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Lemma 20.23. Let \( G \) be a connected Lie group, and let \( H \subseteq G \) be a connected Lie subgroup. Let \( \mathfrak{g} \) and \( \mathfrak{h} \) denote the Lie algebras of \( G \) and \( H \), respectively. Then \( H \) is normal in \( G \) if and only if\n\n\[ \left( {\exp X}\right) \left( {\exp Y}\right) \le... | Proof. Note that \( \exp \left( {-X}\right) = {\left( \exp X\right) }^{-1} \) . Thus if \( H \) is normal, then (20.13) holds by definition. Conversely, suppose (20.13) holds, and choose open subsets \( V \subseteq \mathfrak{g} \) containing 0 and \( U \subseteq G \) containing the identity such that exp: \( V \rightar... | Yes |
Proposition 20.24 (The Adjoint Representation). If \( G \) is a Lie group with Lie algebra \( \mathfrak{g} \), the map \( \operatorname{Ad} : G \rightarrow \mathrm{{GL}}\left( \mathfrak{g}\right) \) is a Lie group representation, called the adjoint representation of \( G \) . | Proof. Because \( {C}_{{g}_{1}{g}_{2}} = {C}_{{g}_{1}} \circ {C}_{{g}_{2}} \) for any \( {g}_{1},{g}_{2} \in G \), it follows immediately that \( \operatorname{Ad}\left( {{g}_{1}{g}_{2}}\right) = \operatorname{Ad}\left( {g}_{1}\right) \circ \operatorname{Ad}\left( {g}_{2}\right) \), and \( \operatorname{Ad}\left( g\rig... | Yes |
Theorem 20.27. Let \( G \) be a Lie group, let \( \mathfrak{g} \) be its Lie algebra, and let Ad: \( G \rightarrow \) \( \mathrm{{GL}}\left( \mathrm{g}\right) \) be the adjoint representation of \( G \). The induced Lie algebra representation \( {\mathrm{{Ad}}}_{ * } : \mathfrak{g} \rightarrow \mathfrak{{gl}}\left( \ma... | Proof. Let \( X \in \mathfrak{g} \) be arbitrary. Then \( {\operatorname{Ad}}_{ * }X \) is determined by its value at the identity, which we can interpret as an element of \( \mathfrak{{gl}}\left( \mathfrak{g}\right) \), the set of all linear maps from \( \mathfrak{g} \) to itself. Because \( t \mapsto \exp {tX} \) is ... | Yes |
Theorem 20.28 (Ideals and Normal Subgroups). Let \( G \) be a connected Lie group, and suppose \( H \subseteq G \) is a connected Lie subgroup. Then \( H \) is a normal subgroup of \( G \) if and only if \( \operatorname{Lie}\left( H\right) \) is an ideal in \( \operatorname{Lie}\left( G\right) \) . | Proof. Write \( \mathfrak{g} = \operatorname{Lie}\left( G\right) \) and \( \mathfrak{h} = \operatorname{Lie}\left( H\right) \), considering \( \mathfrak{h} \) as a Lie subalgebra of \( \mathfrak{g} \) . For any \( g \in G \), the commutative diagram (20.3) applied to the Lie group homomorphism \( {C}_{g}\left( h\right)... | Yes |
Lemma 21.1. For any continuous action of a topological group \( G \) on a topological space \( M \), the quotient map \( \pi : M \rightarrow M/G \) is an open map. | Proof. For any \( g \in G \) and any subset \( U \subseteq M \), we define a set \( g \cdot U \subseteq M \) by\n\n\[ g \cdot U = \{ g \cdot x : x \in U\} . \]\n\nIf \( U \subseteq M \) is open, then \( {\pi }^{-1}\left( {\pi \left( U\right) }\right) \) is equal to the union of all sets of the form \( g \cdot U \) as \... | Yes |
Proposition 21.4. If a Lie group acts continuously and properly on a manifold, then the orbit space is Hausdorff. | Proof. Suppose \( G \) is a Lie group acting continuously and properly on a manifold \( M \) . Let \( \Theta : G \times M \rightarrow M \times M \) be the proper map \( \Theta \left( {g, p}\right) = \left( {g \cdot p, p}\right) \), and let \( \pi : M \rightarrow M/G \) be the quotient map. Define the orbit relation \( ... | Yes |
Proposition 21.5 (Characterizations of Proper Actions). Let \( M \) be a manifold, and let \( G \) be a Lie group acting continuously on \( M \). The following are equivalent.\n\n(a) The action is proper.\n\n(b) If \( \left( {p}_{i}\right) \) is a sequence in \( M \) and \( \left( {g}_{i}\right) \) is a sequence in \( ... | Proof. Throughout this proof, let \( \Theta : G \times M \rightarrow M \times M \) denote the map \( \Theta \left( {g, p}\right) = \) \( \left( {g \cdot p, p}\right) \) ; thus, the action is proper if and only if \( \Theta \) is a proper map. We will prove (a) \( \Rightarrow \) (b) \( \Rightarrow \) (c) \( \Rightarrow ... | Yes |
Corollary 21.6. Every continuous action by a compact Lie group on a manifold is proper. | Proof. If \( \left( {p}_{i}\right) \) and \( \left( {g}_{i}\right) \) are sequences satisfying the hypotheses of Proposition 21.5(b), then a subsequence of \( \left( {g}_{i}\right) \) converges, for the simple reason that every sequence in \( G \) has a convergent subsequence. | No |
Proposition 21.7 (Orbits of Proper Actions). Suppose \( \theta \) is a proper smooth action of a Lie group \( G \) on a smooth manifold \( M \). For any point \( p \in M \), the orbit map \( {\theta }^{\left( p\right) } : G \rightarrow M \) is a proper map, and thus the orbit \( G \cdot p = {\theta }^{\left( p\right) }... | Proof. If \( K \subseteq M \) is compact, then \( {\left( {\theta }^{\left( p\right) }\right) }^{-1}\left( K\right) \) is closed in \( G \) by continuity, and since it is contained in \( {G}_{K\cup \{ p\} } \), it is compact by Proposition 21.5. Therefore, \( {\theta }^{\left( p\right) } \) is a proper map, which impli... | Yes |
Corollary 21.8. If a Lie group \( G \) acts properly on a manifold \( M \), then each orbit is a closed subset of \( M \), and each isotropy group is compact. | Proof. The first statement follows immediately from Proposition 21.7, and the second from Proposition 21.5, using the fact that the isotropy group of a point \( p \in M \) is the set \( {G}_{K} \) for \( K = \{ p\} \). | Yes |
Lemma 21.11. Suppose a discrete Lie group \( \Gamma \) acts continuously and freely on a manifold \( E \) . The action is proper if and only if the following conditions both hold:\n\n(i) Every point \( p \in E \) has a neighborhood \( U \) such that for each \( g \in \Gamma ,\left( {g \cdot U}\right) \cap \) \( U = \va... | Proof. First, suppose that the action is free and proper, and let \( \pi : E \rightarrow E/\Gamma \) denote the quotient map. By Proposition 21.4, \( E/\Gamma \) is Hausdorff. If \( p,{p}^{\prime } \in E \) are not in the same orbit, we can choose disjoint neighborhoods \( W \) of \( \pi \left( p\right) \) and \( {W}^{... | Yes |
Proposition 21.12. Let \( M \) be a smooth manifold, and let \( \pi : E \rightarrow M \) be a smooth covering map. With the discrete topology, the automorphism group \( {\operatorname{Aut}}_{\pi }\left( E\right) \) acts smoothly, freely, and properly on \( E \) . | Proof. We already showed in Proposition 7.23 that the action is smooth and free. To show it is proper, we will show that it satisfies conditions (i) and (ii) of Lemma 21.11. First, if \( p \in E \) is arbitrary, choose \( W \subseteq M \) to be an evenly covered neighborhood of \( \pi \left( p\right) \) . If \( U \) is... | Yes |
