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1075_(GTM233)Topics in Banach Space Theory
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Definition 3.1.1
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Definition 3.1.1. A basis \( {\left( {u}_{n}\right) }_{n = 1}^{\infty } \) of a Banach space \( X \) is unconditional if for each \( x \in X \) the series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}^{ * }\left( x\right) {u}_{n} \) converges unconditionally.
Obviously, \( {\left( {u}_{n}\right) }_{n = 1}^{\infty } \) is an unconditional basis of \( X \) if and only if \( {\left( {u}_{\pi \left( n\right) }\right) }_{n = 1}^{\infty } \) is a basis of \( X \) for every permutation \( \pi : \mathbb{N} \rightarrow \mathbb{N} \) .
Example 3.1.2. The standard unit vector basis \( {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) is an unconditional basis of \( {c}_{0} \) and \( {\ell }_{p} \) for \( 1 \leq p < \infty \) . An example of a basis that is conditional (i.e., not unconditional) is the summing basis of \( {c}_{0} \), defined as
\[
{f}_{n} = {e}_{1} + \cdots + {e}_{n},\;n \in \mathbb{N}.
\]
To see that \( {\left( {f}_{n}\right) }_{n = 1}^{\infty } \) is a basis for \( {c}_{0} \) we prove that for each \( \xi = {\left( \xi \left( n\right) \right) }_{n = 1}^{\infty } \in {c}_{0} \) we have \( \xi = \mathop{\sum }\limits_{{n = 1}}^{\infty }{f}_{n}^{ * }\left( \xi \right) {f}_{n} \), where \( {f}_{n}^{ * } = {e}_{n}^{ * } - {e}_{n + 1}^{ * } \) are the biorthogonal functionals of \( {\left( {f}_{n}\right) }_{n = 1}^{\infty } \) . Given \( N \in \mathbb{N} \) ,
\[
\mathop{\sum }\limits_{{n = 1}}^{N}{f}_{n}^{ * }\left( \xi \right) {f}_{n} = \mathop{\sum }\limits_{{n = 1}}^{N}\left( {{e}_{n}^{ * }\left( \xi \right) - {e}_{n + 1}^{ * }\left( \xi \right) }\right) {f}_{n}
\]
\[
= \mathop{\sum }\limits_{{n = 1}}^{N}\left( {\xi \left( n\right) - \xi \left( {n + 1}\right) }\right) {f}_{n}
\]
\[
= \mathop{\sum }\limits_{{n = 1}}^{N}\xi \left( n\right) {f}_{n} - \mathop{\sum }\limits_{{n = 2}}^{{N + 1}}\xi \left( n\right) {f}_{n - 1}
\]
\[
= \mathop{\sum }\limits_{{n = 1}}^{N}\xi \left( n\right) \left( {{f}_{n} - {f}_{n - 1}}\right) - \xi \left( {N + 1}\right) {f}_{N}
\]
\[
= \left( {\mathop{\sum }\limits_{{n = 1}}^{N}\xi \left( n\right) {e}_{n}}\right) - \xi \left( {N + 1}\right) {f}_{N}
\]
where we have used the convention that \( {f}_{0} = 0 \) . Therefore,
\[
{\begin{Vmatrix}\xi - \mathop{\sum }\limits_{{n = 1}}^{N}{f}_{n}^{ * }\left( \xi \right) {f}_{n}\end{Vmatrix}}_{\infty } = {\begin{Vmatrix}\mathop{\sum }\limits_{{N + 1}}^{\infty }\xi \left( n\right) {e}_{n} + \xi \left( N + 1\right) {f}_{N}\end{Vmatrix}}_{\infty }
\]
\[
\leq {\begin{Vmatrix}\mathop{\sum }\limits_{{N + 1}}^{\infty }{\xi }_{n}{e}_{n}\end{Vmatrix}}_{\infty } + \left| {\xi \left( {N + 1}\right) }\right| {\begin{Vmatrix}{f}_{N}\end{Vmatrix}}_{\infty }\overset{N \rightarrow \infty }{ \rightarrow }0
\]
and \( {\left( {f}_{n}\right) }_{n = 1}^{\infty } \) is a basis.
Now we will identify the set \( S \) of coefficients \( {\left( {\alpha }_{n}\right) }_{n = 1}^{\infty } \) such that the series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{\alpha }_{n}{f}_{n} \) converges. In fact, we have that \( {\left( {\alpha }_{n}\right) }_{n = 1}^{\infty } \in S \) if and only if there exists \( \xi = {\left( \xi \left( n\right) \right) }_{n = 1}^{\infty } \in {c}_{0} \) such that \( {\alpha }_{n} = \xi \left( n\right) - \xi \left( {n + 1}\right) \) for all \( n \) . Then, clearly, unless the series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{\alpha }_{n} \) converges absolutely, the convergence of \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{\alpha }_{n}{f}_{n} \) in \( {c}_{0} \) is not equivalent to the convergence of \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{\epsilon }_{n}{\alpha }_{n}{f}_{n} \) for all choices of signs \( {\left( {\epsilon }_{n}\right) }_{n = 1}^{\infty } \) . Hence \( {\left( {f}_{n}\right) }_{n = 1}^{\infty } \) cannot be unconditional.
Proposition 3.1.3. A basis \( {\left( {u}_{n}\right) }_{n = 1}^{\infty } \) of a Banach space \( X \) is unconditional if and only if there is a constant \( K \geq 1 \) such that for all \( N \in \mathbb{N} \) ,
\[
\begin{Vmatrix}{\mathop{\sum }\limits_{{n = 1}}^{N}{a}_{n}{u}_{n}}\end{Vmatrix} \leq K\begin{Vmatrix}{\mathop{\sum }\limits_{{n = 1}}^{N}{b}_{n}{u}_{n}}\end{Vmatrix}
\]
(3.1)
whenever \( {a}_{1},\ldots ,{a}_{N},{b}_{1},\ldots ,{b}_{N} \) are scalars satisfying \( \left| {a}_{n}\right| \leq \left| {b}_{n}\right| \) for \( n = 1,\ldots, N \) . Proof. Assume \( {\left( {u}_{n}\right) }_{n = 1}^{\infty } \) is unconditional. If \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}{u}_{n} \) converges, then so does \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{t}_{n}{a}_{n}{u}_{n} \) for all \( {\left( {t}_{n}\right) }_{n = 1}^{\infty } \in {\ell }_{\infty } \) by Proposition 2.4.9. By the Banach-Steinhaus theorem, the linear map \( {T}_{\left( {t}_{n}\right) } : X \rightarrow X \) given by \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}{u}_{n} \mapsto \mathop{\sum }\limits_{{n = 1}}^{\infty }{t}_{n}{a}_{n}{u}_{n} \) is continuous. Now the uniform boundedness principle yields \( K \) such that equation (3.1) holds.
Conversely, let us take a convergent series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}{u}_{n} \) in \( X \) . We are going to prove that the subseries \( \mathop{\sum }\limits_{{k = 1}}^{\infty }{a}_{{n}_{k}}{u}_{{n}_{k}} \) is convergent for every increasing sequence of integers \( {\left( {n}_{k}\right) }_{k = 1}^{\infty } \) and appeal to Lemma 2.4.2 to deduce that it is unconditionally convergent. Given \( \epsilon > 0 \) ; there is \( N = N\left( \epsilon \right) \in \mathbb{N} \) such that if \( {m}_{2} > {m}_{1} \geq N \), then
\[
\begin{Vmatrix}{\mathop{\sum }\limits_{{n = {m}_{1} + 1}}^{{m}_{2}}{a}_{n}{u}_{n}}\end{Vmatrix} < \frac{\epsilon }{K}
\]
By hypothesis, if \( N \leq {n}_{k} < \cdots < {n}_{k + l} \), we have
\[
\begin{Vmatrix}{\mathop{\sum }\limits_{{j = k + 1}}^{{k + l}}{a}_{{n}_{j}}{u}_{{n}_{j}}}\end{Vmatrix} \leq K\begin{Vmatrix}{\mathop{\sum }\limits_{{j = {n}_{k} + 1}}^{{n}_{k + l}}{a}_{j}{u}_{j}}\end{Vmatrix} < \epsilon
\]
and so \( \mathop{\sum }\limits_{{k = 1}}^{\infty }{a}_{{n}_{k}}{u}_{{n}_{k}} \) is Cauchy.
Definition 3.1.4. Let \( {\left( {u}_{n}\right) }_{n = 1}^{\infty } \) be an unconditional basis of a Banach space \( X \) . The unconditional basis constant \( {\mathrm{K}}_{\mathrm{u}} \) of \( {\left( {u}_{n}\right) }_{n = 1}^{\infty } \) is the least constant \( K \) such that equation (3.1) holds. We then say that \( {\left( {u}_{n}\right) }_{n = 1}^{\infty } \) is \( K \) -unconditional whenever \( K \geq {\mathrm{K}}_{\mathrm{u}} \) .
Suppose \( {\left( {u}_{n}\right) }_{n = 1}^{\infty } \) is an unconditional basis for a Banach space \( X \) . For each sequence of scalars \( \left( {\alpha }_{n}\right) \) with \( \left| {\alpha }_{n}\right| = 1 \), let \( {T}_{\left( {\alpha }_{n}\right) } : X \rightarrow X \) be the isomorphism defined by \( {T}_{\left( {\alpha }_{n}\right) }\left( {\mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}{u}_{n}}\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }{\alpha }_{n}{a}_{n}{u}_{n} \) . Then
\[
{\mathrm{K}}_{\mathrm{u}} = \sup \left\{ {\begin{Vmatrix}{T}_{\left( {\alpha }_{n}\right) }\end{Vmatrix} : \left( {\alpha }_{n}\right) \text{ scalars,}\left| {\alpha }_{n}\right| = 1\text{ for all }n}\right\} .
\]
Let \( {\left( {u}_{n}\right) }_{n = 1}^{\infty } \) be an unconditional basis of \( X \) . For every \( A \subseteq \mathbb{N} \) there is a linear projection \( {P}_{A} \) from \( X \) onto \( \left\lbrack {{u}_{k} : k \in A}\right\rbrack \) defined for each \( x = \mathop{\sum }\limits_{{k = 1}}^{\infty }{u}_{k}^{ * }\left( x\right) {u}_{k} \) in \( X \) by
\[
{P}_{A}\left( x\right) = \mathop{\sum }\limits_{{k \in A}}{u}_{k}^{ * }\left( x\right) {u}_{k}
\]
The members of the set \( \left\{ {{P}_{A} : A \subseteq \mathbb{N}}\right\} \) are the natural projections associated to the unconditional basis \( {\left( {u}_{n}\right) }_{n = 1}^{\infty } \) .
Proposition 3.1.5. Let \( {\left( {u}_{n}\right) }_{n = 1}^{\infty } \) be a basis of a Banach space \( X \) . The following are equivalent:
(i) The basis \( {\left( {u}_{n}\right) }_{n = 1}^{\infty } \) is unconditional.
(ii) The map \( {P}_{A} \) is well defined for every \( A \subseteq \mathbb{N} \) .
(iii) The map \( {P}_{A} \) is well defined for every \( A \subseteq \mathbb{N} \) and \( \mathop{\sup }\limits_{A}\begin{Vmatrix}{P}_{A}\end{Vmatrix} < \infty \) .
(iv) \( \sup \left\{ {\begin{Vmatrix}{{P}_{F}\left( x\right) }\end{Vmatrix} : F \subseteq \mathbb{N}}\right. \), F finite \( \} < \infty \) .
(v) The map \( {P}_{B} \) is well defined for every cofinite subset \( B \) of \( \mathbb{N} \) and
\[
\sup \left\{ {\begin{Vmatrix}{{P}_{B}\left( x\right) }\end{Vmatrix} : B\text{ cofinite subset of }\mathbb{N}}\right\} < \infty .
\]
Moreover, if any of the above statements holds, then the suprema in (iii), (iv), and (v) coincide.
Proof. The implication \( \left( i\right) \Rightarrow \left( {ii}\right) \) is a consequence of Proposition 3.1.3.
\( \left( {ii}\right) \Rightarrow \left( {iii}\right) \) follows readily from the uniform boundedness principle.
(iii) \( \Rightarrow \left( {iv}\right) \) and \( \left( {iii}\right) \Rightarrow \left( v\right) \) are trivial.
(iv) \( \Rightarrow \) (iii) Let \( A \) be any subset (finite or infinite) of \( \mathbb{N} \) . For every \( x \in X \) with finite support in \( {\left( {u}_{n}\right) }_{n = 1}^{\infty } \) let \( S = \operatorname{supp}\left( x\right) \) . Since \( {P}_{A}\left( x\right) = {P}_{A \cap S}\left( x\right) \) ,
\[
\begin{Vmatrix}{{P}_{A}\left( x\right) }\end{Vmatrix} = \begin{Vmatrix}{{P}_{A \cap S}\left( x\right) }\end{Vmatrix} \leq \begin{Vmatrix}{P}_{A \cap S}\end{Vmatrix}\parallel x\parallel \leq \sup \left\{ {\begin{Vmatrix}{P}_{F}\end{Vmatrix} : F\text{ finite }}\right\} \parallel x\parallel .
\]
By density \( {P}_{A} \) extends to a bounded operator from \( X \) to \( X \) .
To close the cycle of equivalences we will show that \( \left(
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1129_(GTM35)Several Complex Variables and Banach Algebras
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Definition 11.5
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Definition 11.5. Let \( \left( {A, X,\Omega, p}\right) \) be a maximum modulus algebra. Let \( E \) be a subset of \( \Omega \) . For \( n \) an integer \( \geq 1 \), we say that \( \left( {A, X,\Omega, p}\right) \) lies at most \( n \) -sheeted over \( E \) if \( \# {p}^{-1}\left( \lambda \right) \leq n \) for each \( \lambda \in E \) .
EXAMPLE 11.3. Let \( \Omega \) be a region in \( \mathbb{C} \) and let \( {a}_{1},{a}_{2},\ldots ,{a}_{n} \) be analytic functions defined on \( \Omega \) . We let \( X \) be the set in \( {\mathbb{C}}^{2} \) defined by the equation
(28)
\[
{w}^{n} + {a}_{1}\left( z\right) {w}^{n - 1} + {a}_{2}\left( z\right) {w}^{n - 2} + \cdots + {a}_{n}\left( z\right) = 0,
\]
in the sense that \( X = \left\{ {\left( {z, w}\right) \in {\mathbb{C}}^{2} : \left( {z, w}\right) \text{satisfies (28)}}\right\} \) .
Let \( A \) be the algebra consisting of all restrictions to \( X \) of polynomials in \( z \) and \( w \) . Then \( A \) is an algebra of continuous functions on \( X \) . Put \( p\left( {z, w}\right) = z \), for \( \left( {z, w}\right) \in X \) . Then \( p \in A \) and \( p : X \rightarrow \Omega \) is a proper map.
We claim that \( \left( {A, X,\Omega, p}\right) \) is a maximum modulus algebra, and that it lies at most \( n \) -sheeted over \( \Omega \) -provided that the polynomial of (28) satisfies an additional hypothesis on its discriminant, to be formulated below.
For each \( z \in \Omega \), equation (28) has \( n \) roots in \( \mathbb{C} \), and we denote these roots by \( {w}_{1}\left( z\right) ,{w}_{2}\left( z\right) ,\ldots ,{w}_{n}\left( z\right) \), taken in some order. If \( \sigma \) is any symmetric function of \( n \) variables, the number \( \sigma \left( {{w}_{1}\left( z\right) ,{w}_{2}\left( z\right) ,\ldots ,{w}_{n}\left( z\right) }\right) \) is independent of the order of the roots and hence gives a single-valued function of \( z \) on \( \Omega \) .
In particular, if we take \( \sigma \) to be \( \mathop{\prod }\limits_{{i < j}}{\left( {w}_{i} - {w}_{j}\right) }^{2} \), and define \( D\left( z\right) = \) \( \mathop{\prod }\limits_{{i < j}}{\left( {w}_{i}\left( z\right) - {w}_{j}\left( z\right) \right) }^{2} \), then \( D \) is a single-valued function on \( \Omega \) called the discriminant.
Hypothesis. We shall assume that \( D \) is not identically 0 on \( \Omega \) .
The coefficient functions \( {a}_{j} \) in (28) correspond to the elementary symmetric functions. Since \( \mathop{\prod }\limits_{{i < j}}{\left( {w}_{i} - {w}_{j}\right) }^{2} \) is a polynomial in the elementary symmetric functions, it follows that \( D \) is analytic in \( \Omega \) . Since, by hypothesis, \( D \) is not identically 0, the zeros of \( D \) form a discrete subset \( \Lambda \) of \( \Omega ;\Lambda \) is empty, finite, or countably infinite.
Fix \( {z}_{0} \in \Omega \smallsetminus \Lambda \) . Then the roots \( {w}_{1}\left( z\right) ,{w}_{2}\left( z\right) ,\ldots ,{w}_{n}\left( z\right) \) are distinct. Cauchy theory yields that, in some neighborhood \( \mathcal{U} \) of \( {z}_{0} \), there are \( n \) single-valued analytic functions \( {w}_{1},{w}_{2},\ldots ,{w}_{n} \) that provide the roots of (28). For \( z \in \mathcal{U} \) , the points of \( X \) over \( z \), i.e., which are mapped to \( z \) by \( p \), are the points \( \left( {z,{w}_{1}\left( z\right) }\right) ,\left( {z,{w}_{2}\left( z\right) }\right) ,\ldots ,\left( {z,{w}_{n}\left( z\right) }\right) \) . Fix a function \( f \in A \) . Then there exists a polynomial \( Q\left( {z, w}\right) \) such that \( f\left( \left( {z,{w}_{j}\left( z\right) }\right) \right) = Q\left( {z,{w}_{j}\left( z\right) }\right), j = 1,\ldots, n \) . Thus the function \( z \mapsto f\left( \left( {z,{w}_{j}\left( z\right) }\right) \right) \) is analytic on \( \mathcal{U} \) for each \( j \) . We define the symmetric function \( \sigma \left( {{\alpha }_{1},{\alpha }_{2},\cdots ,{\alpha }_{n}}\right) = \mathop{\max }\limits_{{1 \leq j \leq n}}\left| {\alpha }_{j}\right| \) . Hence the function \( u : z \mapsto \mathop{\max }\limits_{{1 \leq j \leq n}}\left| {f\left( \left( {z,{w}_{j}\left( z\right) }\right) \right) }\right| \) is well defined on \( \Omega \smallsetminus \Lambda \) . By the above discussion, \( \left| {f\left( \left( {z,{w}_{j}\left( z\right) }\right) \right) }\right| \) is locally subharmonic at each point \( {z}_{0} \) in \( \Omega \smallsetminus \Lambda \) . Hence \( u \) is subharmonic on \( \Omega \smallsetminus \Lambda \) and has isolated singularities at the points of \( \Lambda \) . In a deleted neighborhood of each point of \( \Lambda, u \) is locally bounded. It follows (see [Tsu] Thm. III.30) that, if we define \( u\left( {z}_{1}\right) = {\overline{\lim }}_{z \rightarrow {z}_{1}}u\left( z\right) \), then \( u \) is subharmonic on all \( \Omega \) . We claim that the equality \( u\left( z\right) = \mathop{\max }\limits_{{1 \leq j \leq n}}\left| {f\left( \left( {z,{w}_{j}\left( z\right) }\right) \right) }\right| \) remains true at points \( z \in \Lambda \) ; by the definition of \( u \), we already know that it holds for points \( z \in \Omega \smallsetminus \Lambda \) . Let \( \lambda \in \Lambda \) and fix one of the roots \( {w}_{{j}_{0}}\left( \lambda \right) \) at \( \lambda \) . Then, by the Cauchy theory, there exists \( {z}_{k} \rightarrow \lambda ,{z}_{k} \in \Omega \smallsetminus \Lambda \) and points \( \left\{ {{w}_{{j}_{k}}\left( {z}_{k}\right) }\right\} \) such that \( {w}_{{j}_{k}}\left( {z}_{k}\right) \rightarrow {w}_{{j}_{0}}\left( \lambda \right) \) . It follows that \( z \mapsto \mathop{\max }\limits_{{1 \leq j \leq n}}\left| {f\left( \left( {z,{w}_{j}\left( z\right) }\right) \right) }\right| \) is continuous at \( z = \lambda \) and so the claim follows.
