T \circ {\pi }_{1}{\left( g\right) }^{-... | Yes |
Proposition 2.9. The constant \( d \) in Theorem 2.4 equals \( \dim \left( V\right) \) . | Proof. Let \( {v}_{1},\ldots ,{v}_{n} \) be an orthonormal basis of \( V, n = \dim \left( V\right) \) . We have\n\n\[ \chi \left( g\right) = \mathop{\sum }\limits_{i}\left\langle {{\pi }_{i}\left( g\right) {v}_{i},{v}_{i}}\right\rangle \]\n\nsince \( \left\langle {\pi \left( g\right) {v}_{j},{v}_{i}}\right\rangle \) is... | Yes |
Proposition 2.10. If \( f \in {\mathcal{M}}_{\pi } \) then so is \( \Theta \left( {{g}_{1},{g}_{2}}\right) f \) . The representations \( \Theta \) and \( \Pi \) are equivalent. | Proof. Let \( L \in {V}^{ * } \) and \( v \in V \) . Define \( {f}_{L, v}\left( g\right) = L\left( {\pi \left( g\right) v}\right) \) . The map \( L, v \mapsto \) \( {f}_{L, v} \) is bilinear, hence induces a linear map \( \sigma : {V}^{ * } \otimes V \rightarrow {\mathcal{M}}_{\pi } \) . It is surjective by the definit... | Yes |
Proposition 2.11. If \( f \) is the matrix coefficient of an irreducible representation \( \left( {\pi, V}\right) \), and if \( f \) is a class function, then \( f \) is a constant multiple of \( {\chi }_{\pi } \) . | Proof. By Schur’s lemma, there is a unique \( G \) -invariant vector in \( {\operatorname{Hom}}_{\mathbb{C}}\left( {V, V}\right) \) ; hence. by Proposition 2.10, the same is true of \( {\mathcal{M}}_{\pi } \) in the action of \( G \) by conjugation. This matrix coefficient is of course \( {\chi }_{\pi } \) . | No |
Theorem 2.6. If \( f \) is a matrix coefficient and also a class function, then \( f \) is a finite linear combination of characters of irreducible representations. | Proof. Write \( f = \mathop{\sum }\limits_{{i = 1}}^{n}{f}_{i} \), where each \( {f}_{i} \) is a class function of a distinct irreducible representation \( \left( {{\pi }_{i},{V}_{i}}\right) \) . Since \( f \) is conjugation-invariant, and since the \( {f}_{i} \) live in spaces \( {\mathcal{M}}_{{\pi }_{i}} \), which a... | Yes |
Proposition 3.1. Let \( X \) and \( Y \) be compact topological spaces with \( Y \) a metric space with distance function \( d \) . Let \( U \) be a set of continuous maps \( X \rightarrow Y \) such that for every \( x \in X \) and every \( \epsilon > 0 \) there exists a neighborhood \( N \) of \( x \) such that \( d\l... | Proof. Let \( {S}_{0} = \left\{ {{f}_{1},{f}_{2},{f}_{3},\ldots }\right\} \) be a sequence in \( U \) . We will show that it has a convergent subsequence. We will construct a subsequence that is uniformly Cauchy and hence has a limit. For every \( n > 1 \), we will construct a subsequence \( {S}_{n} = \left\{ {{f}_{n1}... | Yes |
Proposition 3.2 (Ascoli and Arzela). Suppose that \( X \) is a compact space and that \( U \subset C\left( X\right) \) is a bounded subset such that for each \( x \in X \) and \( \epsilon > 0 \) there is a neighborhood \( N \) of \( x \) such that \( \left| {f\left( x\right) - f\left( y\right) }\right| \leq \epsilon \)... | Proof. Since \( U \) is bounded, there is a compact interval \( Y \subset \mathbb{R} \) such that all functions in \( U \) take values in \( Y \) . The result follows from Proposition 3.1. | No |
Proposition 4.1. If \( \phi \in C\left( G\right) \), then \( {T}_{\phi } \) is a bounded operator on \( {L}^{1}\left( G\right) \) . If \( f \in {L}^{1}\left( G\right) \), then \( {T}_{\phi }f \in {L}^{\infty }\left( G\right) \) and\n\n\[{\left| {T}_{\phi }f\right| }_{\infty } \leq {\left| \phi \right| }_{\infty }{\left... | Proof. If \( f \in {L}^{1}\left( G\right) \), then\n\n\[{\left| {T}_{\phi }f\right| }_{\infty } = \mathop{\sup }\limits_{{g \in G}}\left| {{\int }_{G}\phi \left( {g{h}^{-1}}\right) f\left( h\right) \mathrm{d}h}\right| \leq {\left| \phi \right| }_{\infty }{\int }_{G}\left| {f\left( h\right) }\right| \mathrm{d}h,\]\n\npr... | Yes |