Theorem 21.13. Suppose \( E \) is a connected smooth manifold and \( \Gamma \) is a discrete Lie group acting smoothly, freely, and properly on \( E \) . Then the orbit space \( E/\Gamma \) is a topological manifold and has a unique smooth structure such that \( \pi : E \rightarrow E/\Gamma \) is a smooth normal coveri... | Proof. It follows from the quotient manifold theorem that \( E/\Gamma \) has a unique smooth manifold structure such that \( \pi \) is a smooth submersion. Because a smooth covering map is in particular a smooth submersion, any other smooth manifold structure on \( E \) making \( \pi \) into a smooth covering map must ... | Yes |
Theorem 21.17 (Homogeneous Space Construction Theorem). Let \( G \) be a Lie group and let \( H \) be a closed subgroup of \( G \). The left coset space \( G/H \) is a topological manifold of dimension equal to \( \dim G - \dim H \), and has a unique smooth structure such that the quotient map \( \pi : G \rightarrow G/... | Proof. If we let \( H \) act on \( G \) by right translation, then \( {g}_{1},{g}_{2} \in G \) are in the same \( H \) -orbit if and only if \( {g}_{1}h = {g}_{2} \) for some \( h \in H \), which is the same as saying that \( {g}_{1} \) and \( {g}_{2} \) are in the same coset of \( H \). In other words, the orbit space... | Yes |
Theorem 21.18 (Homogeneous Space Characterization Theorem). Let \( G \) be a Lie group, let \( M \) be a homogeneous \( G \)-space, and let \( p \) be any point of \( M \). The isotropy group \( {G}_{p} \) is a closed subgroup of \( G \), and the map \( F : G/{G}_{p} \rightarrow M \) defined by \( F\left( {g{G}_{p}}\ri... | Proof. For simplicity, let us write \( H = {G}_{p} \). Note that \( H \) is closed by continuity, because \( H = {\left( {\theta }^{\left( p\right) }\right) }^{-1}\left( p\right) \), where \( {\theta }^{\left( p\right) } : G \rightarrow M \) is the orbit map.\n\nTo see that \( F \) is well defined, assume that \( {g}_{... | Yes |
Theorem 21.20. Suppose \( X \) is a set, and we are given a transitive action of a Lie group \( G \) on \( X \) such that for some point \( p \in X \), the isotropy group \( {G}_{p} \) is closed in \( G \) . Then \( X \) has a unique smooth manifold structure with respect to which the given action is smooth. With this ... | Proof. Theorem 21.17 shows that \( G/{G}_{p} \) is a smooth manifold of dimension equal to \( \dim G - \dim {G}_{p} \) . The map \( F : G/{G}_{p} \rightarrow X \) defined by \( F\left( {g{G}_{p}}\right) = g \cdot p \) is an equivariant bijection by exactly the same argument as we used in the proof of the characterizati... | Yes |
Let \( {\mathrm{G}}_{k}\left( {\mathbb{R}}^{n}\right) \) denote the Grassmannian of \( k \) - dimensional subspaces of \( {\mathbb{R}}^{n} \). The general linear group \( \mathrm{{GL}}\left( {n,\mathbb{R}}\right) \) acts transitively on \( {\mathrm{G}}_{k}\left( {\mathbb{R}}^{n}\right) \): given two subspaces \( A \) a... | \[ H = \left\{ {\left( \begin{array}{ll} A & B \\ 0 & D \end{array}\right) : A \in \mathrm{{GL}}\left( {k,\mathbb{R}}\right), D \in \mathrm{{GL}}\left( {n - k,\mathbb{R}}\right), B \in \mathrm{M}\left( {k \times \left( {n - k}\right) ,\mathbb{R}}\right) }\right\} ,\] which is easily seen to be closed in \( \mathrm{{GL}... | Yes |
Theorem 21.26 (Quotient Theorem for Lie Groups). Suppose \( G \) is a Lie group and \( K \subseteq G \) is a closed normal subgroup. Then \( G/K \) is a Lie group, and the quotient map \( \pi : G \rightarrow G/K \) is a surjective Lie group homomorphism whose kernel is \( K \) . | Proof. By the homogeneous space construction theorem, \( G/K \) is a smooth manifold and \( \pi \) is a smooth submersion; and by Theorem 21.23, \( G/K \) is a group and \( \pi : G \rightarrow G/K \) is a surjective homomorphism with kernel \( K \) . Thus, the only thing that needs to be verified is that multiplication... | Yes |
Theorem 21.27 (First Isomorphism Theorem for Lie Groups). If \( F : G \rightarrow H \) is a Lie group homomorphism, then the kernel of \( F \) is a closed normal Lie subgroup of \( G \), the image of \( F \) has a unique smooth manifold structure making it into a Lie subgroup of \( H \), and \( F \) descends to a Lie g... | Proof. By Theorem 21.24, \( \operatorname{Ker}F \) is a normal subgroup, \( \operatorname{Im}F \) is a subgroup, and \( F \) descends to a group isomorphism \( \widetilde{F} : G/\operatorname{Ker}F \rightarrow \operatorname{Im}F \) . By continuity, \( \operatorname{Ker}F \) is closed in \( G \), so it follows from Theo... | Yes |
Proposition 21.28. Every discrete subgroup of a Lie group is a closed Lie subgroup of dimension zero. | Proof. Let \( G \) be a Lie group and \( \Gamma \subseteq G \) be a discrete subgroup. With the subspace topology, \( \Gamma \) is a countable discrete space and thus a zero-dimensional Lie group. By the closed subgroup theorem, \( \Gamma \) is a closed Lie subgroup of \( G \) if and only if it is a closed subset. A di... | Yes |
Theorem 21.29 (Quotients of Lie Groups by Discrete Subgroups). If \( G \) is a connected Lie group and \( \Gamma \subseteq G \) is a discrete subgroup, then \( G/\Gamma \) is a smooth manifold and the quotient map \( \pi : G \rightarrow G/\Gamma \) is a smooth normal covering map. | Proof. The proof of Theorem 21.17 showed that \( \Gamma \) acts smoothly, freely, and properly on \( G \) on the right, and its quotient—which is the coset space \( G/\Gamma \) —is a smooth manifold. The theorem is then an immediate consequence of Theorem 21.13. | Yes |
Theorem 21.31 (Homomorphisms with Discrete Kernels). Let \( G \) and \( H \) be connected Lie groups. For any Lie group homomorphism \( F : G \rightarrow H \), the following are equivalent:\n\n(a) \( F \) is surjective and has discrete kernel.\n\n(b) \( F \) is a smooth covering map.\n\n(c) \( F \) is a local diffeomor... | Proof. We will show that (a) \( \Rightarrow \) (b) \( \Rightarrow \) (c) \( \Rightarrow \) (a) and (c) \( \Leftrightarrow \) (d). First, assume that \( F \) is surjective with discrete kernel \( \Gamma \subseteq G \) . Then Theorem 21.29 implies that the quotient map \( \pi : G \rightarrow G/\Gamma \) is a smooth cover... | Yes |
Theorem 21.32. Let \( \mathfrak{g} \) be a finite-dimensional Lie algebra. The connected Lie groups whose Lie algebras are isomorphic to \( \mathfrak{g} \) are (up to isomorphism) precisely those of the form \( G/\Gamma \), where \( G \) is the simply connected Lie group with Lie algebra \( \mathfrak{g} \), and \( \Gam... | Proof. Given \( \mathfrak{g} \), by Theorem 20.21 there exists a simply connected Lie group \( G \) with Lie algebra isomorphic to \( \mathfrak{g} \) . Suppose \( H \) is any other connected Lie group whose Lie algebra is isomorphic to \( \mathfrak{g} \), and let \( \varphi : \operatorname{Lie}\left( G\right) \rightarr... | Yes |