Now let \( \Delta \) be any closed disk contained in \( \Omega \), with center \( {\lambda }_{0} \), and let \( {x}_{0} \) be a point of \( X \) lying over \( {\lambda }_{0} \) . Since \( {x}_{0} \) can be written \( \left( {{\lambda }_{0},{w}_{j}\left( {\lambda }_{0}\right) }\right) \) for some \( j \) , \( \left| {f\left( {x}_{0}\right) }\right| \leq \mathop{\max }\limits_{{1 \leq j \leq n}}\left| {f\left( {{\lambda }_{0},{w}_{j}\left( {\lambda }_{0}\right) }\right) }\right| = u\left( {\lambda }_{0}\right) \) . Since \( u \) is subharmonic on \( \Omega \) , \( u\left( {\lambda }_{0}\right) \leq \mathop{\max }\limits_{{\lambda \in \partial \Delta }}u\left( \lambda \right) = \mathop{\max }\limits_{{{p}^{-1}\left( \Delta \right) }}\left( \left| f\right| \right) \) . Hence \( \left| {f\left( {x}_{0}\right) }\right| \leq \mathop{\max }\limits_{{{p}^{-1}\left( \Delta \right) }}\left( \left| f\right| \right) \) . Thus \( \left( {A, X,\Omega, f}\right) \) is a maximum modulus algebra, as claimed. That it lies at most \( n \) -sheeted over \( \Omega \) is clear from the definition.
We next show that a maximum modulus algebra \( \left( {A, X,\Omega, f}\right) \) which lies finite-sheeted over a sufficiently large subset \( E \) of \( \Omega \) is an algebra of analytic functions on a certain Riemann surface, in the sense of the following theorem. See the Appendix for the notion of logarithmic capacity.
Theorem 11.8. Let \( \left( {A, X,\Omega, f}\right) \) be a maximum modulus algebra. Assume that, for some integer \( n \), there exists a Borel set \( E \subseteq \Omega \) of logarithmic capacity \( c\left( E\right) > \) 0, such that, for every \( \lambda \in E,\# {f}^{-1}\left( \lambda \right) \leq n \) . Then:
(i) # \( {f}^{-1}\left( \lambda \right) \leq n \) for every \( \lambda \in \Omega \), and
(ii) there exists a discrete subset \( \Lambda \) of \( \Omega \) such that \( {f}^{-1}\left( {\Omega \smallsetminus \Lambda }\right) \) admits the structure of a Riemann surface on which every function in \( A \) is analytic.
Proof. Fix a function \( g \in A \) . By hypothesis, if \( \lambda \in E,\# {f}^{-1}\left( \lambda \right) \leq n \) ; so \( \# g\left( {{f}^{-1}\left( \lambda \right) }\right) \leq n \) and hence \( {d}_{n + 1}\left( {g\left( {{f}^{-1}\left( \lambda \right) }\right) }\right) = 0 \) . We define \( \psi \left( \lambda \right) = \) \( \log {d}_{n + 1}\left( {g\left( {{f}^{-1}\left( \lambda \right) }\right) }\right) \) for \( \lambda \in \Omega \) . Then \( \psi \left( \lambda \right) = - \infty \) on \( E \) . Also by Theorem 11.6, \( \psi \) is subharmonic on \( \Omega \) . By the Appendix, since \( c\left( E\right) > 0 \), this implies that \( \psi \) is identically equal to \( - \infty \) . Hence \( {d}_{n + 1}\left( {g\left( {{f}^{-1}\left( \lambda \right) }\right) }\right) = 0 \) for all \( \lambda \in \Omega \) .
Fix \( {\lambda }_{0} \in \Omega \) . Suppose that \( \# {f}^{-1}\left( {\lambda }_{0}\right) \geq n + 1 \) . Then, because \( A \) separates the points of \( X \), we may choose \( g \in A \) such that the set \( g\left( {{f}^{-1}\left( {\lambda }_{0}\right) }\right) \) contains at least \( n + 1 \) points. Hence \( {d}_{n + 1}\left( {g\left( {{f}^{-1}\left( {\lambda }_{0}\right) }\right) }\right) \neq 0 \) . This is a contradiction; so no such \( {\lambda }_{0} \) exists. Thus \( \# {f}^{-1}\left( \lambda \right) \leq n \) for every \( \lambda \in \Omega \) . Assertion (i) is proved.
Define \( {\Omega }_{n} = \left\{ {\lambda : \# {f}^{-1}\left( \lambda \right) = n}\right\} \) . We clearly may assume that \( {\Omega }_{n} \) is nonempty. Fix \( p \in {f}^{-1}\left( {\Omega }_{n}\right) \) . We shall construct a neighborhood of \( p \) in \( X \) such that \( f \) maps this neighborhood one-one onto a disk in \( \Omega \), centered at \( {\lambda }_{0} = f\left( p\right) \) .
By hypothesis, \( {f}^{-1}\left( {\lambda }_{0}\right) = \left\{ {{p}_{1},{p}_{2},\ldots ,{p}_{n}}\right\} \) . Without loss of generality, \( p = \) \( {p}_{1} \) . We choose disjoint compact neighborhoods \( {\mathcal{U}}_{j} \) of \( {p}_{j},1 \leq j \leq n \), and choose a closed disk \( \Delta = \left\{ {\lambd
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111_111_Three Dimensional Navier-Stokes Equations-James_C._Robinson,_Jos_L._Rodrigo,_Witold_Sadows(z-lib.org
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Definition 14.4
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Definition 14.4 We shall call the map \( \rho \left( {T}_{0}\right) \ni z \rightarrow \gamma \left( z\right) \in \mathbf{B}\left( {\mathcal{K},\mathcal{H}}\right) \) the gamma field and the map \( \rho \left( {T}_{0}\right) \ni z \rightarrow M\left( z\right) \in \mathbf{B}\left( \mathcal{K}\right) \) the Weyl function of the operator \( {T}_{0} \) associated with the boundary triplet \( \left( {\mathcal{K},{\Gamma }_{0},{\Gamma }_{1}}\right) \) .
Basic properties of these operator fields are contained in the next propositions.
Proposition 14.14 For \( z, w \in \rho \left( {T}_{0}\right) \), we have:
(i) \( \gamma {\left( \bar{z}\right) }^{ * } = {\Gamma }_{1}{\left( {T}_{0} - zI\right) }^{-1} \) .
(ii) \( \mathcal{N}\left( {\gamma {\left( z\right) }^{ * }}\right) = {\mathcal{N}}_{z}^{ \bot } \) and \( \gamma {\left( z\right) }^{ * } \) is a bijection of \( {\mathcal{N}}_{z} \) onto \( \mathcal{K} \) .
(iii) \( \gamma \left( w\right) - \gamma \left( z\right) = \left( {w - z}\right) {\left( {T}_{0} - wI\right) }^{-1}\gamma \left( z\right) = \left( {w - z}\right) {\left( {T}_{0} - zI\right) }^{-1}\gamma \left( w\right) \) .
(iv) \( \gamma \left( w\right) = \left( {{T}_{0} - {zI}}\right) {\left( {T}_{0} - wI\right) }^{-1}\gamma \left( z\right) \) .
(v) \( \frac{d}{dz}\gamma \left( z\right) = {\left( {T}_{0} - zI\right) }^{-1}\gamma \left( z\right) \) .
Proof (i): Let \( x \in \mathcal{H} \) . Set \( y = {\left( {T}_{0} - zI\right) }^{-1}x \) . Let \( v \in \mathcal{K} \) . Using the facts that \( {T}^{ * }\gamma \left( \bar{z}\right) v = \bar{z}\gamma \left( \bar{z}\right) v,{\Gamma }_{0}y = 0 \), and \( {\Gamma }_{0}\gamma \left( \bar{z}\right) v = v \) and Definition 14.2(i), we derive
\[
{\left\langle \gamma {\left( \bar{z}\right) }^{ * }\left( {T}_{0} - zI\right) y, v\right\rangle }_{\mathcal{K}} = \left\langle {\left( {{T}_{0} - {zI}}\right) y,\gamma \left( \bar{z}\right) v}\right\rangle
\]
\[
= \left\langle {{T}_{0}y,\gamma \left( \bar{z}\right) v}\right\rangle - \left\langle {y,\bar{z}\gamma \left( \bar{z}\right) v}\right\rangle = \left\langle {{T}^{ * }y,\gamma \left( \bar{z}\right) v}\right\rangle - \left\langle {y,{T}^{ * }\gamma \left( \bar{z}\right) v}\right\rangle
\]
\[
= {\left\langle {\Gamma }_{1}y,{\Gamma }_{0}\gamma \left( \bar{z}\right) v\right\rangle }_{\mathcal{K}} - {\left\langle {\Gamma }_{0}y,{\Gamma }_{1}\gamma \left( \bar{z}\right) v\right\rangle }_{\mathcal{K}} = {\left\langle {\Gamma }_{1}y, v\right\rangle }_{\mathcal{K}}
\]
Since \( v \in \mathcal{K} \) was arbitrary, the latter yields \( \gamma {\left( \bar{z}\right) }^{ * }\left( {{T}_{0} - {zI}}\right) y = {\Gamma }_{1}y \) . Inserting now \( y = {\left( {T}_{0} - zI\right) }^{-1}x \), this gives \( \gamma {\left( \bar{z}\right) }^{ * }x = {\Gamma }_{1}{\left( {T}_{0} - zI\right) }^{-1}x \) .
(ii): Since \( \mathcal{R}\left( {\gamma \left( z\right) }\right) = {\mathcal{N}}_{z} \), we have \( \mathcal{N}\left( {\gamma {\left( z\right) }^{ * }}\right) = {\mathcal{N}}_{z}^{ \bot } \), so \( \gamma {\left( z\right) }^{ * } \mid {\mathcal{N}}_{z} \) is injective. Let \( v \in \mathcal{K} \) . By Definition 14.2(ii) there exists \( y \in \mathcal{D}\left( {T}^{ * }\right) \) such that \( {\Gamma }_{0}y = 0 \) and \( {\Gamma }_{1}y = v \) . Then \( y \in \mathcal{D}\left( {T}_{0}\right) \) . Let \( {y}_{z} \) be the projection of \( \left( {{T}_{0} - \bar{z}I}\right) y \) onto \( {\mathcal{N}}_{z} \) . Using that \( \mathcal{N}\left( {\gamma {\left( z\right) }^{ * }}\right) = {\mathcal{N}}_{z}^{ \bot } \) and (i), we obtain \( \gamma {\left( z\right) }^{ * }{y}_{z} = \gamma {\left( z\right) }^{ * }\left( {{T}_{0} - \bar{z}I}\right) y = {\Gamma }_{1}y = v \) . This shows that \( \gamma {\left( z\right) }^{ * } \upharpoonright {\mathcal{N}}_{z} \) is surjective.
(iii): Let \( v \in \mathcal{K} \) . By Lemma 14.13(ii), \( v = {\Gamma }_{0}u \) for some \( u \in {\mathcal{N}}_{z} \) . Putting
\[
{u}^{\prime } \mathrel{\text{:=}} u + \left( {w - z}\right) {\left( {T}_{0} - wI\right) }^{-1}u,
\]
(14.40)
we compute \( {T}^{ * }{u}^{\prime } = {zu} + \left( {w - z}\right) {T}_{0}{\left( {T}_{0} - wI\right) }^{-1}u = w{u}^{\prime } \), that is, \( {u}^{\prime } \in {\mathcal{N}}_{w} \) . Since \( {\left( {T}_{0} - wI\right) }^{-1}u \in \mathcal{D}\left( {T}_{0}\right) \) and hence \( {\Gamma }_{0}{\left( {T}_{0} - wI\right) }^{-1}u = 0 \), we get \( {\Gamma }_{0}{u}^{\prime } = {\Gamma }_{0}u = v \) . Therefore, \( \gamma \left( z\right) v = \gamma \left( z\right) {\Gamma }_{0}u = u \) and \( \gamma \left( w\right) v = \gamma \left( w\right) {\Gamma }_{0}{u}^{\prime } = {u}^{\prime } \) . Inserting the latter into (14.40), we obtain \( \gamma \left( w\right) v = \gamma \left( z\right) v + \left( {w - z}\right) {\left( {T}_{0} - wI\right) }^{-1}\gamma \left( z\right) v \) . This proves the first equality of (iii). Interchanging \( z \) and \( w \) in the first equality gives the second equality of (iii).
(iv) follows from (iii), since \( \left( {{T}_{0} - {zI}}\right) {\left( {T}_{0} - wI\right) }^{-1} = I + \left( {w - z}\right) {\left( {T}_{0} - wI\right) }^{-1} \) .
(v): We divide the first equality of (iii) by \( w - z \) and let \( w \rightarrow z \) . Using the continuity of the resolvent in the operator norm (see formula (2.7)), we obtain (v).
Proposition 14.15 For arbitrary \( z, w \in \rho \left( {T}_{0}\right) \), we have:
(i) \( M\left( z\right) {\Gamma }_{0}u = {\Gamma }_{1}u \) for \( u \in {\mathcal{N}}_{z} \) .
(ii) \( M{\left( z\right) }^{ * } = M\left( \bar{z}\right) \) .
(iii) \( M\left( w\right) - M\left( z\right) = \left( {w - z}\right) \gamma {\left( \bar{z}\right) }^{ * }\gamma \left( w\right) \) .
(iv) \( \frac{d}{dz}M\left( z\right) = \gamma {\left( \bar{z}\right) }^{ * }\gamma \left( z\right) \) .
Proof (i): Since \( \gamma \left( z\right) = {\left( {\Gamma }_{0} \upharpoonright {\mathcal{N}}_{z}\right) }^{-1} \), we get \( M\left( z\right) {\Gamma }_{0}u = {\Gamma }_{1}\gamma \left( z\right) {\Gamma }_{0}u = {\Gamma }_{1}u \) .
(ii): Let \( u \in {\mathcal{N}}_{z} \) and \( {u}^{\prime } \in {\mathcal{N}}_{\bar{z}} \) . Obviously, \( \left\langle {{T}^{ * }u,{u}^{\prime }}\right\rangle = \left\langle {u,{T}^{ * }{u}^{\prime }}\right\rangle \) . Therefore, by (i) and Definition 14.2(i) we obtain
\[
{\left\langle M\left( z\right) {\Gamma }_{0}u,{\Gamma }_{0}{u}^{\prime }\right\rangle }_{\mathcal{K}} = {\left\langle {\Gamma }_{1}u,{\Gamma }_{0}{u}^{\prime }\right\rangle }_{\mathcal{K}} = {\left\langle {\Gamma }_{0}u,{\Gamma }_{1}{u}^{\prime }\right\rangle }_{\mathcal{K}} = {\left\langle {\Gamma }_{0}u, M\left( \bar{z}\right) {\Gamma }_{0}{u}^{\prime }\right\rangle }_{\mathcal{K}}.
\]
Since \( {\Gamma }_{0}\left( {\mathcal{N}}_{z}\right) = {\Gamma }_{0}\left( {\mathcal{N}}_{\bar{z}}\right) = \mathcal{K} \) by Lemma 14.13, the latter shows that \( M\left( \bar{z}\right) = M{\left( z\right) }^{ * } \) .
(iii): Using Proposition 14.14, (ii) and (i), we derive
\[
M\left( w\right) - M\left( z\right) = {\Gamma }_{1}\left( {\gamma \left( w\right) - \gamma \left( z\right) }\right) = \left( {w - z}\right) {\Gamma }_{1}{\left( {T}_{0} - zI\right) }^{-1}\gamma \left( w\right)
\]
\[
= \left( {w - z}\right) \gamma {\left( \bar{z}\right) }^{ * }\gamma \left( w\right) .
\]
(iv) follows by dividing (iii) by \( w - z \), letting \( w \rightarrow z \), and using the continuity of \( \gamma \left( z\right) \) in the operator norm (by Proposition 14.14(v)).
Propositions 14.14(v) and 14.15(iv) imply that the gamma field \( z \rightarrow \gamma \left( z\right) \) and the Weyl function \( z \rightarrow M\left( z\right) \) are operator-valued holomorphic functions on the resolvent set \( \rho \left( {T}_{0}\right) \) . In particular, both fields are continuous in the operator norm.
Definition 14.5 An operator-valued function \( F : {\mathbb{C}}_{ + } \rightarrow \left( {\mathbf{B}\left( \mathcal{K}\right) ,\parallel \cdot \parallel }\right) \) is called a Nevanlinna function if \( F \) is holomorphic on \( {\mathbb{C}}_{ + } = \{ z \in \mathbb{C} : \operatorname{Im}z > 0\} \) and
\[
\operatorname{Im}F\left( z\right) = \frac{1}{2\mathrm{i}}\left( {F\left( z\right) - F{\left( z\right) }^{ * }}\right) \geq 0\;\text{ for all }z \in {\mathbb{C}}_{ + }.
\]
Each scalar Nevanlinna function admits a canonical integral representation described in Theorem F.1. For general operator Nevanlinna functions and separable Hilbert spaces \( \mathcal{K} \), there is a similar integral representation, where \( a = {a}^{ * } \) and \( b = {b}^{ * } \geq 0 \) are in \( \mathbf{B}\left( \mathcal{K}\right) \), and \( v \) is a positive operator-valued Borel measure on \( \mathbb{R} \) .
Corollary 14.16 The Weyl function \( M\left( z\right) \) is a Nevanlinna function on \( \mathcal{K} \) .
Proof By Proposition 14.15(iv), \( M\left( z\right) \) is a \( \mathbf{B}\left( \mathcal{K}\right) \) -valued holomorphic function on \( {\mathbb{C}}_{ + } \) . Let \( z \in {\mathbb{C}}_{ + } \) and \( y = \operatorname{Im}z \) . From Proposition 14.15,(ii) and (iii), we obtain
\[
M\left( z\right) - M{\left( z\right) }^{ * } = M\left( z\right) - M\left( \bar{z}\right) = \left( {z - \bar{z}}\right) \gamma {\left( z\right) }^{ * }\gamma \left( z\right) = 2\mathrm{i}{y\gamma }{\left( z\right) }^{ * }\gamma \left( z\right) .
\]
Since \( y > 0 \), this implies that \( \operatorname{Im}M\left( z\right) \geq 0 \) .
The next proposition shows how eigenvalues and spectrum of an operator \( {T}_{\mathcal{B}} \) can be detected by means of the Weyl function.
Proposition 14.17 Suppose that \( \mathcal{B} \) is a closed relation on \( \mathcal{K} \) and \( z \in \rho \left( {T}_{0}\right) \) . Then the relation \( \mathcal{B} - M\left( z\right) \) (which is defined by \( \mathcal{B} - \mathcal{G}\left( {M\left( z\right) }\right) \) is also closed, and we have:
(i) \( \gamma \left( z\right) \mathcal{N}\left( {\mathcal{B} - M\left( z\right) }\right) = \mathcal
|
1063_(GTM222)Lie Groups, Lie Algebras, and Representations
|
Definition 4.20
|
Definition 4.20. Let \( G \) be a matrix Lie group and let \( {\Pi }_{1} \) and \( {\Pi }_{2} \) be representations of \( G \), acting on spaces \( {V}_{1} \) and \( {V}_{2} \) . Then the tensor product representation of \( G \) , acting on \( {V}_{1} \otimes {V}_{2} \), is defined by
\[
\left( {{\Pi }_{1} \otimes {\Pi }_{2}}\right) \left( A\right) = {\Pi }_{1}\left( A\right) \otimes {\Pi }_{2}\left( A\right)
\]
for all \( A \in G \) . Similarly, if \( {\pi }_{1} \) and \( {\pi }_{2} \) are representations of a Lie algebra \( \mathfrak{g} \), we define a tensor product representation of \( \mathfrak{g} \) on \( {V}_{1} \otimes {V}_{2} \) by
\[
\left( {{\pi }_{1} \otimes {\pi }_{2}}\right) \left( X\right) = {\pi }_{1}\left( X\right) \otimes I + I \otimes {\pi }_{2}\left( X\right) .