Proposition 4.2. If \( \phi \in C\left( G\right) \), then convolution with \( \phi \) is a bounded operator \( {T}_{\phi } \) on \( {L}^{2}\left( G\right) \) and \( \left| {T}_{\phi }\right| \leq {\left| \phi \right| }_{\infty } \). The operator \( {T}_{\phi } \) is compact, and if \( \phi \left( {g}^{-1}\right) = \ove... | Proof. Using (4.1), \( {L}^{\infty }\left( G\right) \subset {L}^{2}\left( G\right) \subset {L}^{1}\left( G\right) \), and by (4.2), \( {\left| {T}_{\phi }f\right| }_{2} \leq \) \( {\left| {T}_{\phi }f\right| }_{\infty } \leq {\left| \phi \right| }_{\infty }{\left| f\right| }_{1} \leq {\left| \phi \right| }_{\infty }{\l... | Yes |
Proposition 4.3. If \( \phi \in C\left( G\right) \), and \( \lambda \in \mathbb{C} \), the \( \lambda \) -eigenspace\n\n\[ V\left( \lambda \right) = \left\{ {f \in {L}^{2}\left( G\right) \mid {T}_{\phi }f = {\lambda f}}\right\} \]\n\nis invariant under \( \rho \left( g\right) \) for all \( g \in G \) . | Proof. Suppose \( {T}_{\phi }f = {\lambda f} \) . Then\n\n\[ \left( {{T}_{\phi }\rho \left( g\right) f}\right) \left( x\right) = {\int }_{G}\phi \left( {x{h}^{-1}}\right) f\left( {hg}\right) \mathrm{d}h. \]\n\nAfter the change of variables \( h \rightarrow h{g}^{-1} \), this equals\n\n\[ {\int }_{G}\phi \left( {{xg}{h}... | Yes |
Corollary 4.1. The matrix coefficients of \( G \) are dense in \( {L}^{2}\left( G\right) \) . | Proof. Since \( C\left( G\right) \) is dense in \( {L}^{2}\left( G\right) \), this follows from the Peter-Weyl theorem and (4.1). | Yes |
Theorem 4.2. Let \( G \) be a compact group that has no small subgroups. Then \( G \) has a faithful finite-dimensional representation. | Proof. Let \( U \) be a neighborhood of the identity that contains no subgroup but \( \{ 1\} \) . By the Peter-Weyl theorem, we can find a finite-dimensional representation \( \pi \) and a matrix coefficient \( f \) such that \( f\left( 1\right) = 0 \) but \( f\left( g\right) > 1 \) when \( g \notin U \) . The function... | Yes |
If \( F \) is a field, then the general linear group \( \mathrm{{GL}}\left( {n, F}\right) \) is the group of invertible \( n \times n \) matrices with coefficients in \( F \) . It is a Lie group. Assuming that \( F = \mathbb{R} \) or \( \mathbb{C} \), the group \( \operatorname{GL}\left( {n, F}\right) \) is an open set... | The special linear group is the subgroup \( \mathrm{{SL}}\left( {n, F}\right) \) of matrices with determinant 1 . It is a closed Lie subgroup of \( \mathrm{{GL}}\left( {n, F}\right) \) of dimension \( {n}^{2} - 1 \) or \( 2\left( {{n}^{2} - 1}\right) \). | No |
If \( F = \mathbb{R} \) or \( \mathbb{C} \), let \( \mathrm{O}\left( {n, F}\right) = \left\{ {g \in \mathrm{{GL}}\left( {n, F}\right) \mid g \cdot {}^{t}g = I}\right\} \). This is the \( n \times n \) orthogonal group. More geometrically, \( \mathrm{O}\left( {n, F}\right) \) is the group of linear transformations prese... | To see this, if \( \left( x\right) = {}^{t}\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) is represented as a column vector, we have \( Q\left( x\right) = Q\left( {{x}_{1},\ldots ,{x}_{n}}\right) = {}^{t}x \cdot x \), and it is clear that \( Q\left( {gx}\right) = Q\left( x\right) \) if \( g \cdot {}^{t}g = I \). | Yes |