Proposition 21.33. Suppose a topological group \( G \) acts continuously, freely, and properly on a topological space \( M \) . If \( G \) and \( M/G \) are connected, then \( M \) is connected. | Proof. Assume for the sake of contradiction that \( G \) and \( M/G \) are connected but \( M \) is not. Then there are disjoint nonempty open subsets \( U, V \subseteq M \) whose union is \( M \) . Each \( G \) -orbit in \( M \) is the image of \( G \) under an orbit map \( {\theta }^{\left( p\right) } : G \rightarrow... | Yes |
For each \( n \geq 1 \), the Lie groups \( \mathrm{{SO}}\left( n\right) ,\mathrm{U}\left( n\right) \), and \( \mathrm{{SU}}\left( n\right) \) are connected. The group \( \mathrm{O}\left( n\right) \) has exactly two components, one of which is \( \mathrm{{SO}}\left( n\right) \) . | First, we prove by induction on \( n \) that \( \mathrm{{SO}}\left( n\right) \) is connected. For \( n = 1 \) this is obvious, because \( \mathrm{{SO}}\left( 1\right) \) is the trivial group. Now suppose we have shown that \( \operatorname{SO}\left( {n - 1}\right) \) is connected for some \( n \geq 2 \) . Because the h... | Yes |
Suppose \( v = {a}^{i}{A}_{i} + {b}^{i}{B}_{i} \in V \) satisfies \( \omega \left( {v, w}\right) = 0 \) for all \( w \in V \). Then \( 0 = \) \( \omega \left( {v,{B}_{i}}\right) = {a}^{i} \) and \( 0 = \omega \left( {v,{A}_{i}}\right) = - {b}^{i} \), which implies that \( v = 0 \). Thus \( \omega \) is nondegenerate, a... | Then \( 0 = \) \( \omega \left( {v,{B}_{i}}\right) = {a}^{i} \) and \( 0 = \omega \left( {v,{A}_{i}}\right) = - {b}^{i} \), which implies that \( v = 0 \). Thus \( \omega \) is nondegenerate, and so is a symplectic tensor. | Yes |
Proposition 22.5. Let \( \left( {V,\omega }\right) \) be a symplectic vector space, and let \( S \subseteq V \) be a linear subspace.\n\n(a) \( S \) is symplectic if and only if \( {S}^{ \bot } \) is symplectic.\n\n(b) \( S \) is symplectic if and only if \( {\left. \omega \right| }_{S} \) is nondegenerate.\n\n(c) \( S... | Proof. Problem 22-1. | No |
Proposition 22.7 (Canonical Form for a Symplectic Tensor). Let \( \omega \) be a symplectic tensor on an m-dimensional vector space \( V \) . Then \( V \) has even dimension \( m = {2n} \), and there exists a basis for \( V \) in which \( \omega \) has the form (22.1). | Proof. The tensor \( \omega \) has the form (22.1) with respect to a basis \( \left( {{A}_{1},{B}_{1},\ldots ,{A}_{n},{B}_{n}}\right) \) if and only if its action on basis vectors is given by (22.2). We prove the theorem by induction on \( m = \dim V \) by showing that there is a basis with this property.\n\nFor \( m =... | Yes |
Proposition 22.8. Suppose \( V \) is a 2n-dimensional vector space and \( \omega \in {\Lambda }^{2}\left( {V}^{ * }\right) \) . Then \( \omega \) is a symplectic tensor if and only if \( {\omega }^{n} \neq 0 \) . | Proof. Suppose first that \( \omega \) is a symplectic tensor. Let \( \left( {{A}_{i},{B}_{i}}\right) \) be a symplectic basis for \( V \), and write \( \omega = \mathop{\sum }\limits_{i}{\alpha }^{i} \land {\beta }^{i} \) in terms of the dual coframe. Then \( {\omega }^{n} = \) \( \mathop{\sum }\limits_{I}{\alpha }^{{... | Yes |
Proposition 22.11. Let \( M \) be a smooth manifold. The tautological 1-form \( \tau \) is smooth, and \( \omega = - {d\tau } \) is a symplectic form on the total space of \( {T}^{ * }M \) . | Proof. Let \( \left( {x}^{i}\right) \) be smooth coordinates on \( M \), and let \( \left( {{x}^{i},{\xi }_{i}}\right) \) denote the corresponding natural coordinates on \( {T}^{ * }M \) as defined on p. 277. Recall that the coordinates of \( \left( {q,\varphi }\right) \in {T}^{ * }M \) are defined to be \( \left( {{x}... | Yes |
Proposition 22.12. Let \( M \) be a smooth manifold, and let \( \sigma \) be a smooth 1 -form on \( M \) . Thought of as a smooth map from \( M \) to \( {T}^{ * }M,\sigma \) is a smooth embedding, and \( \sigma \) is closed if and only if its image \( \sigma \left( M\right) \) is a Lagrangian submanifold of \( {T}^{ * ... | Proof. Throughout this proof we need to remember that \( \sigma : M \rightarrow {T}^{ * }M \) is playing two roles: on the one hand, it is a 1 -form on \( M \), and on the other hand, it is a smooth map between manifolds. Since they are literally the same map, we do not use different notations to distinguish between th... | Yes |
Proposition 22.15. Let \( M \) be a smooth manifold and \( J \subseteq \mathbb{R} \) be an open interval. Suppose \( V : J \times M \) is a smooth time-dependent vector field on \( M,\psi : \mathcal{E} \rightarrow M \) is its time-dependent flow, and \( A : J \times M \rightarrow {T}^{k}{T}^{ * }M \) is a smooth time-d... | Proof. For sufficiently small \( \varepsilon > 0 \), consider the smooth map \( F : \left( {{t}_{1} - \varepsilon ,{t}_{1} + \varepsilon }\right) \times \left( {{t}_{1} - \varepsilon ,{t}_{1} + \varepsilon }\right) \rightarrow {T}^{k}\left( {{T}_{p}^{ * }M}\right) \) defined by \[ F\left( {u, v}\right) = {\left( {\thet... | Yes |
Proposition 22.16 (Properties of Hamiltonian Vector Fields). Let \( \left( {M,\omega }\right) \) be a symplectic manifold and let \( f \in {C}^{\infty }\left( M\right) \) .\n\n(a) \( f \) is constant along each integral curve of \( {X}_{f} \) .\n\n(b) At each regular point of \( f \), the Hamiltonian vector field \( {X... | Proof. Both assertions follow from the fact that\n\n\[ \n{X}_{f}f = {df}\left( {X}_{f}\right) = \omega \left( {{X}_{f},{X}_{f}}\right) = 0 \n\]\n\nbecause \( \omega \) is alternating. | Yes |
Proposition 22.17 (Hamiltonian and Symplectic Vector Fields). Let \( \left( {M,\omega }\right) \) be a symplectic manifold. A smooth vector field on \( M \) is symplectic if and only if it is locally Hamiltonian. Every locally Hamiltonian vector field on \( M \) is globally Hamiltonian if and only if \( {H}_{\mathrm{{d... | Proof. By Theorem 12.37, a smooth vector field \( X \) is symplectic if and only if \( {\mathcal{L}}_{X}\omega = 0 \) . Using Cartan’s magic formula, we compute\n\n\[ \n{\mathcal{L}}_{X}\omega = d\left( {X\lrcorner \omega }\right) + X\lrcorner \left( {d\omega }\right) = d\left( {X\lrcorner \omega }\right) .\n\]\n\n(22.... | Yes |
Proposition 22.19 (Properties of the Poisson Bracket). Suppose \( \left( {M,\omega }\right) \) is a symplectic manifold, and \( f, g, h \in {C}^{\infty }\left( M\right) \) . (a) BILINEARITY: \( \{ f, g\} \) is linear over \( \mathbb{R} \) in \( f \) and in \( g \) . (b) ANTISYMMETRY: \( \{ f, g\} = - \{ g, f\} \) . (c)... | Proof. Parts (a) and (b) are obvious from the characterization \( \{ f, g\} = \omega \left( {{X}_{f},{X}_{g}}\right) \) together with the fact that \( {X}_{f} = {\widehat{\omega }}^{-1}\left( {df}\right) \) depends linearly on \( f \) . Because of the nondegeneracy of \( \omega \), to prove (d), it suffices to show tha... | Yes |