\]
It is easy to check that \( {\Pi }_{1} \otimes {\Pi }_{2} \) and \( {\pi }_{1} \otimes {\pi }_{2} \) are actually representations of \( G \) and \( \mathfrak{g} \), respectively. The notation is, unfortunately, ambiguous, since if \( {\Pi }_{1} \) and \( {\Pi }_{2} \) are representations of the same group \( G \), we can regard \( {\Pi }_{1} \otimes {\Pi }_{2} \) either as a representation of \( G \) or as a representation of \( G \times G \) . We must, therefore, be careful to specify which way we are thinking about \( {\Pi }_{1} \otimes {\Pi }_{2} \) .
If \( {\Pi }_{1} \) and \( {\Pi }_{2} \) are irreducible representations of a group \( G \), then \( {\Pi }_{1} \otimes {\Pi }_{2} \) will typically not be irreducible when viewed as a representation of \( G \) . One can, then, attempt to decompose \( {\Pi }_{1} \otimes {\Pi }_{2} \) as a direct sum of irreducible representations. This process is called the Clebsch-Gordan theory or, in the physics literature, "addition of angular momentum." See Exercise 12 and Appendix C for more information about this topic.
## 4.3.3 Dual Representations
Suppose that \( \pi \) is a representation of a Lie algebra \( \mathfrak{g} \) acting on a finite-dimensional vector space \( V \) . Let \( {V}^{ * } \) denote the dual space of \( V \), that is, the space of linear
functionals on \( V \) (Sect. A.7). If \( A \) is a linear operator on \( V \), let \( {A}^{tr} \) denote the dual or transpose operator on \( {V}^{ * } \), given by
\[
\left( {{A}^{tr}\phi }\right) \left( v\right) = \phi \left( {Av}\right)
\]
for \( \phi \in {V}^{ * }, v \in V \) . If \( {v}_{1},\ldots ,{v}_{n} \) is a basis for \( V \), then there is a naturally associated "dual basis" \( {\phi }_{1},\ldots ,{\phi }_{n} \) with the property that \( {\phi }_{j}\left( {v}_{k}\right) = {\delta }_{jk} \) . The matrix for \( {A}^{tr} \) in the dual basis is then simply the transpose (not the conjugate transpose!) of the matrix of \( A \) in the original basis. If \( A \) and \( B \) are linear operators on \( V \), it is easily verified that
\[
{\left( AB\right) }^{tr} = {B}^{tr}{A}^{tr}.
\]
(4.8)
Definition 4.21. Suppose \( G \) is a matrix Lie group and \( \Pi \) is a representation of \( G \) acting on a finite-dimensional vector space \( V \) . Then the dual representation \( {\Pi }^{ * } \) to \( \Pi \) is the representation of \( G \) acting on \( {V}^{ * } \) and given by
\[
{\Pi }^{ * }\left( g\right) = {\left\lbrack \Pi \left( {g}^{-1}\right) \right\rbrack }^{tr}.
\]
(4.9)
If \( \pi \) is a representation of a Lie algebra \( \mathfrak{g} \) acting on a finite-dimensional vector space \( V \), then \( {\pi }^{ * } \) is the representation of \( \mathfrak{g} \) acting on \( {V}^{ * } \) and given by
\[
{\pi }^{ * }\left( X\right) = - \pi {\left( X\right) }^{tr}.
\]
(4.10)
Using (4.8), it is easy to check that both \( {\Pi }^{ * } \) and \( {\pi }^{ * } \) are actually representations. (Here the inverse on the right-hand side of (4.9) and the minus sign on the right-hand side of (4.10) are essential.) The dual representation is also called contragredient representation.
Proposition 4.22. If \( \Pi \) is a representation of a matrix Lie group \( G \), then (1) \( {\Pi }^{ * } \) is irreducible if and only if \( \Pi \) is irreducible and (2) \( {\left( {\Pi }^{ * }\right) }^{ * } \) is isomorphic to \( \Pi \) . Similar statements apply to Lie algebra representations.
Proof. See Exercise 6.
## 4.4 Complete Reducibility
Much of representation theory is concerned with studying irreducible representations of a group or Lie algebra. In favorable cases, knowing the irreducible representations leads to a description of all representations.
Definition 4.23. A finite-dimensional representation of a group or Lie algebra is said to be completely reducible if it is isomorphic to a direct sum of a finite number of irreducible representations.
Definition 4.24. A group or Lie algebra is said to have the complete reducibility property if every finite-dimensional representation of it is completely reducible.
As it turns out, most groups and Lie algebras do not have the complete reducibility property. Nevertheless, many interesting example groups and Lie algebras do have this property, as we will see in this section and Sect. 10.3.
Example 4.25. Let \( \Pi : \mathbb{R} \rightarrow \mathrm{{GL}}\left( {2;\mathbb{C}}\right) \) be given by
\[
\Pi \left( x\right) = \left( \begin{array}{ll} 1 & x \\ 0 & 1 \end{array}\right)
\]
Then \( \Pi \) is not completely reducible.
Proof. Direct calculation shows that \( \Pi \) is, in fact, a representation of \( \mathbb{R} \) . If \( \left\{ {{e}_{1},{e}_{2}}\right\} \) is the standard basis for \( {\mathbb{C}}^{2} \), then clearly the span of \( {e}_{1} \) is an invariant subspace. We now claim that \( \left\langle {e}_{1}\right\rangle \) is the only nontrivial invariant subspace for \( \Pi \) . To see this, suppose \( V \) is a nonzero invariant subspace and suppose \( V \) contains a vector not in the span of \( {e}_{1} \), say, \( v = a{e}_{1} + b{e}_{2} \) with \( b \neq 0 \) . Then
\[
\Pi \left( 1\right) v - v = b{e}_{1}
\]
also belongs to \( V \) . Thus, \( {e}_{1} \) and \( {e}_{2} = \left( {v - a{e}_{1}}\right) /b \) belong to \( V \), showing that \( V = {\mathbb{C}}^{2} \) . We conclude, then, that \( {\mathbb{C}}^{2} \) does not decompose as a direct sum of irreducible invariant subspaces.
Proposition 4.26. If \( V \) is a completely reducible representation of a group or Lie algebra, then the following properties hold.
1. For every invariant subspace \( U \) of \( V \), there is another invariant subspace \( W \) such that \( V \) is the direct sum of \( U \) and \( W \) .
2. Every invariant subspace of \( V \) is completely reducible.
Proof. For Point 1, suppose that \( V \) decomposes as
\[
V = {U}_{1} \oplus {U}_{2} \oplus \cdots \oplus {U}_{k}
\]
where the \( {U}_{j} \) ’s are irreducible invariant subspaces, and that \( U \) is any invariant subspace of \( V \) . If \( U \) is all of \( V \), then we can take \( W = \{ 0\} \) and we are done. If \( U \neq V \), there must be some \( {j}_{1} \) such that \( {U}_{{j}_{1}} \) is not contained in \( U \) . Since \( {U}_{{j}_{1}} \) is irreducible, it follows that the invariant subspace \( {U}_{{j}_{1}} \cap U \) must be \( \{ 0\} \) . Suppose now that \( U + {U}_{{j}_{1}} = V \) . If so, the sum is direct (since \( {U}_{{j}_{1}} \cap U = \{ 0\} \) ) and we are done.
If \( U + {U}_{{j}_{1}} \neq V \), there is some \( {j}_{2} \) such that \( U + {U}_{{j}_{1}} \) does not contain \( {U}_{{j}_{2}} \) , in which case, \( \left( {U + {U}_{{j}_{1}}}\right) \cap {U}_{{j}_{2}} = \{ 0\} \) . Proceeding on in the same way, we must eventually obtain some family \( {j}_{1},{j}_{2},\ldots ,{j}_{l} \) such that \( U + {U}_{{j}_{1}} + \cdots + {U}_{{j}_{l}} = V \) and the sum is direct. Then \( W \mathrel{\text{:=}} {U}_{{j}_{1}} + \cdots + {U}_{{j}_{l}} \) is the desired complement to \( U \) .
For Point 2, suppose \( U \) is an invariant subspace of \( V \) . We first establish that \( U \) has the "invariant complement property" in Point 1. Suppose, then, that \( X \) is another invariant subspace of \( V \) with \( X \subset U \) . By Point 1, we can find invariant subspace \( Y \) such that \( V = X \oplus Y \) . Let \( Z = Y \cap U \), which is then an invariant subspace. We want to show that \( U = X \oplus Z \) . For all \( u \in U \), we can write \( u = x + y \) with \( x \in X \) and \( y \in Y \) . But since \( X \subset U \), we have \( x \in U \) and therefore \( y = u - x \in U \) . Thus, \( y \in Z = Y \cap U \) . We have shown, then, that every \( u \in U \) can be written as the sum of an element of \( X \) and an element of \( Z \) . Furthermore, \( X \cap Z \subset X \cap Y = \{ 0\} \), so actually \( U \) is the direct sum of \( X \) and \( Z \) .
We may now easily show that \( U \) is completely reducible. If \( U \) is irreducible, we are done. If not, \( U \) has a nontrivial invariant subspace \( X \) and thus \( U \) decomposes as \( U = X \oplus Z \) for some invariant subspace \( Z \) . If \( X \) and \( Z \) are irreducible, we are done, and if not, we proceed on in the same way. Since \( U \) is finite dimensional, this process must eventually terminate with \( U \) being decomposed as a direct sum of irreducibles.
Proposition 4.27. If \( G \) is a matrix Lie group and \( \Pi \) is a finite-dimensional unitary representation of \( G \), then \( \Pi \) is completely reducible. Similarly, if \( \mathfrak{g} \) is a real Lie algebra and \( \pi \) is a finite-dimensional "unitary"representation of \( \mathfrak{g} \) (meaning that \( \pi {\left( X\right) }^{ * } = - \pi \left( X\right) \) for all \( X \in \mathfrak{g} \) ), then \( \pi \) is completely reducible.
Proof. Let \( V \) denote the Hilbert space on which \( \Pi \) acts and let \( \langle \cdot , \cdot \rangle \) denote the inner product on \( V \) . If \( W \subset V \) is an invariant subspace, let \( {W}^{ \bot } \) be the orthogonal complement of \( W \), so that \( V \) is the direct sum of \( W \) and \( {W}^{ \bot } \) . We claim t
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1068_(GTM227)Combinatorial Commutative Algebra
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Definition 7.17
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Definition 7.17 Let \( C \) be a rational pointed cone in \( {\mathbb{R}}^{d} \) . The pointed semigroup \( Q = C \cap {\mathbb{Z}}^{d} \) has a unique minimal generating set, called the Hilbert basis of the cone \( C \) and denoted by \( {\mathcal{H}}_{C} \) or \( {\mathcal{H}}_{Q} \), afforded by Theorem 7.16 and Proposition 7.15. More generally, a finite subset of \( {\mathbb{Z}}^{d} \) is a Hilbert basis if it coincides with the Hilbert basis of the cone it generates in \( {\mathbb{R}}^{d} \) .
Example 7.18 Let \( C \) be the cone in \( {\mathbb{R}}^{4} \) consisting of all vectors such that the sum of any two distinct coordinates is nonnegative. This is the cone over a 3-dimensional cube. The Hilbert basis \( {\mathcal{H}}_{C} \) equals \( \{ \left( {1,0,0,0}\right) ,\left( {0,1,0,0}\right) \) , \( \left( {0,0,1,0}\right) ,\left( {0,0,0,1}\right) ,\left( {-1,1,1,1}\right) ,\left( {1, - 1,1,1}\right) ,\left( {1,1, - 1,1}\right) ,\left( {1,1,1, - 1}\right) \} \)
It is instructive to examine the Hilbert basis of a cone in the plane.
Example 7.19 (Two-dimensional Hilbert bases) Let \( C \) be a rational pointed cone in \( {\mathbb{R}}^{2} \) . The Hilbert basis \( {\mathcal{H}}_{C} \) is constructed geometrically as follows. Let \( {\mathcal{P}}_{C} \) denote the unbounded polygon in \( {\mathbb{R}}^{2} \) obtained by taking the convex hull of all nonzero integer points in \( C \) . The polygon \( {\mathcal{P}}_{C} \) has two unbounded edges and a finite number of bounded edges. The Hilbert basis \( {\mathcal{H}}_{C} \) is the set of all lattice points that lie on the bounded edges of \( {\mathcal{P}}_{C} \) . We order the elements \( {\mathbf{a}}_{1},{\mathbf{a}}_{2},\ldots ,{\mathbf{a}}_{n} \) of \( {\mathcal{H}}_{C} \) in counterclockwise order. Then \( {\mathbf{a}}_{1} \) and \( {\mathbf{a}}_{n} \) are the primitive lattice points on the boundary of \( C \), and we have
\[
\det \left( {{\mathbf{a}}_{i},{\mathbf{a}}_{i + 1}}\right) = 1\;\text{ for }i = 1,\ldots, n - 1
\]
because the triangle with vertices \( \left\{ {\mathbf{0},{\mathbf{a}}_{i},{\mathbf{a}}_{i + 1}}\right\} \) has no other lattice points in it (Exercise 7.11). It follows that there exists \( {\lambda }_{i} \in \mathbb{N} \) with
\[
{\lambda }_{i} \cdot {\mathbf{a}}_{i} = {\mathbf{a}}_{i - 1} + {\mathbf{a}}_{i + 1}\;\text{ for }i = 2,3,\ldots, n - 1,
\]
(7.5)
which gives rise to the following binomials in the associated lattice ideal:
\[
{x}_{i - 1}{x}_{i + 1} - {x}_{i}^{{\lambda }_{i}} \in {I}_{L}\;\text{ for }i = 2,3,\ldots, n - 1.
\]
(7.6)
We will return to this ideal in the next section.
We next describe an algorithm for computing the Hilbert basis of a rational pointed cone \( C \), which we assume has \( m \) facets ( \( = \) maximal faces). As a first step, we embed \( C \) as the intersection of a linear subspace \( V \) with a positive integer orthant \( {\mathbb{N}}^{m} \) .
Proposition 7.20 Assume \( C \subset {\mathbb{R}}^{d} \) is a pointed cone, and let \( {\nu }_{1},\ldots ,{\nu }_{m} \) be the primitive integer inner normals to the facets of \( C \) . Define the map \( \nu : {\mathbb{R}}^{d} \rightarrow {\mathbb{R}}^{m} \) sending \( \mathbf{a} \in {\mathbb{R}}^{d} \) to \( \left( {{\nu }_{1} \cdot \mathbf{a},\ldots ,{\nu }_{m} \cdot \mathbf{a}}\right) \), and set \( V = \nu \left( {\mathbb{R}}^{d}\right) \) . Then \( \nu \) is injective, and its restriction to \( C \) is an isomorphism to \( \nu \left( C\right) = {\mathbb{R}}_{ \geq 0}^{m} \cap V \) .
Proof. The map \( \nu \) is injective precisely because \( C \) is pointed: the intersection of the kernels of \( {\nu }_{1},\ldots ,{\nu }_{m} \) is by definition the lineality space of \( C \) , which is zero for pointed cones. Moreover, a point \( \mathbf{a} \in {\mathbb{R}}^{d} \) lies in \( C \) if and only if all \( {\nu }_{i} \cdot \mathbf{a} \) are nonnegative.
We wish to compute the Hilbert basis for the pointed semigroup \( {\mathbb{N}}^{m} \cap V \) . Consider the sublattice \( \Lambda = \left\{ {\left( {\mathbf{v}, - \mathbf{v}}\right) \mid \mathbf{v} \in V \cap {\mathbb{Z}}^{m}}\right\} \) of \( {\mathbb{Z}}^{2m} \) . The lattice ideal \( {I}_{\Lambda } \) is an ideal in \( \mathbb{k}\left\lbrack {\mathbf{x},\mathbf{y}}\right\rbrack = \mathbb{k}\left\lbrack {{x}_{1},\ldots ,{x}_{m},{y}_{1},\ldots ,{y}_{m}}\right\rbrack \) . Such ideals are called Lawrence ideals. We can compute a minimal generating set of this ideal using Lemma 7.6. By Theorem 7.21, this solves our problem.
Theorem 7.21 A vector \( \mathbf{a} \in {\mathbb{Z}}^{d} \) lies in the Hilbert basis \( {\mathcal{H}}_{C} \) if and only if the binomial \( {\mathbf{x}}^{\nu \cdot \mathbf{a}} - {\mathbf{y}}^{\nu \cdot \mathbf{a}} \) appears among the minimal generators of \( {I}_{\Lambda } \) .
Proof. We will equivalently prove that \( \mathbf{u} \in {\mathcal{H}}_{\nu \left( C\right) } \) if and only if \( {\mathbf{x}}^{\mathbf{u}} - {\mathbf{y}}^{\mathbf{u}} \) appears among the minimal generators of \( {I}_{\Lambda } \) . Consider a nonzero vector \( \mathbf{u} \) in \( {\mathbb{N}}^{m} \cap V \) . If \( \mathbf{u} \) is not in \( {\mathcal{H}}_{\nu \left( C\right) } \), then we can write \( \mathbf{u} = {\mathbf{u}}_{1} + {\mathbf{u}}_{2} \) for two nonzero vectors \( {\mathbf{u}}_{1} \) and \( {\mathbf{u}}_{2} \) in \( {\mathbb{N}}^{m} \cap V \) . The identity
\[
{\mathbf{x}}^{\mathbf{u}} - {\mathbf{y}}^{\mathbf{u}} = {\mathbf{x}}^{{\mathbf{u}}_{1}} \cdot \left( {{\mathbf{x}}^{{\mathbf{u}}_{2}} - {\mathbf{y}}^{{\mathbf{u}}_{2}}}\right) + {\mathbf{y}}^{{\mathbf{u}}_{2}} \cdot \left( {{\mathbf{x}}^{{\mathbf{u}}_{1}} - {\mathbf{y}}^{{\mathbf{u}}_{1}}}\right)
\]
shows that \( {\mathbf{x}}^{\mathbf{u}} - {\mathbf{y}}^{\mathbf{u}} \) is not a minimal generator of \( {I}_{\Lambda } \) .
For the converse, suppose \( {\mathbf{x}}^{\mathbf{u}} - {\mathbf{y}}^{\mathbf{u}} \) is not a minimal generator of \( {I}_{\Lambda } \) . Then \( {\mathbf{x}}^{\mathbf{u}} - {\mathbf{y}}^{\mathbf{u}} \) is a (nonconstant) monomial linear combination of some binomials \( {\mathbf{x}}^{\mathbf{v}}{\mathbf{y}}^{\mathbf{w}} - {\mathbf{x}}^{\mathbf{w}}{\mathbf{y}}^{\mathbf{v}} \) with \( \mathbf{v},\mathbf{w} \in {\mathbb{N}}^{m} \) and \( \mathbf{v} - \mathbf{w} \in V \) . We may assume that all terms have the same degree. By setting all \( {x}_{i} \) equal to zero, we see that at least one appearing binomial satisfies \( \mathbf{v} = \mathbf{0} \) or \( \mathbf{w} = \mathbf{0} \) . Suppose \( \mathbf{w} = \mathbf{0} \) . Then \( {\mathbf{x}}^{\mathbf{v}} \) properly divides \( {\mathbf{x}}^{\mathbf{u}} \), so \( \mathbf{u} \) is not in the Hilbert basis \( {\mathcal{H}}_{\nu \left( C\right) } \) .
Example 7.22 Let us find all nonnegative integer solutions to the equation
\[
2{u}_{1} + 7{u}_{2} = 3{u}_{3} + 5{u}_{4}
\]
(7.7)
The lattice of all integer solutions to this equation has the basis
\[
\left( {-1,0,1, - 1}\right) ,\left( {-1,1,0,1}\right) ,\left( {2,1,2,1}\right) \text{.}
\]
Using this basis we express the corresponding Lawrence ideal \( {I}_{\Lambda } \) as follows:
\[
\left( {\left\langle {{x}_{1}{x}_{4}{y}_{3} - {x}_{3}{y}_{1}{y}_{4},{x}_{2}{x}_{4}{y}_{1} - {x}_{1}{y}_{2}{y}_{4},{x}_{1}^{2}{x}_{2}{x}_{3}^{2}{x}_{4} - {y}_{1}^{2}{y}_{2}{y}_{3}^{2}{y}_{4}}\right\rangle : {\left\langle {x}_{1}{x}_{2}\cdots {y}_{4}\right\rangle }^{\infty }}\right) .