Let \( F \) be a field and let \( A \) be an \( F \) -algebra. By a derivation of \( A \) we mean a map \( D : A \rightarrow A \) that is \( F \) -linear, and satisfies \( D\left( {fg}\right) = \) \( {fD}\left( g\right) + D\left( f\right) g \). | We have \( D\left( {1 \cdot 1}\right) = {2D}\left( 1\right) \), which implies that \( D\left( 1\right) = 0 \), and therefore \( D\left( c\right) = 0 \) for any \( c \in F \subset A \) . It is easy to check that if \( {D}_{1} \) and \( {D}_{2} \) are derivations, then so is \( \left\lbrack {{D}_{1},{D}_{2}}\right\rbrack... | Yes |
Proposition 5.1. Let \( U \) be an open subset of \( {\mathbb{R}}^{n} \), and let \( x \in U \) . Then we may find a smooth function \( f \) with compact support contained in \( U \) that does not vanish at \( x \) . | Proof. We may assume \( x = \left( {{x}_{1},\ldots ,{x}_{n}}\right) \) is the origin. Define\n\n\[ f\left( {{x}_{1},\ldots ,{x}_{n}}\right) = \left\{ \begin{matrix} {\mathrm{e}}^{-{\left( 1 - {\left| x\right| }^{2}/{r}^{2}\right) }^{-1}} & \text{ if }\left| x\right| \leq r, \\ 0 & \text{ otherwise. } \end{matrix}\right... | Yes |
Proposition 5.2. Let \( G \) be a closed Lie subgroup of \( \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \), and let \( X \in \) \( {\operatorname{Mat}}_{n}\left( \mathbb{C}\right) \) . Then the path \( t \rightarrow \exp \left( {tX}\right) \) is tangent to the submanifold \( G \) of \( \mathrm{{GL}}\left( {n,\mathbb{C}}\... | Proof. If \( \exp \left( {tX}\right) \) is contained in \( G \) for all \( t \), then clearly it is tangent to \( G \) at \( t = 0 \) . We must prove the converse. Suppose that \( \exp \left( {{t}_{0}X}\right) \notin G \) for some \( {t}_{0} > \) 0 . Using Proposition 5.1, Let \( {\phi }_{0} \) be a smooth compactly su... | Yes |
Proposition 5.3. Let \( G \) be a closed Lie subgroup of \( \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \). The set \( \operatorname{Lie}\left( G\right) \) of all \( X \in {\operatorname{Mat}}_{n}\left( \mathbb{C}\right) \) such that \( \exp \left( {tX}\right) \subset G \) is a vector space whose dimension is equal to th... | Proof. This is clear from the characterization of Proposition 5.2. | No |
Proposition 5.4. Let \( G \) be a closed Lie subgroup of \( \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \). The map\n\n\[ \nX \rightarrow \exp \left( X\right)\n\]\n\ngives a diffeomorphism of a neighborhood of the identity in \( \operatorname{Lie}\left( G\right) \) onto a neighborhood of the identity in \( G \) . | Proof. First we note that since \( \exp \left( X\right) = I + X + \frac{1}{2}{X}^{2} + \cdots \), the Jacobian of exp at the identity is 1 , so exp induces a diffeomorphism of an open neighborhood \( U \) of the identity in \( {\operatorname{Mat}}_{n}\left( \mathbb{C}\right) \) onto a neighborhood of the identity in \(... | Yes |
Proposition 5.5. If \( G \) is a closed Lie subgroup of \( \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \), and if \( X, Y \in \) \( \operatorname{Lie}\left( G\right) \), then \( \left\lbrack {X, Y}\right\rbrack \in \operatorname{Lie}\left( G\right) \) . | Proof. It is evident that \( \operatorname{Lie}\left( G\right) \) is mapped to itself under conjugation by elements of \( G \) . Thus, \( \operatorname{Lie}\left( G\right) \) contains\n\n\[ \n\frac{1}{t}\left( {{\mathrm{e}}^{tX}Y{\mathrm{e}}^{-{tX}} - Y}\right) = {XY} - {YX} + \frac{t}{2}\left( {{X}^{2}Y - {2XYX} + Y{X... | Yes |