Proposition 22.21. Let \( \left( {M,\omega, H}\right) \) be a Hamiltonian system.\n\n(a) A function \( f \in {C}^{\infty }\left( M\right) \) is a conserved quantity if and only if \( \{ f, H\} = 0 \) . | Proof. Problem 22-18. | No |
Theorem 22.22 (Noether’s Theorem). Let \( \left( {M,\omega, H}\right) \) be a Hamiltonian system. If \( f \) is any conserved quantity, then its Hamiltonian vector field is an infinitesimal symmetry. Conversely, if \( {H}_{\mathrm{{dR}}}^{1}\left( M\right) = 0 \), then each infinitesimal symmetry is the Hamiltonian vec... | Proof. Suppose \( f \) is a conserved quantity. Proposition 22.21 shows that \( \{ f, H\} = \) 0 . This in turn implies that \( {X}_{f}H = \{ H, f\} = 0 \), so \( H \) is constant along the flow of \( {X}_{f} \) . Since \( \omega \) is invariant along the flow of any Hamiltonian vector field by Proposition 22.17, this ... | Yes |
Theorem 22.23 (Hamiltonian Flowout Theorem). Suppose \( \left( {M,\omega }\right) \) is a symplectic manifold, \( H \in {C}^{\infty }\left( M\right) ,\Gamma \) is an embedded isotropic submanifold of \( M \) that is contained in a single level set of \( H \), and the Hamiltonian vector field \( {X}_{H} \) is nowhere ta... | Proof. Let \( \theta \) be the flow of \( {X}_{H} \) . Recall from Theorem 9.20 that the flowout is parametrized by the restriction of \( \theta \) to a neighborhood \( {\mathcal{O}}_{\delta } \) of \( \{ 0\} \times \Gamma \) in \( \mathbb{R} \times \Gamma \) . First consider a point \( p \in \Gamma \subseteq S \) . If... | Yes |
Theorem 22.28 (The Reeb Field). Let \( \left( {M, H}\right) \) be a contact manifold, and suppose \( \theta \) is a contact form for \( H \) . There is a unique vector field \( T \in \mathfrak{X}\left( M\right) \), called the Reeb field of \( \mathbf{\theta } \), that satisfies the following two conditions:\n\n\[ T\lrc... | Proof. Define a smooth bundle homomorphism \( \Phi : {TM} \rightarrow {T}^{ * }M \) by \( \Phi \left( X\right) = \) \( X\lrcorner {d\theta } \), and for each \( p \in M \), let \( {\Phi }_{p} \) denote the linear map \( {\left. \Phi \right| }_{{T}_{p}M} : {T}_{p}M \rightarrow {T}_{p}^{ * }M \) . The fact that \( d{\the... | Yes |
Theorem 22.31 (Contact Darboux Theorem). Suppose \( \theta \) is a contact form on a \( \left( {{2n} + 1}\right) \) -dimensional manifold \( M \) . For each \( p \in M \), there are smooth coordinates \( \left( {{x}^{1},\ldots ,{x}^{n},{y}^{1},\ldots ,{y}^{n}, z}\right) \) centered at \( p \) in which \( \theta \) has ... | Proof. Let \( p \in M \) be arbitrary. Let \( \left( {U,\left( {u}^{i}\right) }\right) \) be a smooth coordinate cube centered at \( p \) in which the Reeb field of \( \theta \) has the form \( T = \partial /\partial {u}^{1} \), and let \( Y \subseteq U \) be the slice defined by \( {u}^{1} = 0 \) . Because \( T \) is ... | Yes |
Proposition 22.32. Suppose \( \left( {M, H}\right) \) is a contact manifold and \( \theta \) is a contact form for \( H \) . For any function \( f \in {C}^{\infty }\left( M\right) \), there is a unique vector field \( {X}_{f} \) , called the contact Hamiltonian vector field of \( f \), that satisfies \( \theta \left( {... | Proof. Suppose \( f \in {C}^{\infty }\left( M\right) \) . Because the restriction of \( {d\theta } \) to \( H \) is nondegenerate, there is a unique smooth vector field \( B \in \Gamma \left( H\right) \) such that \( B\lrcorner {d\theta }{\left| {}_{H} = df\right| }_{H} \) . If we set \( {X}_{f} = {fT} - B \), where \(... | Yes |
Theorem 22.33 (Characterization of Contact Vector Fields). If (M, H) is a contact manifold and \( \theta \) is a contact form for \( H \), then a smooth vector field on \( M \) is a contact vector field if and only if it is a contact Hamiltonian vector field. | Proof. Problem 22-21. | No |
Theorem 22.34 (Contact Flowout Theorem). Suppose \( \left( {M, H}\right) \) is a contact manifold, \( F \in {C}^{\infty }\left( M\right) ,\Gamma \) is an embedded isotropic submanifold of \( M \) that is contained in the zero set of \( F \), and the contact Hamiltonian vector field \( {X}_{F} \) is nowhere tangent to \... | Proof. Problem 22-23. | No |
Theorem 22.39 (The General First-Order Cauchy Problem). Suppose \( M \) is a smooth manifold, \( W \subseteq {J}^{1}M \) is an open subset, \( F : W \rightarrow \mathbb{R} \) is a smooth function, \( S \subseteq M \) is an embedded hypersurface, and \( \varphi : S \rightarrow \mathbb{R} \) is a smooth function. If the ... | Proof. Problem 22-26. | No |
Theorem 0.2.5 Every number field has an integral basis. | If \( \alpha \) is an algebraic integer such that \( k = \mathbb{Q}\left( \alpha \right) \), then we have seen that \( \mathbb{Z}\left\lbrack \alpha \right\rbrack \subset {R}_{k} \) . If \( \delta \) is the discriminant of the basis \( \left\{ {1,\alpha ,{\alpha }^{2},\ldots ,{\alpha }^{n - 1}}\right\} \), then it can ... | No |
Theorem 0.2.8 For any positive integer \( D \), there are only finitely many fields with \( \left| {\Delta }_{k}\right| \leq D \) . | This theorem can be deduced from Minkowski's theorem in the geometry of numbers on the existence of lattice points in convex bodies in \( {\mathbb{R}}^{n} \) whose volume is large enough relative to a fundamental region for the lattice. | No |
Theorem 0.2.10 Let \( \ell \mid k \) be a finite extension of number fields, with \( \left\lbrack {\ell : k}\right\rbrack = d \) . | \[ \left| {\Delta }_{\ell }\right| = \left| {N\left( {\delta }_{\ell \mid k}\right) {\Delta }_{k}^{d}}\right| \] | No |
Corollary 0.3.13 Let \( \ell \mid k \) be an abelian extension and \( \left( {f, g}\right) \) a splitting type for \( \ell \mid k \) . Then a necessary and sufficient condition that there are infinitely many prime ideals in \( k \) with this splitting type is that \( \mathcal{G} \) contains an element of order \( f \) ... | This is an immediate deduction from the theorem, for if \( {n}_{f} \) is the number of elements in \( \mathcal{G} \) of order \( f \), then the set of unramified primes \( \mathcal{P} \) such that \( \left( \frac{\ell /k}{\mathcal{P}}\right) \) has order \( f \) has density \( {n}_{f}/n \) . However, when this Frobeniu... | No |
Lemma 0.3.14 If \( I = \alpha {R}_{k} \) is a principal ideal in \( {R}_{k} \), then | \[ N\left( I\right) = \left| {{N}_{k \mid \mathbb{Q}}\left( \alpha \right) }\right| \] | No |
Example 0.4.3 Let \( k = \mathbb{Q}\left( t\right) \), where \( t \) satisfies \( {x}^{3} + x + 1 = 0 \) . This field has one complex and one real place so that \( {R}_{k}^{ * } \cong W \times \mathbb{Z} \), where \( o\left( W\right) = 2 \) in this case. Note that \( t \) is a unit. In fact, we will prove that it is a ... | Let \( \widetilde{k} \) denote the Galois closure of \( k \) . Then \( {x}^{n} \pm t \) must split completely in \( \widetilde{k} \) and so \( {e}^{{2\pi i}/n} \in \widetilde{k} \) . If \( \mathbb{Q}\left( {e}^{{2\pi i}/n}\right) \subset \widetilde{k} \), then \( n = 1,2,3,4,6 \) . If \( n \neq 1,2,\widetilde{k} = \mat... | Yes |