\]
This ideal has 30 minimal generators. Eighteen of the generators have the form required in Theorem 7.21 and hence give elements in the Hilbert basis. For example, the generator \( {x}_{2}^{4}{x}_{3}{x}_{4}^{5} - {y}_{2}^{4}{y}_{3}{y}_{4}^{5} \) of \( {I}_{\Lambda } \) gives \( \left( {0,4,1,5}\right) \in {\mathcal{H}}_{C} \) .
We find that the cone \( C \) of nonnegative solutions to (7.7) has Hilbert basis
\[
\left( {0,2,3,1}\right) ,\left( {0,3,2,3}\right) ,\underline{\left( 0,3,7,0\right) },\left( {0,4,1,5}\right) ,\underline{\left( 0,5,0,7\right) },\left( {1,1,3,0}\right) \text{,}
\]
\[
\left( {1,2,2,2}\right) ,\left( {1,3,1,4}\right) ,\left( {1,4,0,6}\right) ,\left( {2,1,2,1}\right) ,\left( {2,2,1,3}\right) ,\left( {2,3,0,5}\right) \text{,}
\]
\[
\left( {3,0,2,0}\right) ,\left( {3,1,1,2}\right) ,\left( {3,2,0,4}\right) ,\left( {4,0,1,1}\right) ,\left( {4,1,0,3}\right) ,\left( {5,0,0,2}\right) \text{.}
\]
These 18 vectors minimally generate the semigroup of solutions to (7.7). The underlined vectors will be explained in Example 7.26.
Proposition 7.20 has another useful consequence.
Corollary 7.23 Every d-dimensional pointed affine semigroup can be embedded inside \( {\mathbb{N}}^{d} \) .
Proof. Given a pointed cone \( C \), define \( V \subseteq {\mathbb{R}}^{m} \) as in Proposition 7.20. Choose \( m - d \) standard basis vectors \( {\mathbf{e}}_{{i}_{1}},\ldots ,{\mathbf{e}}_{{i}_{m - d}} \) so that their images modulo \( V \) form a basis for \( {\mathbb{R}}^{m}/V \) . Then the coordinate subspace \( E \) spanned by \( {\mathbf{e}}_{{i}_{1}},\ldots ,{\mathbf{e}}_{{i}_{m - d}} \) intersects \( V \) trivially. Under the projection \( {\mathbb{R}}^{m} \rightarrow {\mathbb{R}}^{d} \) with kernel \( E \), the subspace \( V \) maps isomorphically to its image, and \( {\mathbb{N}}^{m} \) maps to \( {\mathbb{N}}^{d} \) . Therefore, projection modulo \( E \) takes any subsemigroup of \( C \) isomorphically to its image in \( {\mathbb{N}}^{d} \) .
If \( Q \) is an arbitrary affine semigroup in \( {\mathbb{Z}}^{d} \), then \( {\mathbb{R}}_{ \geq 0}Q \) is the smallest cone in \( {\mathbb{R}}^{d} \) containing \( Q \) . Similarly, there is a smallest subgroup of \( {\mathbb{Z}}^{d} \) containing \( Q \) . Intersecting these yields an affine semigroup closely related to \( Q \) .
Definition 7.24 If \( A \) is the subgroup of \( {\mathbb{Z}}^{d} \) generated by an affine semigroup \( Q \) inside of \( {\mathbb{Z}}^{d} \), then the semigroup \( {Q}_{\text{sat }} = \left( {{\mathbb{R}}_{ \geq 0}Q}\right) \cap A \) is called the saturation of the
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1079_(GTM237)An Introduction to Operators on the Hardy-Hilbert Space
|
Definition 2.2.9
|
Definition 2.2.9. A function \( \phi \in {\mathbf{H}}^{\infty } \) satisfying \( \left| {\widetilde{\phi }\left( {e}^{i\theta }\right) }\right| = 1 \) a.e. is an inner function.
Theorem 2.2.10. If \( \phi \) is a nonconstant inner function, then \( \left| {\phi \left( z\right) }\right| < 1 \) for all \( z \in \mathbb{D} \) .
Proof. This follows immediately from Corollary 1.1.24 and Theorem 1.1.17.
The definition of inner functions requires that the functions be in \( {\mathbf{H}}^{\infty } \) . It is often useful to know that this follows if a function is in \( {\mathbf{H}}^{2} \) and has boundary values of modulus 1 a.e.
Theorem 2.2.11. Let \( \phi \in {\mathbf{H}}^{2} \) . If \( \left| {\widetilde{\phi }\left( {e}^{i\theta }\right) }\right| = 1 \) a.e., then \( \phi \) is an inner function.
Proof. It only needs to be shown that \( \phi \in {\mathbf{H}}^{\infty } \) ; this follows from Corollary 1.1.24.
Corollary 2.2.12 (Beurling's Theorem). Every invariant subspace of the unilateral shift other than \( \{ 0\} \) has the form \( \phi {\mathbf{H}}^{2} \), where \( \phi \) is an inner function.
Proof. The unilateral shift is the restriction of multiplication by \( {e}^{i\theta } \) to \( {\widetilde{\mathbf{H}}}^{2} \), so if \( \mathcal{M} \) is an invariant subspace of the unilateral shift, it is an invariant subspace of the bilateral shift contained in \( {\widetilde{\mathbf{H}}}^{2} \) . Thus, by Theorem 2.2.7, \( \mathcal{M} = \phi {\widetilde{\mathbf{H}}}^{2} \) for some measurable function satisfying \( \left| {\phi \left( {e}^{i\theta }\right) }\right| = 1 \) a.e. (Note that \( \{ 0\} \) is the only reducing subspace of the bilateral shift that is contained in \( {\widetilde{\mathbf{H}}}^{2} \) .) Since \( 1 \in {\widetilde{\mathbf{H}}}^{2},\phi \in {\widetilde{\mathbf{H}}}^{2} \) .
Translating this situation back to \( {\mathbf{H}}^{2} \) on the disk gives \( \mathcal{M} = \phi {\mathbf{H}}^{2} \) with \( \phi \) inner, by Theorem 2.2.11.
Corollary 2.2.13. Every invariant subspace of the unilateral shift is cyclic. (See Definition 1.2.17.)
Proof. If \( \mathcal{M} \) is an invariant subspace of the unilateral shift, it has the form \( \phi {\mathbf{H}}^{2} \) by Beurling’s theorem (Corollary 2.2.12). For each \( n,{U}^{n}\phi = {z}^{n}\phi \), so \( \mathop{\bigvee }\limits_{{n = 0}}^{\infty }\left\{ {{U}^{n}\phi }\right\} \) contains all functions of the form \( \phi \left( z\right) p\left( z\right) \), where \( p \) is a polynomial. Since the polynomials are dense in \( {\mathbf{H}}^{2} \) (as the finite sequences are dense in \( \left. {\ell }^{2}\right) \), it follows that \( \mathop{\bigvee }\limits_{{n = 0}}^{\infty }\left\{ {{U}^{n}\phi }\right\} = \phi {\mathbf{H}}^{2} \) .
## 2.3 Inner and Outer Functions
We shall see that every function in \( {\mathbf{H}}^{2} \), other than the constant function 0, can be written as a product of an inner function and a cyclic vector for the unilateral shift. Such cyclic vectors will be shown to have a special form.
Definition 2.3.1. The function \( F \in {\mathbf{H}}^{2} \) is an outer function if \( F \) is a cyclic vector for the unilateral shift. That is, \( F \) is an outer function if
\[
\mathop{\bigvee }\limits_{{k = 0}}^{\infty }\left\{ {{U}^{k}F}\right\} = {\mathbf{H}}^{2}
\]
Theorem 2.3.2. If \( F \) is an outer function, then \( F \) has no zeros in \( \mathbb{D} \) .
Proof. If \( F\left( {z}_{0}\right) = 0 \), then \( \left( {{U}^{n}F}\right) \left( {z}_{0}\right) = {z}_{0}^{n}F\left( {z}_{0}\right) = 0 \) for all \( n \) . Since the limit of a sequence of functions in \( {\mathbf{H}}^{2} \) that all vanish at \( {z}_{0} \) must also vanish at \( {z}_{0} \) (Theorem 1.1.9),
\[
\mathop{\bigvee }\limits_{{k = 0}}^{\infty }\left\{ {{U}^{k}F}\right\}
\]
cannot be all of \( {\mathbf{H}}^{2} \) . Hence there is no \( {z}_{0} \in \mathbb{D} \) with \( F\left( {z}_{0}\right) = 0 \) .
Recall that a function analytic on \( \mathbb{D} \) is identically zero if it vanishes on a set that has a limit point in \( \mathbb{D} \) . The next theorem is an analogous result for boundary values of functions in \( {\mathbf{H}}^{2} \) .
Theorem 2.3.3 (The F. and M. Riesz Theorem). If \( f \in {\mathbf{H}}^{2} \) and the set
\[
\left\{ {{e}^{i\theta } : \widetilde{f}\left( {e}^{i\theta }\right) = 0}\right\}
\]
has positive measure, then \( f \) is identically 0 on \( \mathbb{D} \) .
Proof. Let \( E = \left\{ {{e}^{i\theta } : \widetilde{f}\left( {e}^{i\theta }\right) = 0}\right\} \) and let
\[
\mathcal{M} = \mathop{\bigvee }\limits_{{k = 0}}^{\infty }\left\{ {{U}^{k}\widetilde{f}}\right\} = \mathop{\bigvee }\limits_{{k = 0}}^{\infty }\left\{ {{e}^{ik\theta }\widetilde{f}}\right\}
\]
Then every function \( \widetilde{g} \in \mathcal{M} \) vanishes on \( E \), since all functions \( {e}^{ik\theta }\widetilde{f} \) do. If \( \widetilde{f} \) is not identically zero, it follows from Beurling's theorem (Theorem 2.2.12) that \( \mathcal{M} = \widetilde{\phi }{\widetilde{\mathbf{H}}}^{2} \) for some inner function \( \phi \) . In particular, this implies that \( \widetilde{\phi } \in \mathcal{M} \) , so \( \widetilde{\phi } \) vanishes on \( E \) . But \( \left| {\widetilde{\phi }\left( {e}^{i\theta }\right) }\right| = 1 \) a.e. This contradicts the hypothesis that \( E \) has positive measure, thus \( \widetilde{f} \), and hence \( f \), must be identically zero.
Another beautiful result that follows from Beurling's theorem is the following factorization of functions in \( {\mathbf{H}}^{2} \) .
Theorem 2.3.4. If \( f \) is a function in \( {\mathbf{H}}^{2} \) that is not identically zero, then \( f = {\phi F} \), where \( \phi \) is an inner function and \( F \) is an outer function. This factorization is unique up to constant factors.
Proof. Let \( f \in {\mathbf{H}}^{2} \) and consider \( \mathop{\bigvee }\limits_{{n = 0}}^{\infty }\left\{ {{U}^{n}f}\right\} \) . If this span is \( {\mathbf{H}}^{2} \), then \( f \) is outer by definition, and we can take \( \phi \) to be the constant function 1 and \( F = f \) to obtain the desired conclusion.
If \( \mathop{\bigvee }\limits_{{n = 0}}^{\infty }\left\{ {{U}^{n}f}\right\} \neq {\mathbf{H}}^{2} \), then, by Beurling’s theorem (Corollary 2.2.12), there must exist a nonconstant inner function \( \phi \) with \( \mathop{\bigvee }\limits_{{n = 0}}^{\infty }\left\{ {{U}^{n}f}\right\} = \phi {\mathbf{H}}^{2} \) . Since \( f \) is in \( \mathop{\bigvee }\limits_{{n = 0}}^{\infty }\left\{ {{U}^{n}f}\right\} = \phi {\mathbf{H}}^{2} \), there exists a function \( F \) in \( {\mathbf{H}}^{2} \) with \( f = {\phi F} \) . We shall show that \( F \) is outer.
The invariant subspace \( \mathop{\bigvee }\limits_{{n = 0}}^{\infty }\left\{ {{U}^{n}F}\right\} \) equals \( \psi {\mathbf{H}}^{2} \) for some inner function \( \psi \) . Then, since \( f = {\phi F} \), it follows that \( {U}^{n}f = {U}^{n}\left( {\phi F}\right) = \phi {U}^{n}F \) for every positive integer \( n \), from which we can conclude, by taking linear spans, that \( \phi {\mathbf{H}}^{2} = {\phi \psi }{\mathbf{H}}^{2} \) . Theorem 2.2.8 now implies that \( \phi \) and \( {\phi \psi } \) are constant multiples of each other. Hence \( \psi \) must be a constant function. Therefore \( \mathop{\bigvee }\limits_{{n = 0}}^{\infty }\left\{ {{U}^{n}F}\right\} = {\mathbf{H}}^{2} \), so \( F \) is an outer function.
Note that if \( f = {\phi F} \) with \( \phi \) inner and \( F \) outer, then \( \mathop{\bigvee }\limits_{{n = 0}}^{\infty }\left\{ {{U}^{n}f}\right\} = \phi {\mathbf{H}}^{2} \) . Thus uniqueness of the factorization follows from the corresponding assertion in Theorem 2.2.8.
Definition 2.3.5. For \( f \in {\mathbf{H}}^{2} \), if \( f = {\phi F} \) with \( \phi \) inner and \( F \) outer, we call \( \phi \) the inner part of \( f \) and \( F \) the outer part of \( f \) .
Theorem 2.3.6. The zeros of an \( {\mathbf{H}}^{2} \) function are precisely the zeros of its inner part.
Proof. This follows immediately from Theorem 2.3.2 and Theorem 2.3.4.
To understand the structure of Lat \( U \) as a lattice requires being able to determine when \( {\phi }_{1}{\mathbf{H}}^{2} \) is contained in \( {\phi }_{2}{\mathbf{H}}^{2} \) for inner functions \( {\phi }_{1} \) and \( {\phi }_{2} \) . This will be accomplished by analysis of a factorization of inner functions.
## 2.4 Blaschke Products
Some of the invariant subspaces of the unilateral shift are those consisting of the functions vanishing at certain subsets of \( \mathbb{D} \) . The simplest such subspaces are those of the form, for \( {z}_{0} \in \mathbb{D} \) ,
\[
{\mathcal{M}}_{{z}_{0}} = \left\{ {f \in {\mathbf{H}}^{2} : f\left( {z}_{0}\right) = 0}\right\} .
\]
The subspace \( {\mathcal{M}}_{{z}_{0}} \) is an invariant subspace for \( U \) . Therefore Beurling’s theorem (Corollary 2.2.12) implies that there is an inner function \( \psi \) such that \( {\mathcal{M}}_{{z}_{0}} = \psi {\mathbf{H}}^{2} \)
Theorem 2.4.1. For each \( {z}_{0} \in \mathbb{D} \), the function
\[
\psi \left( z\right) = \frac{{z}_{0} - z}{1 - \overline{{z}_{0}}z}
\]
is an inner function and \( {\mathcal{M}}_{{z}_{0}} = \left\{ {f \in {\mathbf{H}}^{2} : f\left( {z}_{0}\right) = 0}\right\} = \psi {\mathbf{H}}^{2} \) .
Proof. The function \( \psi \) is clearly in \( {\mathbf{H}}^{\infty } \) . Moreover, it is continuous on the closure of \( \mathbb{D} \) . Therefore, to show that \( \psi \) is inner, it suffices to show that \( \left| {\psi \left( z\right) }\right| = 1 \) when \( \left| z\right| = 1 \) . For this, note that \( \left| z\right| = 1 \) implies \( z\bar{z} = 1 \), so that
\[
\left| \frac{{z}_{0} - z}{1 - \overline{{z}_{0}}z}\right| = \left| \frac{{z}_{0} - z}{z\left( {\bar{z} - \overline{{z}_{0}}}\right) }\right| = \frac{1}{\left| z\right| }\left| \frac{{z}_{0} - z}{\bar{z} - \overline{{z}_{0}}}\right| = 1.
\]
To show that \( {\mathcal{M}}_{{z}_{0}} = \psi {\mathbf{H}}^{2} \), first note that \( \psi
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1057_(GTM217)Model Theory
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Definition 3.1.1
|
Definition 3.1.1 We say that a theory \( T \) has quantifier elimination if for every formula \( \phi \) there is a quantifier-free formula \( \psi \) such that
\[
T \vDash \phi \leftrightarrow \psi
\]
We will start by showing that DLO, the theory of dense linear orders without endpoints, has quantifier elimination. We need a slight variant of the proof of Theorem 2.4.1.
Lemma 3.1.2 Let \( \left( {A, < }\right) \) and \( \left( {B, < }\right) \) be countable dense linear orders, \( {a}_{1},\ldots ,{a}_{n} \in A,{b}_{1},\ldots ,{b}_{n} \in B \), such that \( {a}_{1} < \ldots < {a}_{n} \) and \( {b}_{1} < \ldots < {b}_{n} \) . Then there is an isomorphism \( f : A \rightarrow B \) such that \( f\left( {a}_{i}\right) = {b}_{i} \) for \( i = 1,\ldots, n \) .
Proof Modify the proof of Theorem 2.4.1 starting with \( {A}_{0} = \left\{ {{a}_{1},\ldots ,{a}_{n}}\right\} \) , \( {B}_{0} = \left\{ {{b}_{1},\ldots ,{b}_{n}}\right\} \), and the partial isomorphism \( {f}_{0} : {A}_{0} \rightarrow {B}_{0} \), where \( {f}_{0}\left( {a}_{i}\right) = {b}_{i} \) . The rest of the proof works, and we build \( f : A \rightarrow B \), an isomorphism extending \( {f}_{0} \) .
Theorem 3.1.3 DLO has quantifier elimination.
Proof First, suppose that \( \phi \) is a sentence. If \( \mathbb{Q} \vDash \phi \), then because DLO is complete, DLO \( = \phi \) and
\[
\mathrm{{DLO}} \vDash \phi \leftrightarrow {x}_{1} = {x}_{1}
\]
whereas if \( \mathbb{Q} \vDash \neg \phi \) ,
\[
\mathrm{{DLO}} \vDash \phi \leftrightarrow {x}_{1} \neq {x}_{1}
\]
Next, suppose that \( \phi \) is a formula with free variables \( {x}_{1},\ldots ,{x}_{n} \), where \( n \geq 1 \) . We will show that there is a quantifier-free formula \( \psi \) with free variables from among \( {x}_{1},\ldots ,{x}_{n} \) such that
\[
\mathbb{Q} \vDash \forall \bar{x}\left( {\phi \left( \bar{x}\right) \leftrightarrow \psi \left( \bar{x}\right) }\right)
\]
Because DLO is complete,
\[
\mathrm{{DLO}} \vDash \forall \bar{x}\left( {\phi \left( \bar{x}\right) \leftrightarrow \psi \left( \bar{x}\right) }\right)
\]
so this will suffice.
For \( \sigma : \{ \left( {i, j}\right) : 1 \leq i < j \leq n\} \rightarrow 3 \), let \( {\chi }_{\sigma }\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) be the formula
\[
\mathop{\bigwedge }\limits_{{\sigma \left( {i, j}\right) = 0}}{x}_{i} = {x}_{j} \land \mathop{\bigwedge }\limits_{{\sigma \left( {i, j}\right) = 1}}{x}_{i} < {x}_{j} \land \mathop{\bigwedge }\limits_{{\sigma \left( {i, j}\right) = 2}}{x}_{i} > {x}_{j}.
\]
We call \( {\chi }_{\sigma } \) a sign condition. Each sign condition describes a (possibly inconsistent) arrangement of \( n \) elements in an ordered set.