Let \( \mathfrak{{sl}}\left( {n, F}\right) \) be the subspace of \( X \in \mathfrak{{gl}}\left( {n, F}\right) \) such that \( \operatorname{tr}\left( X\right) = \) 0 . This is a Lie subalgebra, and it is the Lie algebra of \( \mathrm{{SL}}\left( {n, F}\right) \) when \( F = \mathbb{R} \) or \( \mathbb{C} \) . | This follows immediately from the fact that \( \det \left( {\mathrm{e}}^{X}\right) = {\mathrm{e}}^{\operatorname{tr}\left( X\right) } \) for any matrix \( X \) because if \( {x}_{1},\ldots ,{x}_{n} \) are the eigenvalues of \( X \), then \( {\mathrm{e}}^{{x}_{1}},\ldots ,{\mathrm{e}}^{{x}_{n}} \) are the eigenvalues of... | Yes |
If \( F = \mathbb{R} \) or \( \mathbb{C} \), the Lie algebra of \( \mathrm{O}\left( {n, F}\right) \) is \( \mathfrak{o}\left( {n, F}\right) \) . The dimension of \( \mathrm{O}\left( n\right) \) is \( \frac{1}{2}\left( {{n}^{2} - n}\right) \), and the dimension of \( \mathrm{O}\left( {n,\mathbb{C}}\right) \) is \( {n}^{... | Proof. Let \( G = \mathrm{O}\left( {n, F}\right) ,\mathfrak{g} = \operatorname{Lie}\left( G\right) \) . Suppose \( X \in \mathfrak{o}\left( {n, F}\right) \) . Exponentiate the identity \( - {tX} = {t}^{t}X \) to get\n\n\[ \exp {\left( tX\right) }^{-1} = {}^{t}\exp \left( {tX}\right) \]\n\nwhence \( \exp \left( {tX}\rig... | Yes |
Let \( \mathfrak{u}\left( n\right) \) be the set of \( X \in \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \) such that \( X + \overline{{}^{t}X} = 0 \) . One checks easily that this is closed under the \( \mathfrak{{gl}}\left( {n,\mathbb{C}}\right) \) Lie bracket \( \left\lbrack {X, Y}\right\rbrack = \) \( {XY} - {YX} \) ... | Despite the fact that these matrices have complex entries, this is a real Lie algebra, for it is only a real vector space, not a complex one. (It is not closed under multiplication by complex scalars.) It may be checked along the lines of Proposition 5.6 that \( \mathfrak{u}\left( n\right) \) is the Lie algebra of \( \... | No |
Lemma 6.1. Suppose that \( f \) is a smooth function on a neighborhood \( U \) of the origin in \( {\mathbb{R}}^{n} \), and \( f\left( {0,{x}_{2},\ldots ,{x}_{n}}\right) = 0 \) for \( \left( {0,{x}_{2},\ldots ,{x}_{n}}\right) \in U \) . Then\n\n\[ g\left( {{x}_{1},{x}_{2},\ldots ,{x}_{n}}\right) = \left\{ {\begin{matri... | Proof. We show first that \( g \) is continuous. Indeed, with \( {x}_{2},\ldots ,{x}_{n} \) fixed,\n\n\[ \mathop{\lim }\limits_{{{x}_{1} \rightarrow 0}}{x}_{1}^{-1}f\left( {{x}_{1},\ldots ,{x}_{n}}\right) = \left( {\partial f/\partial {x}_{1}}\right) \left( {0,{x}_{2},\ldots ,{x}_{n}}\right) \]\n\nby the definition of ... | Yes |
Proposition 6.1. Let \( m \in M \), where \( M \) is a smooth manifold of dimension \( n \). Let \( \mathcal{O} = {\mathcal{O}}_{m} \) and \( \mathcal{M} = {\mathcal{M}}_{m} \). Let \( {x}_{1},\ldots ,{x}_{n} \) be the germs of a set of local coordinates at \( m \). Then \( {x}_{1},\ldots ,{x}_{n} \) generate the ideal... | Proof. Although this is really a statement about germs of functions, we will work with representative functions defined in some neighborhood of \( m \).\n\nIf \( f \in \mathcal{M} \), we write \( f = {f}_{1} + {f}_{2} \), where \( {f}_{1}\left( {{x}_{1},\ldots ,{x}_{n}}\right) = f\left( {0,{x}_{2},\ldots ,{x}_{n}}\righ... | Yes |
Proposition 6.2. Let \( m \) be a point on an \( n \) -dimensional smooth manifold \( M \) . Every local derivation of \( {\mathcal{O}}_{m} \) is of the form (6.2). The set \( {T}_{m}\left( M\right) \) of such local derivations is an \( n \) -dimensional real vector space. | Proof. If \( f \) and \( g \) both vanish at \( m \), then (6.1) implies that a local derivation \( X \) vanishes on \( {\mathcal{M}}^{2} \), and by Proposition 6.1 it is therefore determined by its values on \( {x}_{1},\ldots ,{x}_{n} \) . If these are \( {a}_{1},\ldots ,{a}_{n} \), then \( X \) agrees with the right-... | No |