Theorem 0.6.6 Let \( k \) be a number field. Any non-Archimedean valuation on \( k \) is equivalent to a \( \mathcal{P} \)-adic valuation \( {v}_{\mathcal{P}} \) for some prime ideal \( \mathcal{P} \) in \( {R}_{k} \). Any Archimedean valuation on \( k \) is equivalent to a valuation \( {v}_{\sigma } \) as described ea... | For prime ideals \( {\mathcal{P}}_{1} \neq {\mathcal{P}}_{2} \), the valuations \( {v}_{{\mathcal{P}}_{1}},{v}_{{\mathcal{P}}_{2}} \) cannot be equivalent. Recall that \( {\mathcal{P}}_{1} + {\mathcal{P}}_{2} = {R}_{k} \) so that \( 1 = x + y \) with \( x \in {\mathcal{P}}_{1} \) and \( y \in {\mathcal{P}}_{2} \). Thus... | No |
Example 0.6.8 Let \( k = \mathbb{Q} \) . Then there is precisely one infinite place represented by the usual absolute value \( v\left( x\right) = \left| x\right| \) . The finite places are in one-to-one correspondence with the rational primes \( p \) of \( \mathbb{Z} \) . For a fixed prime \( p \), the corresponding fi... | \[ R\left( {v}_{p}\right) = \{ a/b \in \mathbb{Q} \mid p \nmid b\} \] and since \( {n}_{p}\left( p\right) = 1 \), the unique maximal ideal is the principal ideal \( {pR}\left( {v}_{p}\right) \) . Note that the field of fractions of \( R\left( {v}_{p}\right) \) is again \( \mathbb{Q} \) and the quotient field \( R\left(... | Yes |
Theorem 0.7.6 The valuation ring \( {R}_{\mathcal{P}} \) of the completion \( {k}_{\mathcal{P}} \) is a discrete valuation ring whose unique maximal ideal is generated by \( {i}_{\mathcal{P}}\left( \pi \right) \) . Furthermore, \( {R}_{\mathcal{P}}/{i}_{\mathcal{P}}\left( \pi \right) {R}_{\mathcal{P}} \cong R\left( {v}... | This result follows because the image of \( {k}_{\mathcal{P}}^{ * } \) under \( {\widehat{v}}_{\mathcal{P}} \) is the same as the image of \( {k}^{ * } \) under \( {v}_{\mathcal{P}} \) . For, if \( \alpha \in {k}_{\mathcal{P}}^{ * } \), then \( \alpha = \left\{ {a}_{n}\right\} + \mathcal{N} \) . Hence\n\n\[ 0 \neq {\wi... | Yes |
In the field \( k = \mathbb{Q}\left( t\right) \) where \( t \) satisfies \( {x}^{3} + x + 1 = 0 \), consider the completions at the prime ideals lying over the rational primes 2, 3 and 31. These were discussed in \( §{0.3} \) and again using Kummer’s Theorem, we can obtain uniformisers in these completions. | Since \( 2{R}_{k} \) is a prime ideal in \( {R}_{k},2 \) will be a uniformiser for \( {k}_{{\mathcal{P}}_{2}} \) and \( \left\lbrack {{k}_{{\mathcal{P}}_{2}} : {\mathbb{Q}}_{2}}\right\rbrack = 3 \) . The two prime ideals \( {\mathcal{P}}_{3}^{\prime } \) and \( {\mathcal{P}}_{3}^{\prime \prime } \) are generated by \( ... | Yes |
Theorem 0.8.1 The complete field \( {k}_{\mathcal{P}} \) is locally compact and its valuation ring \( {R}_{\mathcal{P}} \) is compact. | Proof: We first show that \( {R}_{\mathcal{P}} \) is compact. As earlier, let \( \left\{ {c}_{i}\right\} \) be a (finite) set of coset representatives of \( \widehat{\mathcal{P}} \) in \( {R}_{\mathcal{P}} \) . Let \( \left\{ {{U}_{\lambda },\lambda \in \Omega }\right\} \) be an open cover of \( {R}_{\mathcal{P}} \), w... | Yes |
Theorem 0.8.3 Let \( G \) be a locally compact topological group.\n\n1. There exists a left Haar measure on \( G \) .\n\n2. If \( {\mu }_{1} \) and \( {\mu }_{2} \) are left Haar measures on \( G \), then there exists \( r \in {\mathbb{R}}^{ + } \) such that \( {\mu }_{2} = r{\mu }_{1} \) . | Thus Haar measures are unique up to a scaling and by a suitable choice, a normalised measure can frequently be chosen.\n\nLet \( G = \left( {{k}_{\mathcal{P}}, + }\right) \), in which \( {R}_{\mathcal{P}} \) is compact. Thus we can choose a normalised Haar measure \( \mu \) such that \( \mu \left( {R}_{\mathcal{P}}\rig... | Yes |
If \( V \) is a quadratic space over \( k \) and \( a \in {k}^{ * } \), then \( V \) represents a in \( k \) if and only if \( V \) represents a in all \( {k}_{v} \) where \( v \) ranges over all places of \( k \). | The corollary follows directly from the theorem in view of the remarks following Lemma 0.9.4. | No |
Theorem 0.9.12 Let \( U \) and \( V \) be regular quadratic spaces of the same dimension over a number field \( k \) . Then \( U \) and \( V \) are isometric if and only if \( {U}_{v} \) and \( {V}_{v} \) are isometric over \( {k}_{v} \) for all places \( v \) on \( k \) . | Proof: Any isometry from \( U \) to \( V \) clearly extends to one from \( {U}_{v} \) to \( {V}_{v} \) . The reverse implication is proved by induction on dim \( U \) . Let \( q \) and \( Q \) denote the quadratic maps on \( U \) and \( V \), respectively. Suppose that \( U = < \) \( \mathbf{u} > \) is one-dimensional ... | Yes |
Lemma 1.2.3 Let \( x, y \in \operatorname{PSL}\left( {2,\mathbb{C}}\right) \) . Then \( \langle x, y\rangle \) is reducible if and only if \( \operatorname{tr}\left\lbrack {x, y}\right\rbrack = 2 \) . | Note that the trace of the commutator is well-defined, independent of the choice of pre-images of \( x \) and \( y \) in \( \mathrm{{SL}}\left( {2,\mathbb{C}}\right) \) . More generally, for any \( X, Y \in \) \( \mathrm{{SL}}\left( {2,\mathbb{C}}\right) \), let \( M\left( {X, Y}\right) \) denote the \( 4 \times 4 \) m... | Yes |
Theorem 1.3.2 If \( M \) is a non-compact orientable hyperbolic 3-manifold of finite volume, then \( M \) has finitely many ends and each end (or cusp neighbourhood) is isometric in a Euclidean sense, to \( {T}^{2} \times \lbrack 0,\infty ) \), where \( {T}^{2} \) is a torus. | Note that the classification of discrete subgroups of \( B \) gives that a torsion-free cusp stabiliser is a lattice in Euclidean 2-space generated by a pair of independent translations. | No |
Lemma 1.7.1 The Lobachevski function \( \mathcal{L} \) is periodic of period \( \pi \) and is also odd. | The function \( \mathcal{L} \) has a uniformly convergent Fourier series expansion which can be obtained via its connection with the complex dilogarithm function\n\n\[ \psi \left( z\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{z}^{n}}{{n}^{2}},\;\left| z\right| \leq 1 \]\n\nFor \( \left| z\right| < 1, z{\ps... | No |
Lemma 1.7.2 \( \mathcal{L}\left( \theta \right) \) has the uniformly convergent Fourier series expansion | \[ \mathcal{L}\left( \theta \right) = \frac{1}{2}\mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{\sin \left( {2n\theta }\right) }{{n}^{2}} \] | Yes |
Example 1.7.3 The Coxeter tetrahedron with Coxeter symbol given in Figure 1.7 is the difference of two tetrahedra each with one ideal vertex as shown in Figure 1.8. The number \( n \) labelling an edge indicates a dihedral angle of \( \pi /n \) . The tetrahedron with vertices \( A, B, C \) and \( \infty \) is of the ty... | The tetrahedron with vertices \( D, B, C \) and \( \infty \) is not of the type \( {T}_{\alpha ,\gamma } \) but can be shown to be the union of two such tetrahedra minus one such tetrahedron and so the volume of the compact tetrahedron in Figure 1.8 can be determined (see Exercise 1.7, No. 4). | No |