Let \( \mathcal{L} \) be the language of linear orders and \( \phi \) be an \( \mathcal{L} \) -formula with \( n \geq 1 \) free variables. Let \( {\Lambda }_{\phi } \) be the set of sign conditions \( \sigma : \{ \left( {i, j}\right) : 1 \leq i < j \leq \) \( n\} \rightarrow 3 \) such that there is \( \bar{a} \in \mathbb{Q} \) such that \( \mathbb{Q} \vDash {\chi }_{\sigma }\left( \bar{a}\right) \land \phi \left( \bar{a}\right) \) . There are two cases to consider.
case 1: \( {\Lambda }_{\phi } = \varnothing \) .
Then \( \mathbb{Q} \vDash \forall \bar{x}\neg \phi \left( \bar{x}\right) \) and \( \mathbb{Q} \vDash \phi \left( \bar{x}\right) \leftrightarrow {x}_{1} \neq {x}_{1} \) .
\( \frac{\text{ case }2 : {\Lambda }_{\phi } \neq \varnothing \text{. }}{\text{ Let }} \)
\[
{\psi }_{\phi }\left( \bar{x}\right) = \mathop{\bigvee }\limits_{{\sigma \in {\Lambda }_{\phi }}}{\chi }_{\sigma }\left( \bar{x}\right)
\]
By choice of \( {\Lambda }_{\phi } \) ,
\[
\mathbb{Q} \vDash \phi \left( \bar{x}\right) \rightarrow {\psi }_{\phi }\left( \bar{x}\right)
\]
On the other hand, suppose that \( \bar{b} \in \mathbb{Q} \) and \( \mathbb{Q} \vDash {\psi }_{\phi }\left( \bar{b}\right) \) . Let \( \sigma \in {\Lambda }_{\phi } \) such that \( \mathbb{Q} \vDash {\chi }_{\sigma }\left( \bar{b}\right) \) . There is \( \bar{a} \in \mathbb{Q} \) such that \( \mathbb{Q} \vDash \phi \left( \bar{a}\right) \land {\chi }_{\sigma }\left( \bar{a}\right) \) . By Theorem 2.4.1, there is \( f \), an automorphism of \( \left( {\mathbb{Q}, < }\right) \), such that \( f\left( \bar{a}\right) = \bar{b} \) . By Theorem 1.1.10, \( \mathbb{Q} \vDash \phi \left( \bar{b}\right) \) . Thus \( \phi \left( \bar{b}\right) \leftrightarrow {\psi }_{\phi }\left( \bar{b}\right) \) .
Note that there is a slight anomaly here. If \( \phi \) is not a sentence, then we can find an equivalent quantifier-free sentence using the same variables. Because there are no quantifier-free \( \mathcal{L} \) -sentences, to find a quantifier-free formula equivalent to a sentence, we must introduce a new free variable. If our language has constant symbols, this is unnecessary.
DLO is an example where we can give a direct explicit proof of quantifier elimination. In the exercises, we will look at several more simple examples where there is an easy explicit elimination of quantifiers. For more complicated theories explicit proofs of quantifier elimination are often quite difficult. Next we will give a useful model-theoretic criterion for quantifier elimination.
Theorem 3.1.4 Suppose that \( \mathcal{L} \) contains a constant symbol \( c, T \) is an \( \mathcal{L} \) -theory, and \( \phi \left( \bar{v}\right) \) is an \( \mathcal{L} \) -formula. The following are equivalent:
i) There is a quantifier-free \( \mathcal{L} \) -formula \( \psi \left( \bar{v}\right) \) such that \( T \vDash \forall \bar{v}(\phi \left( \bar{v}\right) \leftrightarrow \) \( \psi \left( \bar{v}\right) ) \) .
ii) If \( \mathcal{M} \) and \( \mathcal{N} \) are models of \( T,\mathcal{A} \) is an \( \mathcal{L} \) -structure, \( \mathcal{A} \subseteq \mathcal{M} \), and \( \mathcal{A} \subseteq \mathcal{N} \), then \( \mathcal{M} \vDash \phi \left( \bar{a}\right) \) if and only if \( \mathcal{N} \vDash \phi \left( \bar{a}\right) \) for all \( \bar{a} \in \mathcal{A} \) .
Proof i) \( \Rightarrow \) ii) Suppose that \( T \vDash \forall \bar{v}\left( {\phi \left( \bar{v}\right) \leftrightarrow \psi \left( \bar{v}\right) }\right) \), where \( \psi \) is quantifier-free. Let \( \bar{a} \in \mathcal{A} \), where \( \mathcal{A} \) is a common substructure of \( \mathcal{M} \) and \( \mathcal{N} \) and the latter two structures are models of \( T \) . In Proposition 1.1.8, we saw that quantifier-free formulas are preserved under substructure and extension. Thus
\[
\mathcal{M} \vDash \phi \left( \bar{a}\right) \; \Leftrightarrow \;\mathcal{M} \vDash \psi \left( \bar{a}\right)
\]
\[
\Leftrightarrow \mathcal{A} \vDash \psi \left( \bar{a}\right) \text{ (because }\mathcal{A} \subseteq \mathcal{M}\text{ ) }
\]
\[
\Leftrightarrow \mathcal{N} \vDash \psi \left( \bar{a}\right) \text{ (because }\mathcal{A} \subseteq \mathcal{N}\text{ ) }
\]
\[
\Leftrightarrow \mathcal{N} \vDash \phi \left( \bar{a}\right) .
\]
ii) \( \Rightarrow \) i) First, if \( T \vDash \forall \bar{v}\phi \left( \bar{v}\right) \), then \( T \vDash \forall \bar{v}\left( {\phi \left( \bar{v}\right) \leftrightarrow c = c}\right) \) . Second, if \( T \vDash \forall \bar{v}\neg \phi \left( \bar{v}\right) \), then \( T \vDash \forall \bar{v}\left( {\phi \left( \bar{v}\right) \leftrightarrow c \neq c}\right) . \)
Thus, we may assume that both \( T \cup \{ \phi \left( \bar{v}\right) \} \) and \( T \cup \{ \neg \phi \left( \bar{v}\right) \} \) are satisfiable.
Let \( \Gamma \left( \bar{v}\right) = \{ \psi \left( \bar{v}\right) : \psi \) is quantifier-free and \( T \vDash \forall \bar{v}\left( {\phi \left( \bar{v}\right) \rightarrow \psi \left( \bar{v}\right) }\right) \} \) . Let \( {d}_{1},\ldots ,{d}_{m} \) be new constant symbols. We will show that \( T \cup \Gamma \left( \bar{d}\right) \vDash \phi \left( \bar{d}\right) \) . Then, by compactness, there are \( {\psi }_{1},\ldots ,{\psi }_{n} \in \Gamma \) such that
\[
T \vDash \forall \bar{v}\left( {\mathop{\bigwedge }\limits_{{i = 1}}^{n}{\psi }_{i}\left( \bar{v}\right) \rightarrow \phi \left( \bar{v}\right) }\right) .
\]
Thus
\[
T \vDash \forall \bar{v}\left( {\mathop{\bigwedge }\limits_{{i = 1}}^{n}{\psi }_{i}\left( \bar{v}\right) \leftrightarrow \phi \left( \bar{v}\right) }\right)
\]
and \( \mathop{\bigwedge }\limits_{{i = 1}}^{n}{\psi }_{i}\left( \bar{v}\right) \) is quantifier-free. We need only prove the following claim.
Claim \( T \cup \Gamma \left( \bar{d}\right) \vDash \phi \left( \bar{d}\right) \) .
Suppose not. Let \( \mathcal{M} \vDash T \cup \Gamma \left( \bar{d}\right) \cup \{ \neg \phi \left( \bar{d}\right) \} \) . Let \( \mathcal{A} \) be the substructure of \( \mathcal{M} \) generated by \( \bar{d} \) .
Let \( \sum = T \cup \operatorname{Diag}\left( \mathcal{A}\right) \cup \phi \left( \bar{d}\right) \) . If \( \sum \) is unsatisfiable, then there are quantifier-free formulas \( {\psi }_{1}\left( \bar{d}\right) ,\ldots ,{\psi }_{n}\left( \bar{d}\right) \in \operatorname{Diag}\left( \mathcal{A}\right) \) such that
\[
T \vDash \forall \bar{v}\left( {\mathop{\bigwedge }\limits_{{i = 1}}^{n}{\psi }_{i}\left( \bar{v}\right) \rightarrow \neg \phi \left( \bar{v}\right) }\right) .
\]
But then
\[
T \vDash \forall \bar{v}\left( {\phi \left( \bar{v}\right) \rightarrow \mathop{\bigvee }\limits_{{i = 1}}^{n}\neg {\psi }_{i}\left( \bar{v}\right) }\right)
\]
so \( \mathop{\bigvee }\limits_{{i = 1}}^{n}\neg {\psi }_{i}\left( \bar{v}\right) \in \Gamma \) and \( \mathcal{A} \vDash \mathop{\bigvee }\limits_{{i = 1}}^{n}\neg {\psi }_{i}\left( \bar{d}\right) \), a contradiction. Thus, \( \sum \) is satisfi-
able.
Let \( \mathcal{N} \vDash \sum \) . Then \( \mathcal{N} \vDash \phi \left( \bar{d}\right) \) . Because \( \sum \supseteq \operatorname{Diag}\left( \mathcal{A}\right) ,\mathcal{A} \subseteq \mathcal{N} \), by Lemma 2.3.3 i). But \( \mathcal{M} \vDash \neg \phi \left( \bar{d}\right) \) ; thus, by ii), \( \mathcal{N} \vDash \neg \phi \left( \
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1042_(GTM203)The Symmetric Group
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Definition 3.8.4
|
Definition 3.8.4 A partial tableau \( P \) is miniature if \( P \) has exactly three elements. -
The miniature tableaux are used to model the dual Knuth relations of the first and second kinds.
Proposition 3.8.5 Let \( P \) and \( Q \) be distinct miniature tableaux of the same shape \( \lambda /\mu \) and content. Then
\[
P\overset{{K}^{ * }}{ \cong }Q \Leftrightarrow P\overset{ * }{ \cong }Q.
\]
Proof. Without loss of generality, let \( P \) and \( Q \) be standard.
" \( \Rightarrow \) " By induction on the number of slides, it suffices to show the following. Let \( c \) be a cell for a slide on \( P, Q \) and let \( {P}^{\prime },{Q}^{\prime } \) be the resultant tableaux. Then we must have
\[
\operatorname{sh}{P}^{\prime } = \operatorname{sh}{Q}^{\prime }\text{ and }{P}^{\prime }\overset{{K}^{ * }}{ \cong }{Q}^{\prime }.
\]
(3.17)
This is a tedious case-by-case verification. First, we must write down all the skew shapes with 3 cells (up to those translations that do not affect slides so the number of diagrams will be finite). Then we must find the possible tableau pairs for each shape (there will be at most two pairs corresponding to \( \cong \) and \( \cong \) ). Finally, all possible slides must be tried on each pair. We leave the details to the reader. However, we will do one of the cases as an illustration.
Suppose that \( \lambda = \left( {2,1}\right) \) and \( \mu = \varnothing \) . Then the only pair of tableaux of this shape is
\[
P = \begin{array}{ll} 1 & 2 \\ 3 & \end{array}\text{ and }Q = \begin{array}{ll} 1 & 3 \\ 2 & \end{array}
\]
or vice versa, and \( P\overset{{1}^{ * }}{ \cong }Q \) . The results of the three possible slides on \( P, Q \) are given in the following table, from which it is easy to verify that (3.17) holds. 
" \( \Leftarrow \) " Let \( N \) be a normal standard tableau of shape \( \mu \) . So \( {P}^{\prime } = {j}^{N}\left( P\right) \) and \( {Q}^{\prime } = {j}^{N}\left( Q\right) \) are normal miniature tableaux. Now \( P\overset{ * }{ \cong }Q \) implies that \( \operatorname{sh}{P}^{\prime } = \operatorname{sh}{Q}^{\prime } \) . This hypothesis also guarantees that \( {v}^{N}\left( P\right) = {v}^{N}\left( Q\right) = V \) , say. Applying equation (3.16),
\[
{j}_{V}\left( {P}^{\prime }\right) = P \neq Q = {j}_{V}\left( {Q}^{\prime }\right)
\]
which gives \( {P}^{\prime } \neq {Q}^{\prime } \) . Thus \( {P}^{\prime } \) and \( {Q}^{\prime } \) are distinct miniature tableaux of the same normal shape. The only possibility is, then,
\[
\left\{ {{P}^{\prime },{Q}^{\prime }}\right\} = \left\{ \begin{array}{llll} 1 & 2, & 1 & 3 \\ 3 & & 2 & \end{array}\right\} .
\]
Since \( {P}^{\prime } \cong {Q}^{\prime } \), we have, by what was proved in the forward direction,
\[
P = {j}_{V}\left( {P}^{\prime }\right) \overset{{K}^{ * }}{ \cong }{j}_{V}\left( {Q}^{\prime }\right) = Q.\blacksquare
\]
To make it more convenient to talk about miniature subtableaux of a larger tableau, we make the following definition.
Definition 3.8.6 Let \( P \) and \( Q \) be standard skew tableaux with
\[
\operatorname{sh}P = \mu /\nu \vdash m\;\text{ and }\;\operatorname{sh}Q = \lambda /\mu \vdash n.
\]
Then \( P \cup Q \) denotes the tableau of shape \( \lambda /\nu \vdash m + n \) such that
\[
{\left( P \cup Q\right) }_{c} = \left\{ \begin{array}{ll} {P}_{c} & \text{ if }c \in \mu /\nu \\ {Q}_{c} + m & \text{ if }c \in \lambda /\mu \end{array}\right.
\]
Using the \( P \) and \( Q \) on page 118, we have
\[
P \cup Q = \begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 7. \\ 6 & & \end{array}
\]
We need one more lemma before the main theorem of this section.
Lemma 3.8.7 ([Hai 92]) Let \( V, W, P \), and \( Q \) be standard skew tableaux with
\[
\operatorname{sh}V = \mu /\nu ,\;\operatorname{sh}P = \operatorname{sh}Q = \lambda /\mu ,\;\operatorname{sh}W = \kappa /\lambda .
\]
Then
\[
P\overset{ * }{ \cong }Q \Rightarrow V \cup P \cup W\overset{ * }{ \cong }V \cup Q \cup W.
\]
Proof. Consider what happens in performing a single forward slide on \( V \cup \) \( P \cup W \), say into cell \( c \) . Because of the relative order of the elements in the \( V \) , \( P \), and \( W \) portions of the tableau, the slide can be broken up into three parts. First of all, the slide travels through \( V \), creating a new tableau \( {V}^{\prime } = {j}_{c}\left( V\right) \) and vacating some inner corner \( d \) of \( \mu \) . Then \( P \) becomes \( {P}^{\prime } = {j}_{d}\left( P\right) \), vacating cell \( e \), and finally \( W \) is transformed into \( {W}^{\prime } = {j}_{e}\left( W\right) \) . Thus \( {j}_{c}\left( {V \cup P \cup W}\right) = \) \( {V}^{\prime } \cup {P}^{\prime } \cup {W}^{\prime } \) .
Now perform the same slide on \( V \cup Q \cup W \) . Tableau \( V \) is replaced by \( {j}_{c}\left( V\right) = {V}^{\prime } \), vacating \( d \) . If \( {Q}^{\prime } = {j}_{d}\left( Q\right) \), then, since \( P \cong Q \), we have \( \operatorname{sh}{P}^{\prime } = \) sh \( {Q}^{\prime } \) . So \( e \) is vacated as before, and \( W \) becomes \( {W}^{\prime } \) . Thus \( {j}_{c}\left( {V \cup Q \cup W}\right) = \) \( {V}^{\prime } \cup {Q}^{\prime } \cup {W}^{\prime } \) with \( {P}^{\prime } \cong {Q}^{\prime } \) by Lemma 3.8.3.
Now the preceding also holds, mutatis mutandis, to backward slides. Hence applying the same slide to both \( V \cup P \cup W \) and \( V \cup P \cup W \) yields tableaux of the same shape that still satisfy the hypotheses of the lemma. By induction, we are done. -
We can now show that Proposition 3.8.5 actually holds for all pairs of tableaux.
Theorem 3.8.8 ([Hai 92]) Let \( P \) and \( Q \) be standard tableaux of the same shape \( \lambda /\mu \) . Then
\[
P\overset{{K}^{ * }}{ \cong }Q \Leftrightarrow P\overset{ * }{ \cong }Q.
\]
Proof. " \( \Rightarrow \) " We need to consider only the case where \( P \) and \( Q \) differ by a single dual Knuth relation, say the first (the second is similar). Now \( Q \) is obtained from \( P \) by switching \( k + 1 \) and \( k + 2 \) for some \( k \) . So
\[
P = V \cup {P}^{\prime } \cup W\text{ and }Q = V \cup {Q}^{\prime } \cup W,
\]
where \( {P}^{\prime } \) and \( {Q}^{\prime } \) are the miniature subtableaux of \( P \) and \( Q \), respectively, that contain \( k, k + 1 \), and \( k + 2 \) . By hypothesis, \( {P}^{\prime }\overset{{1}^{ * }}{ \cong }{Q}^{\prime } \), which implies \( {P}^{\prime }\overset{ * }{ \cong }{Q}^{\prime } \) by Proposition 3.8.5. But then the lemma just proved applies to show that \( P\overset{ * }{ \cong }Q \) .
" \( \Leftarrow \) " Let tableau \( N \) be of normal shape \( \mu \) . Let
\[
{P}^{\prime } = {j}^{N}\left( P\right) \text{ and }{Q}^{\prime } = {j}^{N}\left( Q\right) .
\]
Since \( P \cong Q \), we have \( {P}^{\prime } \cong {Q}^{\prime } \) (Lemma 3.8.3) and \( {v}^{N}\left( P\right) = {v}^{N}\left( Q\right) = V \) for some tableau \( V \) . Thus, in particular, \( \operatorname{sh}{P}^{\prime } = \operatorname{sh}{Q}^{\prime } \), so that \( {P}^{\prime } \) and \( {Q}^{\prime } \) are dual Knuth equivalent by Proposition 3.8.1. Now, by definition, we have a sequence of dual Knuth relations
\[
{P}^{\prime } = {P}_{1}\overset{{i}^{\prime * }}{ \cong }{P}_{2}\overset{{j}^{\prime * }}{ \cong }\cdots \overset{{l}^{\prime * }}{ \cong }{P}_{k} = {Q}^{\prime },
\]
where \( {i}^{\prime },{j}^{\prime },\ldots ,{l}^{\prime } \in \{ 1,2\} \) . Hence the proof of the forward direction of Proposition 3.8.5 and equation (3.16) show that
\[
P = {j}_{V}\left( {P}^{\prime }\right) \overset{{i}^{ * }}{ \cong }{j}_{V}\left( {P}_{2}\right) \overset{{j}^{ * }}{ \cong }\cdots \overset{{l}^{ * }}{ \cong }{j}_{V}\left( {Q}^{\prime }\right) = Q
\]
for some \( i, j,\ldots, l \in \{ 1,2\} \) . This finishes the proof of the theorem. ∎
## 3.9 Evacuation
We now return to our project of determining the effect that a reflection or rotation of the permutation matrix for \( \pi \) has on the tableaux \( P\left( \pi \right) \) and \( Q\left( \pi \right) \) . We have already seen what happens when \( \pi \) is replaced by \( {\pi }^{-1} \) (Theorem 3.6.6). Also, Theorem 3.2.3 tells us what the \( P \) -tableau of \( {\pi }^{r} \) looks like. Since these two operations correspond to reflections that generate the dihedral group of the square, we will be done as soon as we determine \( Q\left( {\pi }^{r}\right) \) . To do this, another concept, called evacuation [Scii 63], is needed.
Definition 3.9.1 Let \( Q \) be a partial skew tableau and let \( m \) be the minimal element of \( Q \) . Then the delta operator applied to \( Q \) yields a new tableau, \( {\Delta Q} \) , defined as follows.
D1 Erase \( m \) from its cell, \( c \), in \( Q \) .
D2 Perform the slide \( {j}^{c} \) on the resultant tableau.
If \( Q \) is standard with \( n \) elements, then the evacuation tableau for \( Q \) is the vacating tableau \( V = \operatorname{ev}Q \) for the sequence
\[
Q,{\Delta Q},{\Delta }^{2}Q,\ldots ,{\Delta }^{n}Q.