Proposition 6.3. There is a one-to-one correspondence between vector fields on a smooth manifold \( M \) and derivations of \( {C}^{\infty }\left( {M,\mathbb{R}}\right) \) . Specifically, if \( D \) is any derivation of \( {C}^{\infty }\left( {M,\mathbb{R}}\right) \), there is a unique vector field \( X \) on \( M \) s... | Proof. We show first that if \( m \in M \), and if \( f \in A = {C}^{\infty }\left( {M,\mathbb{R}}\right) \) has germ zero at \( m \), then the function \( {Df} \) vanishes at \( m \) . This implies that \( D \) induces a well-defined map \( {X}_{m} : {\mathcal{O}}_{m} \rightarrow \mathbb{R} \) that is a local derivati... | No |
The vector space of left-invariant vector fields is closed under \( \left\lbrack \text{,}\right\rbrack {andisaLiealgebraofdimensiondim}\left( G\right) \) . If \( {X}_{e} \in {T}_{e}\left( G\right) \), there is a unique left-invariant vector field \( X \) on \( G \) with the prescribed tangent vector at the identity. | Given a tangent vector \( {X}_{e} \) at the identity element \( e \) of \( G \), we may define a left-invariant vector field by \( {X}_{g} = {L}_{g, * }\left( {X}_{e}\right) \), and conversely any left-invariant vector field must satisfy this identity, so the space of left-invariant vector fields is isomorphic to the t... | Yes |
Lemma 7.1. Let \( f \) be a smooth map from a neighborhood of the origin in \( {\mathbb{R}}^{n} \) into a finite-dimensional vector space. We may write\n\n\[ f\left( x\right) = {c}_{0} + {c}_{1}\left( x\right) + B\left( {x, x}\right) + r\left( x\right) ,\]\n\nwhere \( {c}_{1} : {\mathbb{R}}^{n} \rightarrow V \) is line... | Proof. This is just the familiar Taylor expansion. Denoting \( u = \left( {{u}_{1},\ldots ,{u}_{n}}\right) \) , let \( {c}_{0} = f\left( 0\right) \),\n\n\[ {c}_{1}\left( u\right) = \mathop{\sum }\limits_{i}\frac{\partial f}{\partial {x}_{i}}\left( 0\right) {u}_{i} \]\n\nand\n\n\[ B\left( {u, v}\right) = \frac{1}{2}\mat... | Yes |
Proposition 7.2. If \( X, Y \in {\operatorname{Mat}}_{n}\left( \mathbb{C}\right) \), and if \( f \) is a smooth function on \( G = \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \), then \( \mathrm{d}\left\lbrack {X, Y}\right\rbrack f = \mathrm{d}X\left( {\mathrm{\;d}{Yf}}\right) - \mathrm{d}Y\left( {\mathrm{\;d}{Xf}}\right... | Proof. We fix a function \( f \in {C}^{\infty }\left( G\right) \) and an element \( g \in G \) . By Lemma 7.1, we may write, for \( X \) near 0,\n\n\[ f\left( {g\left( {I + X}\right) }\right) = {c}_{0} + {c}_{1}\left( X\right) + B\left( {X, X}\right) + r\left( X\right) ,\]\n\nwhere \( {c}_{1} \) is linear in \( X, B \)... | Yes |
Proposition 7.3. If \( \phi : G \rightarrow H \) is a Lie group homomorphism, then \( \operatorname{Lie}\left( \phi \right) \) : \( \operatorname{Lie}\left( G\right) \rightarrow \operatorname{Lie}\left( H\right) \) is a Lie algebra homomorphism. | Proof. If \( X, Y \in G \), then \( {X}_{e} \) and \( {Y}_{e} \) are local derivations of \( {\mathcal{O}}_{e}\left( G\right) \), and it is clear from the definitions that \( {\phi }_{ * }\left( \left\lbrack {{X}_{e},{Y}_{e}}\right\rbrack \right) = \left\lbrack {{\phi }_{ * }\left( {X}_{e}\right) ,{\phi }_{ * }\left( {... | Yes |
Proposition 8.1. Suppose that \( M \) is a smooth manifold, \( m \in M \), and \( X \) is a vector field on \( M \). Then, for sufficiently small \( \epsilon > 0 \), there exists a path \( p : \left( {-\epsilon ,\epsilon }\right) \rightarrow M \) such that \( p\left( 0\right) = m \) and \( {p}_{ * }\left( {\mathrm{\;d}... | Proof. In terms of local coordinates \( {x}_{1},\ldots ,{x}_{n} \) on \( M \), the vector field \( X \) is\n\n\[ \sum {a}_{i}\left( {{x}_{1},\ldots ,{x}_{n}}\right) \frac{\partial }{\partial {x}_{i}} \]\n\nwhere the \( {a}_{i} \) are smooth functions in the coordinate neighborhood. If a path \( p\left( t\right) \) is s... | Yes |