1. \( \left( \frac{a, b}{F}\right) \cong \left( \frac{a{x}^{2}, b{y}^{2}}{F}\right) \) for any \( a, b, x, y \in {F}^{ * } \) | 1. Let \( A = \left( \frac{a, b}{F}\right) \) and \( {A}^{\prime } = \left( \frac{a{x}^{2}, b{y}^{2}}{F}\right) \) have bases \( \{ 1, i, j, k\} \) and \( \{ 1,{i}^{\prime },{j}^{\prime },{k}^{\prime }\} \) , respectively. Define \( \phi : {A}^{i} \rightarrow A \) by \( \phi \left( 1\right) = 1,\phi \left( {i}^{\prime ... | Yes |
Lemma 2.1.4 \( x \in A\left( {x \neq 0}\right) \) is a pure quaternion if and only if \( x \notin Z\left( A\right) \) and \( {x}^{2} \in Z\left( A\right) \) . | Thus each \( x \in A \) has a unique decomposition as \( x = a + \alpha \), where \( a \in \) \( Z\left( A\right) = F \) and \( \alpha \in {A}_{0} \) . Define the conjugate \( \bar{x} \) of \( x \) by \( \bar{x} = a - \alpha \) . This defines an anti-involution of the algebra such that \( \overline{\left( x + y\right) ... | No |
Theorem 2.1.8 Every four-dimensional central simple algebra over a field \( F \) of characteristic \( \neq 2 \) is a quaternion algebra. | Proof: Let \( A \) be a four-dimensional central simple algebra over \( F \) . If \( A \) is isomorphic to \( {M}_{2}\left( F\right) \), it is a quaternion algebra, so by Theorem 2.1.7 we can assume that \( A \) is a division algebra. For \( w \notin Z\left( A\right) \), the subalgebra \( F\left( w\right) \) will be co... | Yes |
Corollary 2.1.9 Let \( A \) be a quaternion division algebra over \( F \) . If \( w \in \) \( A \smallsetminus F \) and \( E = F\left( w\right) \), then \( A{ \otimes }_{F}E \cong {M}_{2}\left( E\right) \) . | Proof: As in the above theorem, \( E \) is a quadratic extension field of \( F \) . Furthermore, there exists a standard basis \( \{ 1, y, z,{yz}\} \) of \( A \) with \( E = F\left( y\right) \) and \( {y}^{2} = a \in F \) . Thus there exists \( x \in A{ \otimes }_{F}E \) such that \( {x}^{2} = 1 \) . However, then \( A... | No |
Lemma 2.2.4 An element \( \alpha \in A \) is an integer if and only if the reduced trace \( \operatorname{tr}\left( \alpha \right) \) and the reduced norm \( n\left( \alpha \right) \) lie in \( R \) . | Proof: Any \( \alpha \) in \( A \) satisfies the polynomial\n\n\[ \n{x}^{2} - \operatorname{tr}\left( \alpha \right) x + n\left( \alpha \right) = 0.\n\]\n\nThus if the trace and norm lie in \( R \), then \( \alpha \) is clearly an integer in \( A \) .\n\nSuppose conversely that \( \alpha \) is an integer in \( A \) . I... | Yes |
Lemma 2.2.8 Let \( \mathcal{O} \) be an order in \( \operatorname{End}\left( V\right) \) . Then \( \mathcal{O} \subset \operatorname{End}\left( L\right) \) for some complete \( R \) -lattice \( L \) in \( V \) . | Proof: Let \( L = \left\{ {\ell \in {L}_{0} \mid \mathcal{O}\ell \subset {L}_{0}}\right\} \) . Then \( L \) is an \( R \) -submodule of \( {L}_{0} \) . Also, if \( a\operatorname{End}\left( {L}_{0}\right) \subset \mathcal{O} \subset {a}^{-1}\operatorname{End}\left( {L}_{0}\right) \), then \( a{L}_{0} \subset L \) . Thu... | Yes |
Theorem 2.2.9 Let \( L \) be a complete \( R \) -lattice in \( V \) . Then there exists a basis \( \{ x, y\} \) of \( V \) and a fractional ideal \( J \) such that \( L = {Rx} + {Jy} \) . | Proof: For any non-zero element \( y \in V, L \cap {ky} = {I}_{y}y \), where \( {I}_{y} = \{ \alpha \in \) \( k \mid {\alpha y} \in L\} \) . Since \( L \) is a complete \( R \) -lattice, there exists \( \beta \in R \) such that \( \beta {I}_{y} \subset R \) so that \( {I}_{y} \) is a fractional ideal.\n\nWe first show ... | Yes |
Theorem 2.3.1 For \( A = \left( \frac{a, b}{F}\right) \), the following are equivalent:\n\n(a) \( A \cong \left( \frac{1,1}{F}\right) \left( { \cong {M}_{2}\left( F\right) }\right) \).\n\n(b) \( A \) is not a division algebra.\n\n(c) \( A \) is isotropic as a quadratic space with the norm form.\n\n(d) \( {A}_{0} \) is ... | Proof: The equivalence of \( \left( a\right) \) and \( \left( b\right) \) is just a restatement of Theorem 2.1.7.\n\n\( \left( b\right) \Rightarrow \left( c\right) \) . If \( A \) is not a division algebra, it contains a non-zero non-invertible element \( x \) . Thus \( n\left( x\right) = 0 \) and \( A \) is isotropic.... | Yes |
Corollary 2.3.3 \( \; \) The quaternion algebras \( \left( \frac{1, a}{F}\right) \; \) and \( \left( \frac{a, - a}{F}\right) \; \) are isomorphic to \( {M}_{2}\left( F\right) \) . | Proof: For \( \left( \frac{1, a}{F}\right) \), the result follows immediately from \( \left( e\right) \) . For \( \left( \frac{a, - a}{F}\right) \), the norm form on \( {A}_{0} \) is \( - a{x}^{2} + a{y}^{2} + {a}^{2}{z}^{2} \), which is clearly isotropic. | No |
Theorem 2.3.4 Let \( A \) and \( {A}^{\prime } \) be quaternion algebras over \( F \) . Then \( A \) and \( {A}^{\prime } \) are isomorphic if and only if the quadratic spaces \( {A}_{0} \) and \( {A}_{0}^{\prime } \) are isometric. | Proof: With norm forms \( n \) and \( {n}^{\prime } \), this last statement means that there exists a linear isomorphism \( \phi : {A}_{0} \rightarrow {A}_{0}^{\prime } \) such that \( {n}^{\prime }\left( {\phi \left( x\right) }\right) = n\left( x\right) \) for all \( x \in {A}_{0} \) . Thus suppose that \( \psi : A \r... | Yes |
Corollary 2.3.5 If \( A = \left( \frac{a, b}{F}\right) \) and \( {A}^{\prime } = \left( \frac{{a}^{\prime },{b}^{\prime }}{F}\right) \), then \( A \) and \( {A}^{\prime } \) are isomorphic if and only if the quadratic forms \( a{x}^{2} + b{y}^{2} - {ab}{z}^{2} \) and \( {a}^{\prime }{x}^{2} + \) \( {b}^{\prime }{y}^{2}... | Proof: The norm form on \( {A}_{0} \) with respect to the restriction of the standard basis is \( - a{x}^{2} - b{y}^{2} + {ab}{z}^{2} \) . Thus the equivalence of the quadratic forms in this corollary is a restatement of the fact that \( {A}_{0} \) and \( {A}_{0}^{\prime } \) are isometric (see \( §{0.9} \) ). | No |
Theorem 2.4.1 Let \( A \) be a quaternion algebra defined over a field \( F \) . The homomorphism \( c \) defined at (2.6) induces an isomorphism\n\n\[ \n{A}^{ * }/Z\left( {A}^{ * }\right) \cong \mathrm{{SO}}\left( {{A}_{0}, n;F}\right) \n\] | A well-known example of this result shows, by taking \( A = {M}_{2}\left( \mathbb{R}\right) \), that \( \operatorname{PGL}\left( {2,\mathbb{R}}\right) \) is isomorphic to the group \( \operatorname{SO}\left( {2,1;\mathbb{R}}\right) \) . \n\nIf we restrict to \( {A}^{1} = \{ x \in A \mid n\left( x\right) = 1\} \), then ... | No |