\]
That is,
\( {V}_{d} = n - i \) if cell \( d \) was vacated when passing from \( {\Delta }^{i}Q \) to \( {\Delta }^{i + 1}Q \) . ∎
Taking
\[
Q = \begin{array}{l} {1347} \\ {25} \\ 6 \end{array}
\]
we compute ev \( Q \) as follows, using dots as placeholders for cells not yet filled.
\[
\begin{matrix} 1\;3\;4\;7, & 2\;3\;4\;7, & 3\;4\;7, & 4\;7, & 5\;7, & 6\;7, & 2\;5\;7, & 6\;7, & 2\;5\;5 & 6\;6 & 6\;6 \end{matrix}
\]
<table><thead><tr><th></th><th>···6,</th><th>0056,</th><th>-\( {56} \) ,</th><th>--\( {56}, \)</th><th>-26</th></tr></thead><tr><td>ev \( Q \)</td><td></td><td></td><td></td><td></td><td>3 7</td></tr><tr><td>·</td><td></td><td></td><td></td><td></td><td></td></tr></table>
Completing the last slide we obtain
\[
\operatorname{ev}Q = \begin{array}{llll} 1 & 2 & 5 & 6 \\ 3 & 7 & & \\ 4 & & & \end{array}.
\]
Note that \( Q \) was the \( Q \) -tableau of our
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1234_[丁一文] Number Theory 1
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Definition 1.1.2
|
Definition 1.1.2. Let \( R \) be a commutative ring (with unit). We call \( R \) is euclidean if there exists \( f : R \rightarrow {\mathbb{Z}}_{ \geq 0} \) such that for all \( \alpha ,\beta \in R \) ,
1. \( f\left( \alpha \right) = 0 \Leftrightarrow \alpha = 0 \) ,
2. \( f\left( {\alpha \beta }\right) = f\left( \alpha \right) f\left( \beta \right) \) ,
3. if \( \beta \neq 0 \), then there exist \( \gamma ,\delta \in R \) such that \( \alpha = {\beta \gamma } + \delta \) and \( f\left( \delta \right) < f\left( \beta \right) \) .
Example 1.1.3. \( \mathbb{Z} \) is euclidean with \( f \mathrel{\text{:=}} \left| \cdot \right| \) .
Proposition 1.1.4. If \( R \) is euclidean, then \( R \) is a principal ideal domain.
Proof. Using the conditions \( 1\& 2 \), one sees \( R \) is a domain. Let \( I \) be an ideal of \( R \), and let \( \beta \in I \) such that \( f\left( \beta \right) = \mathop{\min }\limits_{{0 \neq \alpha \in I}}f\left( x\right) \) . For \( \alpha \in I \), there exists \( \delta \) such that \( \alpha = {\beta \gamma } + \delta \) and \( f\left( \delta \right) < f\left( \beta \right) \) . Since \( \delta \in I \), we deduce by the choice of \( \beta \) that \( \delta = 0 \) . Hence \( I = \left( \beta \right) \) .
Proposition 1.1.5. \( \mathbb{Z}\left\lbrack i\right\rbrack \) is euclidean.
Proof. Put \( f : \mathbb{Z}\left\lbrack i\right\rbrack \rightarrow {\mathbb{Z}}_{ \geq 0}, a + {bi} \mapsto {a}^{2} + {b}^{2} \) . We check \( f \) satisfies the conditions in the definition. The conditions 1 and 2 are clear. Let \( \alpha ,\beta \in \mathbb{Z}\left\lbrack i\right\rbrack ,\beta \neq 0 \) . Consider
\( \alpha /\beta = x + {yi} \in \mathbb{C} \) . There exists thus \( \gamma = a + {bi} \in \mathbb{Z}\left\lbrack i\right\rbrack \), such that \( \left| {x - a}\right| \leq 1/2 \) and \( \left| {y - b}\right| \leq 1/2 \) . We deduce \( {\left| \alpha /\beta - \gamma \right| }^{2} \leq 1/2 \) . Putting \( \delta \mathrel{\text{:=}} \alpha - {\beta \gamma } \), we have \( \left| \delta \right| < \left| \beta \right| \) . The proposition follows.
Corollary 1.1.6. \( \mathbb{Z}\left\lbrack i\right\rbrack \) is a PID.
Proposition 1.1.7. There exist \( a, b \in \mathbb{Z} \) such that \( p = {a}^{2} + {b}^{2} \) if and only if \( \left( p\right) \) is not a prime ideal in \( \mathbb{Z}\left\lbrack i\right\rbrack \) . (Recall an ideal \( I \) is called a prime ideal if \( I \supset {I}_{1}{I}_{2} \Rightarrow I \supset {I}_{1} \) or \( I \supset {I}_{2}) \) .
Proof. If \( p = {a}^{2} + {b}^{2} \), then \( p = \left( {a + {bi}}\right) \left( {a - {bi}}\right) \) . If \( \left( p\right) \) is a prime ideal then \( \left( p\right) \supset \left( {a + {bi}}\right) \) or \( \left( p\right) \supset \left( {a - {bi}}\right) \) . Replacing \( b \) by \( - b \) if needed, we assume \( \left( p\right) \supset \left( {a + {bi}}\right) \) . Then there exists \( x \in \mathbb{Z}\left\lbrack i\right\rbrack \) such that \( a + {bi} = {px} \) . This implies \( p = {px}\left( {a - {bi}}\right) \) and hence \( p\left( {1 - x\left( {a - {bi}}\right) }\right) = 0 \) . Since \( \mathbb{Z}\left\lbrack i\right\rbrack \) is a domain, we deduce \( x\left( {a - {bi}}\right) = 1 \) . Hence \( {\left| x\right| }^{2}\left| {{a}^{2} + {b}^{2}}\right| = 1 \) a contradiction (noting \( {\left| x\right| }^{2} \in {\mathbb{Z}}_{ > 0} \) ).
Now assume \( p \) is not a prime ideal. By definition (and the fact that \( \mathbb{Z}\left\lbrack i\right\rbrack \) is a PID), there exist \( \alpha ,\beta \in \mathbb{Z}\left\lbrack i\right\rbrack \) such that \( p = {\alpha \beta } \) and that \( \alpha ,\beta \) are not units. This implies \( {\left| p\right| }^{2} = {\left| \alpha \right| }^{2}{\left| \beta \right| }^{2} \) . Since \( \alpha ,\beta \) are not units, \( {\left| \alpha \right| }^{2} > 1 \) and \( {\left| \beta \right| }^{2} > 1 \) (say, if \( {\left| \alpha \right| }^{2} = 1 \), then \( \alpha \bar{\alpha } = 1 \) and hence \( \alpha \) is a unit). We deduce then \( {\left| \alpha \right| }^{2} = {\left| \beta \right| }^{2} = p \) . Writing \( \alpha = a + {bi} \), we see the "if" part follows.
Proof of Theorem 1.1.1. We only need to prove the "if" part. By the above proposition, it suffices to show if \( p = 1 + {4n} \), then \( p \) is not prime in \( \mathbb{Z}\left\lbrack i\right\rbrack \) . Consider the finite field \( {\mathbb{F}}_{p} \) . Recall that the multiplicative group of a finite field is a cyclic group (Exercise). In particular, \( {\mathbb{F}}_{p}^{ * } \) is a cyclic group of order \( p - 1 = {4n} \) . This implies that there exists \( x \in \mathbb{Z} \) such that \( {x}^{4} \equiv 1 \) \( \left( {\;\operatorname{mod}\;p}\right) ,{x}^{2} \neq 1\left( {\;\operatorname{mod}\;p}\right) \) . So \( {x}^{2} \equiv - 1\left( {\;\operatorname{mod}\;p}\right) \) . Hence \( p\left| {\left( {{x}^{2} + 1}\right) \Rightarrow p}\right| \left( {x + i}\right) \left( {x - i}\right) \) . If \( p \) is prime, without loss of generality, we have \( p \mid \left( {x + i}\right) \) . So there exists \( \alpha = y + {zi} \in \mathbb{Z}\left\lbrack i\right\rbrack \) such that \( x + i = p\left( {y + {zi}}\right) = {py} + {pzi} \), hence \( {pz} = 1 \), a contradiction. This concludes the proof.
Remark 1.1.8. Some key words in the proof: ideals, units, finite field (residue field).
Definition 1.1.9. We call a finite extension of \( \mathbb{Q} \) a number field.
Exercise 1.1.10. Let \( K \) be a number field, show that there exists \( \alpha \in K \) such that \( K = \) \( \mathbb{Q}\left( \alpha \right) \) .
The ring \( \mathbb{Z}\left\lbrack i\right\rbrack \) plays a key role in the proof of the theorem. A natural question is that what is the analogue of \( \mathbb{Z}\left\lbrack i\right\rbrack \) for an arbitrary number field?
Exercise 1.1.11. Show that \( \mathbb{Z}\left\lbrack {2i}\right\rbrack \) is not a PID.
## 1.2 Integral extensions (commutative algebra)
Definition 1.2.1. Let \( A \hookrightarrow B \) be commutative rings with 1 .
(1) An element \( b \in B \) is called integral over \( A \) if there exists a monic polynomial \( f\left( x\right) \in A\left\lbrack x\right\rbrack \) such that \( f\left( b\right) = 0 \) .
(2) The ring \( B \) is called integral over \( A \) if all elements of \( B \) are integral over \( A \) .
Example 1.2.2. \( \mathbb{Z}\left\lbrack i\right\rbrack \) is integral over \( \mathbb{Z} \) : For any \( \alpha = a + {bi} \in \mathbb{Z}\left\lbrack i\right\rbrack ,{\alpha }^{2} - {2a\alpha } + \left( {{a}^{2} + {b}^{2}}\right) = 0 \) .
Definition 1.2.3. Let \( K/\mathbb{Q} \) be a number field, \( \alpha \in K \) is called an algebraic integer, if \( \alpha \) is integral over \( \mathbb{Z} \) .
Let \( {\mathcal{O}}_{K} \mathrel{\text{:=}} \{ \) algebraic integers of \( K\} \) .
Proposition 1.2.4. Let \( A \subset B \) be commutative rings with 1, let \( b \in B \) . Then \( b \) is integral over \( A \) if and only if there exists an \( A \) -subalgebra \( C \) of \( B \) that is finitely generated as \( A \) -module such that \( A\left\lbrack b\right\rbrack \subseteq C \) .
Proof. "Only if": Suppose \( b \) is integral over \( A \) . There exists thus \( f\left( x\right) = {x}^{n} + {a}_{n - 1}{x}^{n - 1} + \) \( \cdots + {a}_{0} \in A\left\lbrack x\right\rbrack \) such that \( f\left( b\right) = 0 \) . Then we see \( A\left\lbrack b\right\rbrack \) is generated (as an \( A \) -module) by \( \left\{ {1,\cdots ,{b}^{n - 1}}\right. \) . We can then choose \( C \mathrel{\text{:=}} A\left\lbrack b\right\rbrack \) .
"If": Since \( C \) is a finitely generated \( A \) -module, there exist \( {v}_{1},\cdots ,{v}_{k} \in C \) such that \( C = A{v}_{1} + \cdots A{v}_{k} \) . Since \( {bC} \subseteq C \) (using \( A\left\lbrack b\right\rbrack \subset C \) ), there exists \( M = \left( {a}_{ij}\right) \in {M}_{k \times k}\left( A\right) \)
such that
\[
\left( \begin{matrix} b{v}_{1} \\ \vdots \\ b{v}_{k} \end{matrix}\right) = M\left( \begin{matrix} {v}_{1} \\ \vdots \\ {v}_{k} \end{matrix}\right)
\]
This implies
\[
N\left( \begin{matrix} {v}_{1} \\ \vdots \\ {v}_{k} \end{matrix}\right) \mathrel{\text{:=}} \left( \begin{matrix} b - {a}_{11} & \cdots & - {a}_{1k} \\ \vdots & \ddots & \vdots \\ - {a}_{k1} & \cdots & b - {a}_{kk} \end{matrix}\right) \left( \begin{matrix} {v}_{1} \\ \vdots \\ {v}_{k} \end{matrix}\right) = 0.
\]
Let \( {N}^{ * } \in {M}_{k \times k}\left( {A\left\lbrack b\right\rbrack }\right) \) be the adjoint matrix of \( N \), we have \( {N}^{ * }N\left( \begin{matrix} {v}_{1} \\ \vdots \\ {v}_{k} \end{matrix}\right) = 0 \), and \( {N}^{ * }N = \) \( \operatorname{diag}\left( {f\left( b\right) ,\cdots, f\left( b\right) }\right) \) where \( f\left( b\right) = \det \left( N\right) \) is of the form \( {b}^{k} + {a}_{k - 1}{b}^{k - 1} + \cdots + {a}_{0} \) . Thus \( f\left( b\right) {v}_{i} = 0 \) for all \( {v}_{i} \), implying \( f\left( b\right) C = 0 \Rightarrow f\left( b\right) 1 = 0 \Rightarrow f\left( b\right) = 0 \) . So \( b \) is integral over A.
Corollary 1.2.5. Let \( A \hookrightarrow B \) be commutative rings with 1, let \( \alpha ,\beta \in B \) be integral over A. Then \( \alpha + \beta ,{\alpha \beta } \) are also integral over \( A \) .
Proof. Since \( \beta \) is integral over \( A,\beta \) is integral over \( A\left\lbrack \alpha \right\rbrack \) . By the above proposition (and the proof), \( A\left\lbrack \alpha \right\rbrack \left\lbrack \beta \right\rbrack \) is a fintiely generated \( A\left\lbrack \alpha \right\rbrack \) -module. Since \( \alpha \) is integral over \( A, A\left\lbrack \alpha \right\rbrack \) is a finitely generated \( A \) -module. Hence \( A\left\lbrack \alpha \right\rbrack \left\lbrack \beta \right\rbrack \) is a finitely generated \( A \) -module. Thus \( \alpha + \beta \) , \( {\alpha \beta } \) are integral.
Corollary 1.2.6. \( {\mathcal{O}}_{K} \) is a \( \mathbb{Z} \) -algebra.
Corollary 1.2.7. Let \( A \subset B \subset
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18_Algebra Chapter 0
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Definition 4.4
|
Definition 4.4. An \( R \) -module \( M \) is cyclic if it is generated by a singleton, that is, if \( M \cong R/I \) for some ideal \( I \) of \( R \) .
The equivalence in the definition is hopefully clear to our reader, as an immediate consequence of the first isomorphism theorem for modules (Corollary 11115.16). If not, go back and (re)do Exercise III 6.16
Cyclic modules are witness to the difference between fields and more general rings: over a field \( k \), a cyclic module is just a 1-dimensional vector space, that is a ’copy of \( k \) ’; over more general rings, cyclic modules may be very interesting (think of the many hours spent contemplating cyclic groups). In fact, we can tell that a ring is a field by just looking at its cyclic modules:
Lemma 4.5. Let \( R \) be an integral domain. Assume that every cyclic \( R \) -module is torsion-free. Then \( R \) is a field.
Proof. Let \( c \in R, c \neq 0 \) ; then \( M = R/\left( c\right) \) is a cyclic module. Note that \( \operatorname{Tor}\left( M\right) = \) \( M \) : indeed, the class of 1 generates \( R/\left( c\right) \) and belongs to \( \operatorname{Tor}\left( M\right) \) since \( c \cdot 1 \) is 0 \( {\;\operatorname{mod}\;\left( c\right) } \) and \( c \neq 0 \) . However, by hypothesis \( M \) is torsion-free; that is, \( \operatorname{Tor}\left( M\right) = \) \( \{ 0\} \) . Therefore \( M = \operatorname{Tor}\left( M\right) \) is the zero module.
This shows \( R/\left( c\right) \) is the zero \( R \) -module; that is, \( \left( c\right) = \left( 1\right) \) . Therefore, \( c \) is a unit. Thus every nonzero element of \( R \) is a unit, proving that \( R \) is a field.
Lemma 4.5 is a simple-minded illustration of the fact that we can study a ring \( R \) by studying the module structure over \( R \), that is, the category \( R \) -Mod, and that we may not even need to look at the whole of \( R \) -Mod to be able to draw strong conclusions about \( R \) .
4.2. Finitely presented modules and free resolutions. The 'right' way to think of a cyclic \( R \) -module \( M \) is as a module which admits an epimorphism from \( R \) ,
viewing the latter as the free rank-1 \( R \) -module 21:
\[
{R}^{1} \rightarrow M \rightarrow 0.
\]
The fact that \( M \) is surjected upon by a free \( R \) -module is nothing special. In fact, every module \( M \) admits such an epimorphism:
\[
{R}^{\oplus A} \rightarrow M \rightarrow 0
\]
provided that we are willing to take \( A \) large enough; if we are desperate, \( A = M \) will surely do. This is immediate from the universal property of free modules; if the reader does not agree, it is time to go back and review [111,6.3] What makes cyclic modules special is that \( A \) can be chosen to be a singleton.
We are now going to focus on a case which is also special, but not quite as special as cyclic modules: finitely generated modules are modules for which we can choose \( A \) to be a finite set (cf. [1116.4). Thus, we will assume that \( M \) admits an epimorphism from a finite-rank free module:
\[
{R}^{m}\overset{\pi }{ \rightarrow }M \rightarrow 0
\]
for some integer \( m \) . The image by \( \pi \) of the \( m \) vectors in a basis of \( {R}^{m} \) is a set of generators for \( M \) .
Finitely generated modules are much easier to handle than arbitrary modules. For example, an ideal of \( R \) can tell us whether a finitely generated module is torsion.
Definition 4.6. The annihilator of an \( R \) -module \( M \) is
\[
{\operatorname{Ann}}_{R}\left( M\right) \mathrel{\text{:=}} \{ r \in R \mid \forall m \in M,{rm} = 0\} .
\]
The subscript is usually omitted. The reader will check (Exercise 4.4) that \( \operatorname{Ann}\left( M\right) \) is an ideal of \( R \) and that if \( M \) is a finitely generated module and \( R \) is an integral domain, then \( M \) is torsion if and only if \( \operatorname{Ann}\left( M\right) \neq 0 \) .
We would like to develop tools to deal with finitely generated modules. It turns out that matrices allow us to describe a comfortably large collection of such modules.
Definition 4.7. An \( R \) -module \( M \) is finitely presented if for some positive integers \( m, n \) there is an exact sequence
\[
{R}^{n}\overset{\varphi }{ \rightarrow }{R}^{m} \rightarrow M \rightarrow 0
\]
Such a sequence is called a presentation of \( M \) .
In other words, finitely presented modules are cokernels (cf. 1116.2) of homomorphisms between finitely generated free modules. Everything about \( M \) must be encoded in the homomorphism \( \varphi \) ; therefore, we should be able to describe the module \( M \) by studying the matrix corresponding to \( \varphi \) .
There is a gap between finitely presented modules and finitely generated modules, but on reasonable rings the two notions coincide:
\( {}^{21} \) In context the exactness of a sequence of \( R \) -modules will be understood, so the displayed sequence is a way to denote the fact that there exists a surjective homomorphism of \( R \) -modules from \( R \) to \( M \) ; cf. Example III 7.2 Also note the convention of denoting \( R \) by \( {R}^{1} \) when it is viewed as a module over itself.
Lemma 4.8. If \( R \) is a Noetherian ring, then every finitely generated \( R \) -module is finitely presented.
Proof. If \( M \) is a finitely generated module, there is an exact sequence
\[
{R}^{m}\overset{\pi }{ \rightarrow }M \rightarrow 0
\]
for some \( m \) . Since \( R \) is Noetherian, \( {R}^{m} \) is Noetherian as an \( R \) -module (Corollary III 6.8). Thus \( \ker \pi \) is finitely generated; that is, there is an exact sequence
\[
{R}^{n} \rightarrow \ker \pi \rightarrow 0
\]
for some \( n \) . Putting together the two sequences gives a presentation of \( M \) .