Theorem 8.1. Let \( G \) be a Lie group and \( \mathfrak{g} \) its Lie algebra. There exists a map \( \exp : \mathfrak{g} \rightarrow G \) that is a local homeomorphism in a neighborhood of the origin in \( \mathfrak{g} \) such that, for any \( X \in \mathfrak{g}, t \rightarrow \exp \left( {tX}\right) \) is an integral... | Proof. Let \( X \in \mathfrak{g} \) . We know that for sufficiently small \( \epsilon > 0 \) there exists an integral curve \( p : \left( {-\epsilon ,\epsilon }\right) \rightarrow G \) for the left-invariant vector field \( X \) with \( p\left( 0\right) = 1 \) . We show first that if \( p : \left( {a, b}\right) \righta... | Yes |
Proposition 8.2. Let \( G, H \) be Lie groups and let \( \mathfrak{g},\mathfrak{h} \) be their respective Lie algebras. Let \( f : G \rightarrow H \) be a homomorphism. Then the following diagram is commutative: | Proof. It is clear from the definitions that \( f \) takes an integral curve for a left-invariant vector field \( X \) on \( G \) to an integral curve for \( \mathrm{d}f\left( X\right) \), and the statement follows. | No |
If \( \pi : G \rightarrow \mathrm{{GL}}\left( V\right) \) is a representation, where \( V \) is a real or complex vector space, then the Lie algebra of \( \mathrm{{GL}}\left( V\right) \) is \( \operatorname{End}\left( V\right) \), so the differential \( \operatorname{Lie}\left( \pi \right) : \operatorname{Lie}\left( G\... | By the universal property of \( U\left( \mathfrak{g}\right) \) in Theorem 10.1, A Lie algebra representation \( \pi : \mathfrak{g} \rightarrow \operatorname{End}\left( V\right) \) extends to a ring homomorphism \( U\left( \mathfrak{g}\right) \rightarrow \operatorname{End}\left( V\right) \) , which we continue to denote... | Yes |
Theorem 8.2. Let \( G \) be a Lie group, \( \mathfrak{g} \) its Lie algebra, and \( \operatorname{Ad} : G \rightarrow \mathrm{{GL}}\left( \mathfrak{g}\right) \) the adjoint representation. Then the Lie group representation \( \mathfrak{g} \rightarrow \operatorname{End}\left( \mathfrak{g}\right) \) corresponding to Ad b... | Proof. It will be most convenient for us to think of elements of the Lie algebra as tangent vectors at the identity or as local derivations of the local ring there. Let \( X, Y \in \mathfrak{g} \) . If \( f \in {C}^{\infty }\left( G\right) \), define \( c\left( g\right) f\left( h\right) = f\left( {{g}^{-1}{hg}}\right) ... | Yes |
Lemma 9.1. In any category, any two initial objects are isomorphic. Any two terminal objects are isomorphic. | Proof. If \( {X}_{0} \) and \( {X}_{1} \) are initial objects, there exist unique morphisms \( f \) : \( {X}_{0} \rightarrow {X}_{1} \) (since \( {X}_{0} \) is initial) and \( g : {X}_{1} \rightarrow {X}_{0} \) (since \( {X}_{1} \) is initial). Then \( g \circ f : {X}_{0} \rightarrow {X}_{0} \) and \( {1}_{{X}_{0}} : {... | Yes |
Theorem 9.1. The tensor product \( M{ \otimes }_{R}N \), if it exists, is determined up to isomorphism by the universal property. | Proof. Let \( \mathcal{C} \) be the following category. An object in \( \mathcal{C} \) is an ordered pair \( \left( {P, p}\right) \), where \( P \) is an \( R \) -module and \( p : M \times N \rightarrow P \) is a bilinear map. If \( X = \left( {P, p}\right) \) and \( Y = \left( {Q, q}\right) \) are objects, then a mor... | Yes |