Lemma 2.6.1 The function \( w \) just defined has the following properties:\n\n(a) \( w\left( {xy}\right) = w\left( x\right) + w\left( y\right) \) for all \( x, y \in {A}^{ * } \) .\n\n(b) \( w\left( {x + y}\right) \geq \operatorname{Min}\{ w\left( x\right), w\left( y\right) \} \) with equality when \( w\left( x\right)... | Proof: The equation (a) follows immediately from the definition of \( \nu \) since \( n \) is multiplicative. Now consider the inequality (b). Let \( x \in A \smallsetminus K \) so that \( K\left( x\right) \) is a quadratic extension of \( K \) by Lemma 2.1.6 and the restriction of \( n \) to \( K\left( x\right) \) is ... | Yes |
Theorem 2.6.6 Let \( K \) be a non-dyadic \( \mathcal{P} \) -adic field, with integers \( R \) and maximal ideal \( \mathcal{P} \) . Let \( A = \left( \frac{a, b}{K}\right) \), where \( a, b \in R \) .\n\n1. If \( a, b \notin \mathcal{P} \), then \( A \) splits.\n\n2. If \( a \notin \mathcal{P}, b \in \mathcal{P} \smal... | Proof: Recall from Hensel’s Lemma that \( c \in R \smallsetminus \mathcal{P} \) is a square in \( R \) if and only if \( c \) is a square \( {\;\operatorname{mod}\;\mathcal{P}} \) . Thus if \( a \) is a square \( {\;\operatorname{mod}\;\mathcal{P}} \), we can certainly solve \( a{x}^{2} + b{y}^{2} = 1 \) in \( K \) and... | Yes |
Decide for which fields \( {\mathbb{Q}}_{p} \), where \( p \) is an odd prime, the quaternion algebra \( \left( \frac{-{15},5}{{\mathbb{Q}}_{p}}\right) \) splits. | By part 1 of the Theorem 2.6.6, the quaternion algebra will certainly split for all odd primes \( \neq 3 \) and 5 . Since 5 is not a square \( {\;\operatorname{mod}\;3} \), the quaternion algebra does not split over \( {\mathbb{Q}}_{3} \) . For \( p = 5 \), consider \( - \left( {-{15}}\right) /5 = 3 \), which is not a ... | Yes |
Theorem 2.7.2 Let \( A \) be a quaternion algebra over a number field \( k \) . Then \( A \) splits over \( k \) if and only if \( A{ \otimes }_{k}{k}_{v} \) splits over \( {k}_{v} \) for all places \( v \) . | Proof: Let \( A = \left( \frac{a, b}{k}\right) \) . Then, by Theorem 2.3.1, \( A \) splits over \( k \) if and only if \( a{x}^{2} + b{y}^{2} = 1 \) has a solution in \( k \) . By the Hasse-Minkowski Theorem (see Corollary 0.9.9), \( a{x}^{2} + b{y}^{2} = 1 \) has a solution in \( k \) if and only if it has a solution ... | Yes |
Theorem 2.7.5 Let \( A \) and \( {A}^{\prime } \) be quaternion algebras over a number field \( k \) . Then \( A \cong {A}^{\prime } \) if and only if \( \operatorname{Ram}\left( A\right) = \operatorname{Ram}\left( {A}^{\prime }\right) \) . | Proof: By Theorem 2.3.4, \( A \) and \( {A}^{\prime } \) are isomorphic if and only if the quadratic spaces \( {A}_{0} \) and \( {A}_{0}^{\prime } \) are isometric. However, by Theorem 0.9.12, \( {A}_{0} \) and \( {A}_{0}^{\prime } \) are isometric if and only if \( {\left( {A}_{0}\right) }_{v} \) and \( {\left( {A}_{0... | Yes |
Lemma 2.8.2 The left regular representation \( \lambda \) induces an isomorphism \( A \cong {\operatorname{End}}_{A}\left( A\right) \) | Proof: For \( a \in A,{\lambda }_{a} \in {\operatorname{End}}_{A}\left( A\right) \) and \( \lambda : A \rightarrow {\operatorname{End}}_{A}\left( A\right) \) is an algebra homomorphism. Since \( A \) has an identity element, the kernel of \( \lambda \) is necessarily trivial. Further, if \( \phi \in {\operatorname{End}... | Yes |
Proposition 2.8.4 Let \( A \) and \( B \) be \( F \) -algebras.\n\n1. If \( {A}^{\prime } \) and \( {B}^{\prime } \) are subalgebras of \( A \) and \( B \), respectively,\n\n\[ \n{C}_{\left( A \otimes B\right) }\left( {{A}^{\prime } \otimes {B}^{\prime }}\right) = {C}_{A}\left( {A}^{\prime }\right) \otimes {C}_{B}\left... | Proof: Let \( E = A \otimes B \) .\n\n1. A routine calculation shows that\n\n\[ \n{C}_{A}\left( {A}^{\prime }\right) \otimes {C}_{B}\left( {B}^{\prime }\right) \subset {C}_{E}\left( {{A}^{\prime } \otimes {B}^{\prime }}\right) .\n\]\n\nChoose a basis \( \left\{ {b}_{j}\right\} \) of \( B \) . Then, if \( e \in {C}_{E}\... | Yes |
Corollary 2.8.6 If \( A \) is a central simple algebra, so is \( {A}^{o} \) and \( A \otimes {A}^{o} \cong \) \( {\operatorname{End}}_{F}\left( A\right) \) . | Proof: The first part is obvious. Define \( \theta : A \otimes {A}^{o} \rightarrow {\operatorname{End}}_{F}\left( A\right) \) by \( \theta \left( {a \otimes b}\right) \left( c\right) = {acb} \) for \( a, b, c \in A \) . Then \( \theta \) defines an algebra homomorphism. By Proposition 2.8.4, \( A \otimes {A}^{o} \) is ... | Yes |
Lemma 2.9.2 (Schur’s Lemma) Let \( M \) and \( N \) be \( A \) -modules and \( \phi : M \rightarrow \) \( N \) a non-zero homomorphism.\n\n1. If \( M \) is simple, \( \phi \) is injective.\n\n2. If \( N \) is simple, \( \phi \) is surjective. | Proof: The kernel and image of \( \phi \) are submodules of \( M \) and \( N \) respectively. \( ▱ \) | No |
Lemma 2.9.4 Let \( M \) be a module such that \( M = \mathop{\sum }\limits_{{j \in J}}{N}_{j} \), where each \( {N}_{j} \) is a simple submodule of \( M \) . Then if \( P \) is any submodule of \( M \), there exists a subset \( I \) of \( J \) such that \( M = \oplus \mathop{\sum }\limits_{{i \in I}}{N}_{i} \oplus P \)... | Proof: By Zorn’s Lemma, there is a subset \( I \) of \( J \) such that the collection \( \left\{ {{N}_{i};i \in I}\right\} \cup \{ P\} \) is maximal with respect to the property \( \mathop{\sum }\limits_{{i \in I}}{N}_{i} + \) \( P = \oplus \mathop{\sum }\limits_{{i \in I}}{N}_{i} \oplus P \) . Let \( {M}_{1} = \oplus ... | Yes |
Proposition 2.9.5 Let \( A \) be a finite-dimensional simple algebra over \( F \) . Then the following two conditions hold:\n\n1. \( {A}_{A} \) is semi-simple.\n\n2. All non-zero minimal right ideals of \( A \) are isomorphic. | Proof: The finite-dimensionality shows that \( A \) will have a non-zero minimal right ideal \( N \) of finite dimension. Now \( {AN} = \mathop{\sum }\limits_{{x \in A}}{xN} \) is a two-sided ideal of \( A \) and so \( {AN} = A \) . By Schur’s Lemma, using \( {\lambda }_{x} \), each \( {xN} \) is either 0 or simple. Th... | Yes |
Theorem 2.9.6 (Wedderburn’s Structure Theorem) Let \( A \) be a simple algebra of finite dimension over the field \( F \) . Then \( A \) is isomorphic to the matrix algebra \( {M}_{n}\left( D\right) \), where \( D \cong {\operatorname{End}}_{A}\left( N\right) \) is a division algebra with \( N \) a minimal right ideal ... | Proof: By Lemma 2.8.2, \( A \cong {\operatorname{End}}_{A}\left( A\right) \), and by Proposition 2.9.5, \( {A}_{A} \) is isomorphic to a direct sum of a number of copies, say \( n \), of a minimal right ideal \( N \) . It thus follows that \( A \cong {M}_{n}\left( {{\operatorname{End}}_{A}\left( N\right) }\right) \) (s... | No |
Proposition 2.9.7 If \( {M}_{1} \) and \( {M}_{2} \) are right A-modules, then \( {M}_{1} \) and \( {M}_{2} \) are isomorphic if and only if they have the same \( F \) -dimension. | Proof: By the remarks preceding this proposition, \( {M}_{1} \) and \( {M}_{2} \) are semisimple, thus direct sums of simple \( A \) -modules and so isomorphic to direct sums of minimal right ideals of \( A \) . However, all these minimal right ideals \( N \) are isomorphic by Proposition 2.9.5. Thus the isomorphism cl... | Yes |