Once we have gone one step to obtain generators and two steps to get a presentation, we should hit upon the idea to keep going:
Definition 4.9. A resolution of an \( R \) -module \( M \) by finitely generated free modules is an exact complex
\[
\ldots \rightarrow {R}^{{m}_{3}} \rightarrow {R}^{{m}_{2}} \rightarrow {R}^{{m}_{1}} \rightarrow {R}^{{m}_{0}} \rightarrow M \rightarrow 0.
\]
Iterating the argument proving Lemma 4.8 shows that if \( R \) is Noetherian, then every finitely generated module has a resolution as in Definition 4.9
It is an important conceptual step to realize that \( M \) may be studied by studying an exact complex of free modules
\[
\ldots \rightarrow {R}^{{m}_{3}} \rightarrow {R}^{{m}_{2}} \rightarrow {R}^{{m}_{1}} \rightarrow {R}^{{m}_{0}}
\]
resolving \( M \), that is, such that \( M \) is the cokernel of the last map. The \( {R}^{{m}_{0}} \) piece keeps track of the generators of \( M;{R}^{{m}_{1}} \) accounts for the relations among these generators; \( {R}^{{m}_{2}} \) records relations among the relations; and so on.
Developing this idea in full generality would take us too far for now: for example, we would have to deal with the fact that every module admits many different resolutions (for example, we can bump up every \( {m}_{i} \) by one by direct-summing each term in the complex with a copy of \( {R}^{1} \), sent to itself by the maps in the complex). We will do this very carefully later on, in Chapter IX
However, we can already learn something by considering coarse questions, such as ’how long’ a resolution can be. A priori, there is no reason to expect a free resolutions to be ’finite’, that is, such that \( {m}_{i} = 0 \) for \( i \gg 0 \) . Such finiteness conditions tell us something special about the base ring \( R \) .
The first natural question of this type is, for which rings \( R \) is it the case that every finitely generated \( R \) -module \( M \) has a free resolution ’of length 0 ’, that is, stopping at \( {m}_{0} \) ? That would mean that there is an exact sequence
\[
0 \rightarrow {R}^{{m}_{0}} \rightarrow M \rightarrow 0.
\]
Therefore, \( M \) itself must be free. What does this say about \( R \) ?
Proposition 4.10. Let \( R \) be an integral domain. Then \( R \) is a field if and only if every finitely generated \( R \) -module is free.
Proof. If \( R \) is a field, then every \( R \) -module is free, by Proposition 1.7. For the converse, assume that every finitely generated \( R \) -module is free; in particular, every cyclic module is free; in particular, every cyclic module is torsion-free. But then \( R \) is a field, by Lemma 4.5
The next natural question concerns rings for which finitely generated modules admit free resolutions of length 1 . It is convenient to phrase the question in stronger terms, that is, to require that for every finitely generated \( R \) -module \( M \) and every beginning of a free resolution
\[
{R}^{{m}_{0}}\overset{\pi }{ \rightarrow }M \rightarrow 0
\]
the resolution can be completed to a length 1 free resolution. This would amount to demanding that there exist an integer \( {m}_{1} \) and an \( R \) -module homomorphism \( {R}^{{m}_{1}} \rightarrow {R}^{{m}_{0}} \) such that the sequence
\[
0 \rightarrow {R}^{{m}_{1}} \rightarrow {R}^{{m}_{0}}\xrightarrow[]{\pi }M \rightarrow 0
\]
is exact. Equivalently, this condition requires that the module \( \ker \pi \) of relations among the \( {m}_{0} \) generators necessarily be free.
Claim 4.11. Let \( R \) be an integral domain satisfying this property. Then \( R \) is a PID.
Proof. Let \( I \) be an ideal of \( R \), and apply the condition to \( M = R/I \) . Since we have an epimorphism
\[
{R}^{1}\overset{\pi }{ \rightarrow }R/I \rightarrow 0
\]
the condition says that \( \ker \pi \) is free; that is, \( I \) is free. Since \( I \) is a free submodule of \( R \), which is free of rank \( 1, I \) must be free of rank \( \leq 1 \) by Proposition 1.9, Therefore \( I \) is generated by one element, as needed.
The classification result for finitely generated modules over PIDs (Theorem 5.6), which I keep bringing up, will essentially be a converse to Claim 4.11 the mysterious condition requiring free resolu
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1234_[丁一文] Number Theory 1
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Definition 3.1.3
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Definition 3.1.3. Put \( \mathop{\prod }\limits_{{j \in J}}^{\prime }{G}_{j} = \left\{ {\left( {x}_{j}\right) \mid {x}_{j} \in {G}_{j}}\right. \) and \( \left. {{x}_{j} \in {H}_{j}\text{for all but finitely many}j}\right\} \) , called the restricted direct product of \( {\left\{ {G}_{j}\right\} }_{j \in J} \) with respect to \( \left\{ {H}_{j}\right\} \) .
It is clear that \( \mathop{\prod }\limits_{{j \in J}}^{\prime }{G}_{j} \) has natural group structure. We equip \( \mathop{\prod }\limits_{{j \in J}}^{\prime }{G}_{j} \) with the topology such that an open basis at \( 1 \in \mathop{\prod }\limits_{{j \in J}}^{\prime }{G}_{j} \) given by
\[
{\Pi }_{j \in S}{U}_{j} \times {\Pi }_{j \notin S}{H}_{j} \subset {\Pi }_{j \in J}^{\prime }{G}_{j}
\]
where \( S \) runs though finite subset of \( J \) containing \( {J}_{\infty } \), and \( {U}_{j} \) runs through open neigh-bourhoods of 1 in \( {G}_{j} \) .
Remark 3.1.4. Let \( S \supset {J}_{\infty } \) be a finite set of \( J \), we have a natural injection
\[
{\Pi }_{j \in S}{G}_{j} \times {\Pi }_{j \notin S}{H}_{j} \hookrightarrow {\Pi }_{j \in J}^{\prime }{G}_{j}
\]
One can easily check that the induced topology on \( \mathop{\prod }\limits_{{j \in S}}{G}_{j} \times \mathop{\prod }\limits_{{j \notin S}}{H}_{j} \) coincides with the product topology.
Exercise 3.1.5. Prove that the topology on \( \mathop{\prod }\limits_{{j \in J}}^{\prime }{G}_{j} \) is finer than the topology induced from the product topology on \( \mathop{\prod }\limits_{{j \in J}}{G}_{j} \) via the natural injection \( \mathop{\prod }\limits_{{j \in J}}^{\prime }{G}_{j} \hookrightarrow \mathop{\prod }\limits_{{j \in J}}{G}_{j} \) .
Proposition 3.1.6. \( \mathop{\prod }\limits_{{j \in J}}^{\prime }{G}_{j} \) is a locally compact group.
Proof. Exercise.
Now let \( K \) be a number field, \( J \) be the set of all places of \( K,{J}_{\infty } \) be the set of archimedean places of \( K \), and \( {G}_{v} \mathrel{\text{:=}} {K}_{v} \) for \( v \in J \), and \( {H}_{v} \mathrel{\text{:=}} {\mathcal{O}}_{{K}_{v}} \) for \( v \in J \smallsetminus {J}_{\infty } \) . Let \( {\mathbb{A}}_{K} \mathrel{\text{:=}} \mathop{\prod }\limits_{{v \in J}}^{\prime }{K}_{v} \) , called the ring of adeles of \( K \) .
Exercise 3.1.7. Prove that the multiplication map \( {\mathbb{A}}_{K} \times {\mathbb{A}}_{K} \rightarrow {\mathbb{A}}_{K} \) is continuous.
Lemma 3.1.8. The diagonal map \( K \hookrightarrow \mathop{\prod }\limits_{{v \in J}}{K}_{v} \) factors though an injection \( K \hookrightarrow {\mathbb{A}}_{K} \) .
Proof. For any \( x \in K \), there are only finitely many \( v \in J \smallsetminus {J}_{\infty } \) such that \( x \notin {\mathcal{O}}_{{K}_{v}} \) . The lemma follows.
For a finite set \( S \supset {J}_{\infty } \), denote by \( {\mathbb{A}}_{S} \mathrel{\text{:=}} \mathop{\prod }\limits_{{v \in S}}{K}_{v} \times \mathop{\prod }\limits_{{v \notin S}}{\mathcal{O}}_{{K}_{v}} \) .
Proposition 3.1.9. We have \( K + {\mathbb{A}}_{{J}_{\infty }} = {\mathbb{A}}_{K} \) .
Proof. We need to show that for all \( {\left( {x}_{v}\right) }_{v \in J} \in {\mathbb{A}}_{K} \), there exists \( x \in K \) such that \( \left( {x - {x}_{v}}\right) \in \) \( {\mathbb{A}}_{{J}_{\infty }} \) . Let \( m \in {\mathcal{O}}_{K} \smallsetminus \{ 0\} \) such that \( m{x}_{v} \in {\mathcal{O}}_{{K}_{v}} \) for all \( v \) . Let \( S \) be the set of finite places \( v \) such that \( m \in {\mathfrak{p}}_{v} \), and for \( v \in S \), denote by \( {e}_{v} \) the maximal integer such that \( m \in {\mathfrak{p}}_{v}^{{e}_{v}} \) . Let \( x \in {\mathcal{O}}_{K} \) such that \( x \equiv m{x}_{v}\left( {\;\operatorname{mod}\;{\mathfrak{p}}_{v}^{{e}_{v}}}\right) \) for all \( v \in S \) (the existence following from Chinese reminder theorem). Then we have \( x - m{x}_{v} \in {\mathcal{O}}_{{K}_{v}} \) for all fintie places \( v \) and \( x - m{x}_{v} \in m{\mathcal{O}}_{{K}_{v}} \) for \( v \in S \) . Hence \( \frac{x}{m} - {x}_{v} \in {\mathcal{O}}_{{K}_{v}} \) for all finite places \( v \) . The proposition follows.
Let \( L/K \) be a finite extension. Denote by \( \iota \) the injection \( \iota : {\mathbb{A}}_{K} \hookrightarrow {\mathbb{A}}_{L},\left( {a}_{v}\right) \mapsto \left( {x}_{w}\right) \) , \( {x}_{w} = {a}_{v} \) for \( w \mid v \) .
Proposition 3.1.10. Let \( {e}_{1},\cdots ,{e}_{d} \) be a basis of \( L \) over \( K \) . The morphism
\[
f : {\mathbb{A}}_{K}{e}_{1} \oplus \cdots \oplus {\mathbb{A}}_{k}{e}_{d} \rightarrow {\mathbb{A}}_{L},\sum {a}_{i}{e}_{i} \mapsto \sum \iota \left( {a}_{i}\right) {e}_{i}
\]
is well defined and is an isomorphism of topological groups.
Proof. Recall for all places \( v \) of \( K \), the morphism \( {\iota }_{v} : {K}_{v}{e}_{1} \oplus \cdots {K}_{v}{e}_{d} \rightarrow \mathop{\prod }\limits_{{w \mid v}}{L}_{w} \) is an isomorphism. Moreover, for all but finitely many finite places \( v,{\iota }_{v} \) induces an isomorphism \( {\mathcal{O}}_{{K}_{v}}{e}_{1} \oplus \cdots \oplus {\mathcal{O}}_{{K}_{v}}{e}_{d}\overset{ \sim }{ \rightarrow }\mathop{\prod }\limits_{{w \mid v}}{\mathcal{O}}_{{L}_{w}} \) . So the map \( f \) is well defined. Since \( {\iota }_{v} \) is injective for all \( v \), we deduce \( f \) is injective. For any \( {\left( {x}_{w}\right) }_{w} = {\left( {\left( {x}_{w}\right) }_{w \mid v}\right) }_{v} \in {\mathbb{A}}_{L} \), since \( {\iota }_{v} \) is surjective, there exist \( {a}_{v, i} \in {K}_{v} \) such that \( {\iota }_{v}\left( {\sum {a}_{v, i}{e}_{i}}\right) = {\left( {x}_{w}\right) }_{w \mid v} \in \mathop{\prod }\limits_{{w \mid v}}{L}_{w} \) . Since \( {\iota }_{v} \) induces an isomorphism \( {\mathcal{O}}_{{K}_{v}}{e}_{1} \oplus \cdots \oplus {\mathcal{O}}_{{K}_{v}}{e}_{d}\overset{ \sim }{ \rightarrow }\mathop{\prod }\limits_{{w \mid v}}{\mathcal{O}}_{{L}_{w}} \) for all but finitely many \( v \), we see \( {a}_{v, i} \in {\mathcal{O}}_{{K}_{v}} \) for all \( i \) for all but finitely many places \( v \) . In particular, the element \( {\left( {a}_{v, i}\right) }_{v} \) lies in \( {\mathbb{A}}_{K} \) for any \( i \) . So \( f \) is surjective.
Let \( {S}_{0} \) be a finite set of places of \( K \) containing all the archimedean places, and let \( {S}_{L} \mathrel{\text{:=}} \) \( \{ w \mid v \mid w \in S\} \) . Let \( {S}_{0} \) be large enough such that for all \( v \notin S, L{ \otimes }_{K}{K}_{v}\overset{ \sim }{ \rightarrow }\mathop{\prod }\limits_{{w \mid v}}{L}_{w} \) induces an isomorphism \( {\mathcal{O}}_{{K}_{v}}{e}_{1} \oplus \cdots \oplus {\mathcal{O}}_{{K}_{v}}{e}_{d}\overset{ \sim }{ \rightarrow }\mathop{\prod }\limits_{{w \mid v}}{\mathcal{O}}_{{L}_{w}} \) . We see \( \left\{ {U = {U}_{S} \times \mathop{\prod }\limits_{{v \notin S}}\left( {{\mathcal{O}}_{{K}_{v}}{e}_{1} \oplus \cdots \oplus }\right. }\right. \) \( \left. \left. {{\mathcal{O}}_{{K}_{v}}{e}_{d}}\right) \right\} {S}_{ \supset {S}_{0}} \), with \( {U}_{S} \) running though open neighbourhood of 0 in \( \mathop{\prod }\limits_{{v \in S}}\left( {{K}_{v}{e}_{1} \oplus \cdots \oplus {K}_{v}{e}_{d}}\right) \) , form an open basis of \( {\mathbb{A}}_{K}{e}_{1} \oplus \cdots \oplus {\mathbb{A}}_{K}{e}_{d} \) of 0 . Using \( L{ \otimes }_{K}{K}_{v} \cong \mathop{\prod }\limits_{{w \mid v}}{L}_{w}, f\left( {U}_{S}\right) \) form an open basis of \( \mathop{\prod }\limits_{{v \in S}}\mathop{\prod }\limits_{{w \mid v}}{L}_{w} \) . We see \( \left\{ {f\left( U\right) = f\left( {U}_{S}\right) { \times }_{w \notin {S}_{L}}{\mathcal{O}}_{{L}_{w}}}\right\} \) form an open basis of 0 in \( {\mathbb{A}}_{L} \) . So \( f \) is a homemorphism.
Theorem 3.1.11. \( K \) is a discrete, cocompact subgroup of \( {\mathbb{A}}_{K} \) (i.e. \( {\mathbb{A}}_{K}/K \) is compact).
Proof. Let \( {e}_{1},\cdots ,{e}_{d} \) be a basis of \( K \) over \( \mathbb{Q} \) . We have by the above proposition
It thus suffices to show the statement for \( K = \mathbb{Q} \) .
Let \( U \mathrel{\text{:=}} \left\{ {\left( {x}_{v}\right) \in {\mathbb{A}}_{\mathbb{Q}}\left| \right| {\left. {x}_{\infty }\right| }_{\infty } \leq 1/2,{\left| {x}_{v}\right| }_{v} \leq 1}\right. \), for all finite places \( \left. v\right\} \), that is an open neighbourhood of 0 in \( {\mathbb{A}}_{\mathbb{Q}} \) . It is clear that \( U \cap \mathbb{Q} = \{ 0\} \) . We deduce \( \mathbb{Q} \) is discrete in \( {\mathbb{A}}_{\mathbb{Q}} \) . By Proposition 3.1.9, for any \( x \in {\mathbb{A}}_{\mathbb{Q}} \), there exists \( y \in \mathbb{Q} \) such that \( {\left| x - y\right| }_{v} \leq 1 \) for all finite places \( v \) of \( \mathbb{Q} \) . Replacing \( y \) by \( y - n \) for a certain integer \( n \), we can and do assume \( x - y \in U \) . Hence \( U\xrightarrow[]{ \sim }{\mathbb{A}}_{\mathbb{Q}}/\mathbb{Q} \) . Since \( U \) is compact, we see \( \mathbb{Q} \) is cocompact in \( {\mathbb{A}}_{\mathbb{Q}} \) .
## 3.2 Ideles
Let \( {I}_{K} \mathrel{\text{:=}} \mathop{\prod }\limits_{{v \in J}}^{\prime }{K}_{v}^{ \times } \) where the restricted product is with respect to \( {\left\{ {\mathcal{O}}_{{K}_{v}}^{ \times }\right\} }_{v \in J \smallsetminus {J}_{\infty }} \) .
Lemma 3.2.1. As a set, we have \( {I}_{K} = {\mathbb{A}}_{K}^{ \times } \) .
Proof. For \( \left( {x}_{v}\right) \in {I}_{K} \subset {\mathbb{A}}_{K} \), we have \( {x}_{v}^{-1} \in {\mathcal{O}}_{{K}_{v}} \) for all but finitely many places \( v \) . Hence \( \left( {x}_{v}^{-1}\right) \in {\mathbb{A}}_{K} \), so \( {I}_{K} \subset {\mathbb{A}}_{K}^{ \times } \) . Conversely, if \( \left( {x}_{v}\right) \in {\mathbb{A}}_{K} \) and \( \left( {x}_{v}^{-1}\right) \in {\mathbb{A}}_{K} \), there exists a finite set \( S \) of places of \( K \) such that for all \( v \notin S,{x}_{v} \in {\mathcal{O}}_{{K}_{v}} \) and \( {x}_{v}^{-1} \in {\mathcal{O}}_{{K}_{v}} \) . Hence \( \left( {x}_{v}\right) \in {I}_{K} \) .
It is clear that the inje
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1065_(GTM224)Metric Structures in Differential Geometry
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Definition 1.1
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Definition 1.1. Let \( \left( {{\xi }_{i},\langle ,{\rangle }_{i}}\right), i = 1,2 \), be Euclidean bundles over \( {M}_{i} \) . A map \( h : E\left( {\xi }_{1}\right) \rightarrow E\left( {\xi }_{2}\right) \) is said to be isometric if
(1) \( h \) maps each fiber \( {\pi }_{1}^{-1}\left( {p}_{1}\right) \) linearly into a fiber \( {\pi }_{2}^{-1}\left( {p}_{2}\right) \), for \( {p}_{i} \in {M}_{i} \) ; and
(2) \( \langle {hu},{hv}{\rangle }_{2} = \langle u, v{\rangle }_{1} \) for \( u, v \in {\pi }_{1}^{-1}\left( p\right), p \in {M}_{1} \) .
Given Riemannian manifolds \( \left( {{M}_{i},{g}_{i}}\right) \), a map \( f : {M}_{1} \rightarrow {M}_{2} \) is said to be isometric if \( {f}_{ * } : T{M}_{1} \rightarrow T{M}_{2} \) is isometric. An isometric diffeomorphism is called an isometry.
EXAMPLES AND REMARKS 1.1. (i) A parallelization \( {X}_{1},\ldots ,{X}_{n} \) of \( {M}^{n} \) induces a Riemannian metric on \( M \) by defining \( \left\langle {{X}_{i},{X}_{j}}\right\rangle = {\delta }_{ij} \) . The canonical metric on \( {\mathbb{R}}^{n} \) is the one induced by the parallelization \( {D}_{1},\ldots ,{D}_{n} \) .