Proposition 9.1. The universal property of the tensor algebra is satisfied. | Proof. If \( \phi : V \rightarrow A \) is any linear map of \( V \) into an \( F \) -algebra, define a map \( \Phi : \bigotimes V \rightarrow A \) by \( \Phi \left( {{v}_{1} \otimes \cdots \otimes {v}_{k}}\right) = \phi \left( {v}_{1}\right) \cdots \phi \left( {v}_{k}\right) \) on \( { \otimes }^{k}V \) . It is easy to... | Yes |
Theorem 10.1. Let \( \mathfrak{g} \) be a Lie algebra over a field \( F \) . There exists an associative \( F \) -algebra \( U\left( \mathfrak{g}\right) \) with a Lie algebra homomorphism \( i : \mathfrak{g} \rightarrow \) \( \operatorname{Lie}\left( {U\left( \mathfrak{g}\right) }\right) \) such that if \( A \) is any ... | Proof. Let \( \mathcal{K} \) be the ideal in \( \bigotimes \mathfrak{g} \) generated by elements of the form \( \left\lbrack {x, y}\right\rbrack \) - \( \left( {x \otimes y - y \otimes x}\right) \) for \( x, y \in \mathfrak{g} \), and let \( U\left( \mathfrak{g}\right) \) be the quotient \( \bigotimes V/\mathcal{K} \) ... | Yes |
Proposition 10.1. If \( \mathfrak{g} \) is the Lie algebra of a Lie group \( G \), then the natural map \( i : \mathfrak{g} \rightarrow U\left( \mathfrak{g}\right) \) is injective. | Proof. Let \( A \) be the ring of endomorphisms of \( {C}^{\infty }\left( G\right) \) . Regarding \( X \in \mathfrak{g} \) as a derivation of \( {C}^{\infty }\left( G\right) \) acting by (10.1), we have a Lie algebra homomorphism \( \mathfrak{g} \rightarrow \operatorname{Lie}\left( A\right) \), which by Theorem 10.1 in... | No |
Proposition 10.2. Let \( \pi : \mathfrak{g} \rightarrow \operatorname{End}\left( V\right) \) be an irreducible representation of the Lie algebra \( \mathfrak{g} \) . If \( c \) is in the center of \( U\left( \mathfrak{g}\right) \), then there exists a scalar \( \lambda \) such that \( \pi \left( c\right) = \lambda {I}_... | Proof. Let \( \lambda \) be any eigenvalue of \( \pi \left( c\right) \) . Let \( U \) be the \( \lambda \) -eigenspace of \( \pi \left( c\right) \) . Since \( \pi \left( c\right) \) commutes with \( \pi \left( x\right) \) for all \( x \in \mathfrak{g} \), we see that \( \pi \left( x\right) U \subseteq U \) for all \( x... | Yes |
Proposition 10.3. Suppose that \( G \) is a Lie group, \( \mathfrak{g} \) its Lie algebra, and \( \pi : G \rightarrow \mathrm{{GL}}\left( V\right) \) a representation admitting an invariant bilinear form \( B \) . Then \( B \) is invariant for the differential of \( \pi \) . | Proof. Invariance under \( \pi \) means that\n\n\[ B\left( {\pi \left( {\mathrm{e}}^{tX}\right) v,\pi \left( {\mathrm{e}}^{tX}\right) w}\right) = B\left( {v, w}\right) . \]\n\nThe derivative of this with respect to \( t \) is zero. By (8.5), this derivative is\n\n\[ B\left( {\pi \left( X\right) v, w}\right) + B\left( {... | Yes |
Proposition 10.4. Suppose that \( \left( {\pi, V}\right) \) is a representation of \( \mathfrak{g} \) . Then the trace bilinear form on \( \mathfrak{g} \) is invariant for the adjoint representation ad : \( \mathfrak{g} \rightarrow \) \( \operatorname{End}\left( \mathfrak{g}\right) \) . | Proof. Invariance under ad means\n\n\[ B\left( {\left\lbrack {x, y}\right\rbrack, z}\right) + B\left( {y,\left\lbrack {x, z}\right\rbrack }\right) = 0. \]\n\n(10.3)\n\nSince \( \pi \) is a representation, \( \pi \left( \left\lbrack {x, y}\right\rbrack \right) = \pi \left( x\right) \pi \left( y\right) - \pi \left( y\rig... | Yes |