Theorem 2.9.8 (Skolem Noether Theorem) Let \( A \) be a finite-dimensional central simple algebra over \( F \) and let \( B \) be a finite-dimensional simple algebra over \( F \) . If \( \phi ,\psi : B \rightarrow A \) are algebra homomorphisms, then there exists an invertible element \( c \in A \) such that \( \phi \l... | Proof: Suppose first that \( A \) is a matrix algebra over \( F \) [i.e., \( A = {\operatorname{End}}_{F}\left( V\right) \) for a vector space \( V\rbrack \) . Using \( \phi, V \) becomes a right \( {B}^{o} \) -module, \( {V}_{\phi } \), by defining \( {\alpha b} = \phi \left( b\right) \left( \alpha \right) \) for \( \... | Yes |
Lemma 3.1.3 If \( X \in \mathrm{{SL}}\left( {2,\mathbb{C}}\right) \), then \( {X}^{n} = {p}_{n}\left( {\operatorname{tr}X}\right) X - {q}_{n}\left( {\operatorname{tr}X}\right) I \), where \( {p}_{n} \) and \( {q}_{n} \) are monic integral polynomials of degrees \( n - 1 \) and \( n - 2 \), respectively. | Proof: The result follows from repeated use of\n\n\[ \n{X}^{2} = \left( {\operatorname{tr}X}\right) X - I \]\n\n(3.1)\n\nfrom which we see that \( {p}_{n}\left( x\right) = x{p}_{n - 1}\left( x\right) - {q}_{n - 1}\left( x\right) \) and \( {q}_{n}\left( x\right) = {p}_{n - 1}\left( x\right) \) . | No |
Lemma 3.1.5 Let \( V \) be an algebraic variety defined over an algebraic number field \( k \) and let \( V \) have dimension 0 . Then \( V \) is a single point and its coordinates are algebraic numbers. | Proof: In this case, \( \mathbb{C}\left( V\right) = \mathbb{C} \) since \( \mathbb{C} \) is algebraically closed. Hence \( \mathbb{C}\left\lbrack V\right\rbrack = \mathbb{C} \) . Let \( \mathbf{x} = \left( {{x}_{1},{x}_{2},\ldots ,{x}_{n}}\right) \in V \) . The maximal ideal defined by \( {m}_{\mathbf{x}} = \{ f \in \m... | Yes |
Theorem 3.2.1 \( {A}_{0}\Gamma \) is a quaternion algebra over \( \mathbb{Q}\left( {\operatorname{tr}\Gamma }\right) \) . | Proof: It is clear that \( {A}_{0}\Gamma \) is an algebra and so, by Theorem 2.1.8, we need to show that \( {A}_{0}\Gamma \) is four-dimensional, central and simple over \( \mathbb{Q}\left( {\operatorname{tr}\Gamma }\right) \) . Since \( \Gamma \) is non-elementary, it contains a pair of loxodromic elements, say \( g \... | Yes |
Corollary 3.2.3 Let the subgroup \( \Gamma \) of \( \mathrm{{SL}}\left( {2,\mathbb{C}}\right) \) contain two elements \( g \) and \( h \) such that \( \langle g, h\rangle \) is irreducible. Then \( {A}_{0}\Gamma \) is a quaternion algebra over \( \mathbb{Q}\left( {\operatorname{tr}\Gamma }\right) \) and\n\n\[ \n{A}_{0}... | Proof: Note that in Theorem 3.2.1, the assumption that the group \( \Gamma \) is non-elementary was only used to exhibit elements \( g \) and \( h \) such that \( \{ I, g, h,{gh}\} \) is a linearly independent set over \( \mathbb{C} \) . Given any such pair of elements in \( \Gamma \), like those guaranteed by the cond... | Yes |
Corollary 3.2.4 With \( \Gamma, g, h \), and \( \lambda \) as described above, \( \Gamma \) is conjugate to a subgroup of \( \mathrm{{SL}}\left( {2, k\left( \lambda \right) }\right) \) . | It should be noted that since \( g \) satisfies the same minimum polynomial as \( \lambda \), the field \( k\left( \lambda \right) \) embeds in \( {A}_{0}\Gamma \) . The above is thus a direct exhibition of the result that \( k\left( \lambda \right) \) splits the algebra \( {A}_{0}\Gamma \) as given in Corollary 2.1.9.... | No |
Corollary 3.2.5 If \( \Gamma \) is a non-elementary subgroup of \( \mathrm{{SL}}\left( {2,\mathbb{C}}\right) \) such that \( \mathbb{Q}\left( {\operatorname{tr}\Gamma }\right) \) is a subset of \( \mathbb{R} \), then \( \Gamma \) is conjugate to a subgroup of \( \mathrm{{SL}}\left( {2,\mathbb{R}}\right) \) . | Proof: If we choose \( g \) to be loxodromic, then as it has real trace, \( g \) will be hyperbolic. Thus \( \lambda \in \mathbb{R} \) and the result follows from Corollary 3.2.4. | No |
Example 3.3.1 Let \( \Gamma \) be the subgroup of \( \operatorname{PSL}\left( {2,\mathbb{C}}\right) \) generated by the images of \( A \) and \( B \), where\n\n\[ A = \left( \begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right) ,\;B = \left( \begin{matrix} 1 & 0 \\ - \omega & 1 \end{matrix}\right) . \]\n\nHere \( \omega ... | It should also be remarked that \( \Gamma \) is, in addition, of finite covolume, as it is a subgroup of index 12 in the arithmetic group \( \operatorname{PSL}\left( {2,{O}_{3}}\right) \) . (See \( §{1.4.3} \) .) | No |
Lemma 3.3.3 \( {\Gamma }^{\left( 2\right) } \) is a finite index normal subgroup of \( \Gamma \) whose quotient is an elementary abelian 2-group. | Proof: \( {\Gamma }^{\left( 2\right) } \) is obviously normal in \( \Gamma \) and such that all elements in the quotient have order 2. Since \( \Gamma \) is finitely generated, it follows that \( \Gamma /{\Gamma }^{\left( 2\right) } \) is a finite elementary abelian 2-group. | Yes |
Corollary 3.3.5 If \( \Gamma \) is a finitely generated non-elementary subgroup of \( \mathrm{SL}\left( {2,\mathbb{C}}\right) \), then the quaternion algebra \( {A}_{0}{\Gamma }^{\left( 2\right) } \) is an invariant of the commensurability class of \( \Gamma \) . | Proof: If \( \Gamma \) and \( \Delta \) are commensurable, then \( \mathbb{Q}\left( {\operatorname{tr}{\Gamma }^{\left( 2\right) }}\right) = \mathbb{Q}\left( {\operatorname{tr}{\Delta }^{\left( 2\right) }}\right) \). Now choose an irreducible pair of loxodromic elements in \( {\Gamma }^{\left( 2\right) } \cap {\Delta }... | Yes |
Theorem 3.3.7 If \( \Gamma \) is a Kleinian group of finite covolume, then its invariant trace field is a finite non-real extension of \( \mathbb{Q} \) . | Proof: That \( {k\Gamma } \) is a finite extension of \( \mathbb{Q} \) follows from Theorem 3.1.2. Suppose that \( {k\Gamma } \) is a real field. By Corollary 3.2.5, \( {\Gamma }^{\left( 2\right) } \) is conjugate to a subgroup of \( \mathrm{{SL}}\left( {2,\mathbb{R}}\right) \) . However, \( {\Gamma }^{\left( 2\right) ... | Yes |
Theorem 3.3.8 If \( \Gamma \) is a non-elementary group which contains parabolic elements, then \( {A}_{0}\Gamma = {M}_{2}\left( {\mathbb{Q}\left( {\operatorname{tr}\Gamma }\right) }\right) \) . In particular, if \( \Gamma \) is a Kleinian group such that \( {\mathbf{H}}^{3}/\Gamma \) has finite volume but is non-compa... | Proof: If \( \Gamma \) has a parabolic element \( \gamma \), then \( \gamma - I \) is non-invertible in the quaternion algebra. Thus \( {A}_{0}\Gamma \) cannot be a division algebra. The result then follows from Theorem 2.1.7. | No |
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