Theorem 10.2. Suppose that the Lie algebra \( \mathfrak{g} \) admits a nondegenerate invariant bilinear form \( B \) . Let \( {x}_{1},\ldots ,{x}_{d} \) be a basis of \( \mathfrak{g} \), and let \( {y}_{1},\ldots ,{y}_{d} \) be the dual basis, so that \( B\left( {{x}_{i},{y}_{j}}\right) = {\delta }_{ij} \) (Kronecker \... | Proof. Let \( z \in \mathfrak{g} \) . There exist constants \( {\alpha }_{ij} \) and \( {\beta }_{ij} \) such that \( \left\lbrack {z,{x}_{i}}\right\rbrack = \) \( \mathop{\sum }\limits_{j}{\alpha }_{ij}{x}_{j} \) and \( \left\lbrack {z,{y}_{i}}\right\rbrack = \mathop{\sum }\limits_{j}{\beta }_{ij}{y}_{j} \) . Since \(... | Yes |
Theorem 11.1. Every complex representation of the Lie algebra \( \mathfrak{u}\left( n\right) \) or the Lie algebra \( \mathfrak{{gl}}\left( {n,\mathbb{R}}\right) \) extends uniquely to a complex representation of \( \mathfrak{{gl}}\left( {n,\mathbb{C}}\right) \) . Every complex representation of the Lie algebra \( \mat... | Proof. This follows from Proposition 11.3 since the complexification of \( \mathfrak{u}\left( n\right) \) or \( \mathfrak{{gl}}\left( {n,\mathbb{R}}\right) \) is \( \mathfrak{{gl}}\left( {n,\mathbb{C}}\right) \), while the complexification of \( \mathfrak{{su}}\left( n\right) \) or \( \mathfrak{{sl}}\left( {n,\mathbb{R... | Yes |
Proposition 12.1. We have\n\n\[ H \cdot {v}_{k - {2l}} = \left( {k - {2l}}\right) {v}_{k - {2l}},\;\left( {0 \leq l \leq k}\right) ,\] | Proof. For example, let us compute the effect of \( { \vee }^{k}R \) on \( {v}_{i} \) . In \( {\mathbb{C}}^{2} \),\n\n\[ \exp \left( {tR}\right) : \left\{ \begin{array}{l} \mathbf{x} \mapsto \mathbf{x}, \\ \mathbf{y} \mapsto \mathbf{y} + t\mathbf{x}. \end{array}\right. \]\n\nSo\n\n\[ R \cdot {v}_{k - {2l}} = {\left. \f... | No |
Proposition 12.2. The representation \( { \vee }^{k}{\mathbb{C}}^{2} \) of \( \mathfrak{{sl}}\left( {2,\mathbb{R}}\right) \) is irreducible. | Proof. Suppose that \( U \) is a nonzero invariant subspace. Choose a nonzero element \( \sum {a}_{k - {2l}}{v}_{k - {2l}} \) of \( U \) . Let \( k - {2l} \) be the smallest integer such that \( {a}_{k - {2l}} \neq 0 \) . Applying \( R \) to this vector \( l \) times shifts each \( {v}_{r} \rightarrow {v}_{r + 2} \) ti... | Yes |
Proposition 12.3. Suppose that \( \left( {\pi, V}\right) \) is an irreducible representation of \( \mathfrak{{sl}}\left( {2,\mathbb{R}}\right) \) . Assume that there exists a vector \( {v}_{k} \) in \( V \) such that \( {v}_{k} \neq 0 \) but \( R{v}_{k} = 0 \) . Then \( {\Delta v} = {\lambda v} \) for all \( v \in V \)... | Proof. By Proposition 10.2 there exists \( \lambda \) such that \( {\Delta v} = {\lambda v} \) for all \( v \) . To calculate \( \lambda \), we use the identity \( \left\lbrack {R, L}\right\rbrack = H \) to write\n\n\[ \Delta = {H}^{2} + {2H} + {4LR} \]\n\n(12.5)\n\nUsing \( R{v}_{k} = 0 \) and \( H{v}_{k} = k{v}_{k} \... | Yes |
Proposition 12.4. The element \( \Delta \) acts by the scalar \( \lambda = {k}^{2} + {2k} \) on \( { \vee }^{k}{\mathbb{C}}^{2} \) . | Proof. This follows from Proposition 12.3. | No |
Lemma 12.1. Suppose \( v \) is an \( H \) -eigenvector in some module for \( \mathfrak{{sl}}\left( {2,\mathbb{R}}\right) \) with eigenvalue \( k \) . Then \( {Rv} \) (if nonzero) is also an eigenvector with eigenvalue \( k + 2 \) , and \( {Lv} \) (if nonzero) is an eigenvector with eigenvalue \( k - 2 \) . | Proof. In the enveloping algebra, we have \( {HR} - {RH} = \left\lbrack {H, R}\right\rbrack = {2R} \), so \( {HRv} = {RHv} + {2Rv} = \left( {r + 2}\right) {Rv} \) . This proves the statement for \( R \), and \( L \) is handled similarly. | Yes |
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