(ii) A left-invariant metric on a Lie group \( G \) is one induced by a parallelization consisting of left-invariant vector fields; alternatively, it is a metric for which each left translation \( {L}_{g} : G \rightarrow G \) is an isometry. Such metrics are therefore in bijective correspondence with inner products on \( {G}_{e} \) . When in addition, each right translation \( {R}_{g} : G \rightarrow G \) is an isometry, the metric is called bi-invariant. In general, bi-invariant metrics are in bijective correspondence with inner products on \( {G}_{e} \cong \mathfrak{g} \) which are Ad-invariant: If \( \langle \) , \( \rangle {isaleft} \) -invariant metric on \( G \), then for \( X, Y \in \mathfrak{g} \) ,
\[
\left\langle {{R}_{g * }X,{R}_{g * }Y}\right\rangle = \left\langle {{L}_{{g}^{-1} * } \circ {R}_{g * }X,{L}_{{g}^{-1} * } \circ {R}_{g * }Y}\right\rangle = \left\langle {{\operatorname{Ad}}_{{g}^{-1}}X,{\operatorname{Ad}}_{{g}^{-1}}Y}\right\rangle .
\]
Thus, a left-invariant metric on \( G \) is right-invariant iff the induced inner product on \( {G}_{e} \) is Ad-invariant.
It follows for example that any compact Lie group admits a bi-invariant metric: Fix an inner product \( \langle \) , \( {\rangle }_{0} \) on \( \mathfrak{g} \), and define for \( X, Y \in \mathfrak{g} \) ,
\[
\langle X, Y\rangle \mathrel{\text{:=}} {\int }_{G}f,\;f\left( g\right) \mathrel{\text{:=}} {\left\langle {\operatorname{Ad}}_{g}X,{\operatorname{Ad}}_{g}Y\right\rangle }_{0}.
\]
\( \langle \) , \( \rangle {isclearlyaninnerproduct},{andfor}\;a \in G, \)
\[
\left\langle {{\operatorname{Ad}}_{a}X,{\operatorname{Ad}}_{a}Y}\right\rangle = {\int }_{G}f \circ {R}_{a} = {\int }_{G}f = \langle X, Y\rangle .
\]
(iii) A Riemannian metric on a homogeneous space \( M = G/H \) is said to be \( G \) -invariant if
\[
{\mathbb{L}}_{g} : M \rightarrow M
\]
\[
{aH} \mapsto {gaH}
\]
is an isometry for every \( g \in G \) . Notice that if \( \pi : G \rightarrow M \) is the projection, then \( {\mathbb{L}}_{g} \circ \pi = \pi \circ {L}_{g} \) . If \( g = h \in H \), then \( {\mathbb{L}}_{h} \circ \pi = \pi \circ {\mathbb{L}}_{h} \circ {R}_{{h}^{-1}} \), so that
(1.1)
\[
{\mathbb{L}}_{h * } \circ {\pi }_{ * } = {\pi }_{ * } \circ {\operatorname{Ad}}_{h}
\]
This implies that the \( G \) -invariant metrics on \( M \) are in bijective correspondence with the inner products on \( \mathfrak{g}/\mathfrak{h} \) which are \( {\operatorname{Ad}}_{H} \) -invariant (and in particular, any bi-invariant metric on \( G \) induces a \( G \) -invariant metric on \( M \) ): In fact, \( {\pi }_{*e} : \mathfrak{g}/\mathfrak{h} \rightarrow {M}_{p} \) is an isomorphism (here \( p = \pi \left( e\right) \) ), and for each \( h \in H \) , \( {\operatorname{Ad}}_{h} \) induces a map \( {\operatorname{Ad}}_{h} : \mathfrak{g}/\mathfrak{h} \rightarrow \mathfrak{g}/\mathfrak{h} \), since \( {\operatorname{Ad}}_{h}\left( \mathfrak{h}\right) \subset \mathfrak{h} \) . Thus, by (1.1), a \( G \) -invariant metric on \( M \) induces via \( {\pi }_{*e} \) an \( {\operatorname{Ad}}_{H} \) -invariant inner product on \( \mathfrak{g}/\mathfrak{h} \) .
Conversely, any such inner product defines one on \( {M}_{p} \) by requiring \( {\pi }_{ * } \) to be a linear isometry. By (1.1), the latter is invariant under each \( {\mathbb{L}}_{h * p} \) . It may then be extended to all of \( M \) by setting \( \left\langle {{\mathbb{L}}_{g * }u,{\mathbb{L}}_{g * }v}\right\rangle = \langle u, v\rangle \) .
(iv) Although the group of diffeomorphisms of a manifold is not, in general, a Lie group, Myers and Steenrod have shown that the isometry group of a Riemannian manifold with the compact-open topology admits a Lie group structure.
(v) Let \( c : \left\lbrack {a, b}\right\rbrack \rightarrow M \) be a differentiable curve on a Riemannian manifold \( M \) . Since the function \( \left| \dot{c}\right| : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) is continuous, we may define the length of \( c \) to be \( L\left( c\right) \mathrel{\text{:=}} {\int }_{a}^{b}\left| \dot{c}\right| \) . If \( f : M \rightarrow N \) is an isometry, then \( L\left( {f \circ c}\right) = L\left( c\right) \) .
(vi) Suppose \( {\xi }_{i} = {\pi }_{i} : \left( {{E}_{i},\langle ,{\rangle }_{i}}\right) \rightarrow {M}_{i} \) are Euclidean vector bundles, \( i = \) 1,2. The product metric on \( {\xi }_{1} \times {\xi }_{2} \) is defined by
\[
\left\langle {\left( {{u}_{1},{v}_{1}}\right) ,\left( {{u}_{2},{v}_{2}}\right) }\right\rangle \mathrel{\text{:=}} {\left\langle {u}_{1},{v}_{1}\right\rangle }_{1} + {\left\langle {u}_{2},{v}_{2}\right\rangle }_{2}.
\]
When \( {\xi }_{i} \) is the tangent bundle \( \tau {M}_{i} \) of \( {M}_{i} \), it is called the Riemannian product metric on \( {M}_{1} \times {M}_{2} \) (after identifying the tangent space of \( {M}_{1} \times {M}_{2} \) at \( \left( {{m}_{1},{m}_{2}}\right) \) with \( {\left( {M}_{1}\right) }_{{m}_{1}} \times {\left( {M}_{2}\right) }_{{m}_{2}} \) via \( \left( {{p}_{1 * },{p}_{2 * }}\right) \), where \( {p}_{i} : {M}_{1} \times {M}_{2} \rightarrow {M}_{i} \) is the projection). Similarly, the tensor product metric on \( {\xi }_{1} \otimes {\xi }_{2} \) is given by
\[
\left\langle {{u}_{1} \otimes {u}_{2},{v}_{1} \otimes {v}_{2}}\right\rangle \mathrel{\text{:=}} {\left\langle {u}_{1},{v}_{1}\right\rangle }_{1} \cdot {\left\langle {u}_{2},{v}_{2}\right\rangle }_{2},
\]
on decomposable elements.
If \( M = {M}_{1} = {M}_{2} \), the Whitney sum metric on \( {\xi }_{1} \oplus {\xi }_{2} \) is the Euclidean metric for which \( {\pi }_{2} : E\left( {{\xi }_{1} \oplus {\xi }_{2}}\right) \rightarrow {E}_{1} \times {E}_{2} \) becomes isometric.
(vii) Since a Euclidean metric is a nonsingular pairing of \( E = E\left( \xi \right) \) with itself (cf. Section 10 in Chapter 1), there are induced equivalences
\[
\flat : E \rightarrow {E}^{ * },\;\sharp : {E}^{ * } \rightarrow E,
\]
where \( {u}^{\mathfrak{b}}\left( v\right) = \langle u, v\rangle \), and \( {\alpha }^{\sharp } \) is the unique element of \( E \) satisfying \( \left\langle {{\alpha }^{\sharp }, v}\right\rangle = \alpha \left( v\right) \) for all \( v \in E \) . The Euclidean metric on the dual \( {\xi }^{ * } \) is that metric for which the above musical equivalences become isometric.
If \( {\xi }_{i} \) are Euclidean vector bundles over \( M \), the Euclidean metric on the bundle \( \operatorname{Hom}\left( {{\xi }_{1},{\xi }_{2}}\right) \) is the metric for which the equivalence \( {\xi }_{1}^{ * } \otimes {\xi }_{2} \cong \operatorname{Hom}\left( {{\xi }_{1},{\xi }_{2}}\right) \) becomes isometric.
(viii) The Euclidean metric on \( {\Lambda }_{k}\left( \xi \right) \) is the one given on decomposable elements by \( \left\langle {{u}_{1} \land \cdots \land {u}_{k},{v}_{1} \land \ldots {v}_{k}}\right\rangle = \det \left( \left\langle {{u}_{i},{v}_{j}}\right\rangle \right) \) .
EXERCISE 103. Show that \( f : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \) is an isometry (with respect to the canonical metric) iff there exist some \( A \in O\left( n\right) \) and \( b \in {\mathbb{R}}^{n} \) such that \( f\left( a\right) = {Aa} + b \) for all \( a \in {\mathbb{R}}^{n} \) .
EXERCISE 104. The length function of a curve \( c : J = \left\lbrack {a, b}\right\rbrack \rightarrow M \) in a Riemannian manifold \( M \) is given by \( {l}_{c}\left( t\right) = L\left( {c}_{\mid \left\lbrack {a, t}\right\rbrack }\right), a \leq t \leq b \) . If \( \phi : I \rightarrow J \) is a differentiable monotone function onto \( J \), the curve \( c \circ \phi : I \rightarrow M \) is called a reparametrization of \( c \) .
(a) Show that \( {l}_{c \circ \phi } = {l}_{c} \circ \phi \) if \( {\phi }^{\prime } \geq 0 \), and \( {l}_{c \circ \phi } = L\left( c\right) - {l}_{c} \circ \phi \) if \( {\phi }^{\prime } \leq 0 \) . In particular, the length of a curve is invariant under reparametrization.
(b) Suppose that \( c \) is a regular curve; i.e., \( \dot{c}\left( t\right) \neq 0 \) for all \( t \) . Prove that \( c \) may be reparametrized by arc-length, meaning there exists a reparametrization \( \widetilde{c} \) of \( c \) with \( {l}_{\widetilde{c}}\left( t\right) = t - a \) .
EXERCISE 105. Let \( {\xi }_{i} \) be Euclidean vector bundles over \( M, i = 1,2 \), and suppose \( L : E{\left( {\xi }_{1}\right) }_{p} \rightarrow E{\left( {\xi }_{2}\right) }_{p} \in \operatorname{Hom}\left( {{\xi }_{1},{\xi }_{2}}\right) \) . Show that \( {\left| L\right| }^{2} = \mathop{\sum }\limits_{i}{\left| L{v}_{i}\right| }^{2} \) , where \( \left\{ {v}_{i}\right\} \) denotes an orthonormal basis of \( E{\left( {\xi }_{1}\right) }_{p} \) .
## 2. Riemannian Connections
Recall from Examples and Remarks 2.1(vi) in Chapter 4 that a connection on a Euclidean vector bundle \( \left( {\xi ,\langle \rangle }\right)
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117_《微积分笔记》最终版_by零蛋大
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Definition 4.13
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Definition 4.13. An invertible measure-preserving transformation \( T \) of a probability space \( \left( {X,\mathcal{B}, m}\right) \) is a Kolmogorov automorphism \( \left( {K\text{-automorphism}}\right) \) if there exists a sub- \( \sigma \) -algebra \( \mathcal{K} \) of \( \mathcal{B} \) such that:
(i) \( \mathcal{N} \subset T\mathcal{K} \) .
(ii) \( \mathop{\bigvee }\limits_{{n = 0}}^{\infty }{T}^{n}\mathcal{N} \doteq \mathcal{B} \) .
(iii) \( \mathop{\bigcap }\limits_{{n = 0}}^{\infty }{T}^{-n}\mathcal{K} \doteq \mathcal{N} = \{ X,\phi \} \) .
We always assume \( {\mathcal{N}}^{r} \neq \mathcal{D} \) (since if not the identity is the only measure-algebra automorphism). Hence \( \mathcal{K} \neq T\mathcal{K} \) . In fact the space \( \left( {X,\mathcal{B}, m}\right) \) is usually taken to be a Lebesgue space
Theorem 4.30. Every Bernoulli automorphism is a Kolmogorov automorphism.
Proof. Let the state space for \( T \) be \( \left( {Y,\mathcal{F},\mu }\right) \) . If \( F \in \mathcal{F} \), let \( \widetilde{F} = \) \( \left\{ {\left\{ {x}_{n}\right\} \in X : {x}_{0} \in F}\right\} \in \mathcal{B} \) . Let \( \mathcal{G} = \{ \widetilde{F} : F \in \mathcal{F}\} \), which is called the time- \( {0\sigma } \) - algebra. Let \( \mathcal{K} = \mathop{\bigvee }\limits_{{i = - \infty }}^{0}{T}^{i}\mathcal{G} \) . We now verify that \( \mathcal{K} \) satisfies the conditions for a Kolmogorov automorphism.
(i) \( \mathcal{K} = \mathop{\bigvee }\limits_{{i = - \infty }}^{0}{T}^{i}\mathcal{G} \subset \mathop{\bigvee }\limits_{{i = - \infty }}^{1}{T}^{i}\mathcal{G} = T\mathcal{K} \) .
(ii) \( \mathop{\bigvee }\limits_{{n = 0}}^{\infty }{T}^{n}\mathcal{K} = \mathop{\bigvee }\limits_{{n = 0}}^{\infty }\mathop{\bigvee }\limits_{{i = - n}}^{n}{T}^{i}\mathcal{G} = \mathop{\bigvee }\limits_{{-\prime }}^{\infty }{T}^{i}\mathcal{G} = \mathcal{B} \) by definition of \( \mathcal{B} \) .
(iii) We have to show \( {\bigcap }_{0}^{\infty }{T}^{-n}\mathcal{K} \doteq .\mathcal{V} = \{ X,\phi \} \) . Fix \( A \in {\bigcap }_{0}^{\infty }{T}^{-n}\mathcal{K} = \) \( \mathop{\bigcap }\limits_{{n = 0}}^{\prime }{}^{\prime } = 0\bigvee {}^{-n}{}_{f}{T}^{i}\mathcal{G} \) . Let \( B \in \mathop{\bigvee }\limits_{{k = j}}^{\prime }{}^{n}{}_{f}{T}^{k}\mathcal{G} \), for some fixed \( j \in Z \) . Since \( A \in \mathop{\bigvee }\limits_{{i < j}} \) \( {T}^{i}\mathcal{G}, A \) and \( B \) are independent, and therefore \( m\left( {A \cap B}\right) = m\left( A\right) m\left( B\right) \) . The collection of all sets \( B \) for which \( m\left( {A \cap B}\right) = m\left( A\right) m\left( B\right) \) is a monotone class, and, by the above, contains \( \mathop{\bigcup }\limits_{{j = - \alpha }}^{\infty }\mathop{\bigvee }\limits_{{k = j}}^{\infty }{T}^{k}\mathcal{G} \) . Therefore \( \forall B \in \mathcal{B}, m\left( {A \cap B}\right) = \) \( m\left( A\right) m\left( B\right) \) . Put \( B = A \), then \( m\left( A\right) = m{\left( A\right) }^{2} \) which implies \( m\left( A\right) = 0 \) or 1 . Hence
\[
\mathop{\bigcap }\limits_{{n = 0}}^{\infty }{T}^{-n}\mathcal{K} \doteq \mathcal{N}
\]
It was an open problem from 1958 to 1969 as to whether the converse of Theorem 4.30 was true, i.e., whether a Kolmogorov automorphism acting on a Lebesgue space is a Bernoulli automorphism. This was shown to be false by Ornstein.
Theorem 4.31 (Ornstein). There is an example of a Kolmogorov automorphism \( T \) which is not a Bernoulli automorphism.
Corollary 4.31.1. Entropy is not a complete invariant for the class of Kolmogorov automorphisms.
Proof. Let \( T \) be the example of Ornstein. By Corollary 4.14.4 \( h\left( T\right) > 0 \) . Choose a Bernoulli automorphism \( S \) with \( h\left( S\right) = h\left( T\right) \) . \( S \) and \( T \) are not isomorphic.
The following results show that the class of Kolmogorov automorphisms does not share all the properties the class of Bernoulli automorphisms enjoys. The proofs are given in the references cited.
## Theorem 4.32
(i) There are uncountably many non-conjugate Kolmogorov automorphisms with the same entropy (Ornstein and Shields [1]).
(ii) There is a Kolmogorov automorphism \( T \) not conjugate to its inverse \( {T}^{-1} \) (Ornstein and Shields [1]).
(iii) There is a Kolmogorov automorphism which has no n-th roots for any \( n \geq 2 \) (Clark [1]).
(iv) There are non-conjugate Kolmogorov automorphisms \( T, S \) with \( {T}^{2} = {S}^{2} \) (Rudolf [1]).
(v) There are two non-conjugate Kolmogorov automorphisms each of which is a factor of the other (Polit [1] and Rudolf [2]).
## Remarks
(1) Statement (ii) of Theorem 4.32 contrasts with the behaviour of ergodic transformations with pure point spectrum (see Corollary 3.4.1).
(2) Ornstein's example for Theorem 4.31 is defined by induction and so is fairly complicated to describe. It is therefore important to check whether the more "natural" examples of Kolmogorov automorphisms are Bernoulli automorphisms or not. We consider some of these at the end of this section and give an (casy to describe) example of a Kolmogorov automorphism that was recently shown not to be a Bernoulli automorphism.
(3) Sinai has proved that if \( T \) is an ergodic invertible measure-preserving transformation of a Lebesgue space \( \left( {X,\mathcal{B}, m}\right) \) with \( h\left( T\right) > 0 \) and if \( S \) is a Bernoulli automorphism with \( h\left( S\right) \leq h\left( T\right) \) then there exists a measure-preserving transformation \( \phi \) such that \( {\phi T} = {S\phi } \), i.e., \( S \) is a factor of \( T \) (see Rohlin [3], p. 45).
The next theorem shows that all Kolmogorov automorphisms are spectrally the same.
Theorem 4.33 (Rohlin). If \( \left( {X,\mathcal{B}, m}\right) \) is a probability space with a countable basis then any Kolmogorov automorphism \( T : X \rightarrow X \) has countable Lebesgue spectrum.
Proof. Recall that we are assuming \( \mathcal{A} \neq \{ X,\phi \} = \mathcal{N} \) . We have (i) \( \mathcal{K} \subset T\mathcal{K} \) , (ii) \( \bigvee {T}^{n}\mathcal{K} \circeq \mathcal{A} \) ,(iii) \( \bigcap {T}^{-n}\mathcal{K} \circeq \mathcal{N} \) . We split the proof into three parts:
(a) We first show that \( \mathcal{K} \) has no atoms, i.e., if \( C \in \mathcal{K} \) and \( m\left( C\right) > 0 \) then \( \exists D \in \mathcal{K} \) with \( D \subset C \) and \( m\left( D\right) < m\left( C\right) \) .
Suppose \( C \) is an atom of \( \mathcal{K} \) with \( m\left( C\right) > 0 \) . Then \( {TC} \) is an atom of \( T\mathcal{K} \) and since \( \mathcal{K} \subset T\mathcal{K} \) either \( {TC} \subset C \) or \( m\left( {C \cap {TC}}\right) = 0 \) . If \( {TC} \subset C \) then \( {TC} \doteq C \) since both sets have the same measure so that \( C \in \mathop{\bigcap }\limits_{{n = 0}}^{\infty }{T}^{-n}\mathcal{K} \) and therefore \( m\left( C\right) = 1 \) . Hence \( \mathcal{K} \doteq \mathcal{N} \) so \( \mathcal{D} \doteq \mathcal{N} \), a contradiction. On the other hand, suppose \( m\left( {{TC} \cap C}\right) = 0 \) . Then either for some \( k > 0{T}^{k}C \in C \) (and we use the above proof to get a contradiction) or \( m\left( {{T}^{k}C \cap C}\right) = 0\forall k > 0 \) and then \( C \cup {TC} \cup {T}^{2}C \cup \) has infinite measure, a contradiction.
(b) Let \( \mathcal{K} = \left\{ {f \in {L}^{2}\left( m\right) : f}\right. \) is \( \mathcal{K} \) -measurable \( \} \) . Then \( {U}_{T}\mathcal{K} \subset \mathcal{H} \) . Let \( \mathcal{M} = V \oplus {U}_{T}\mathcal{H} \) . From \( {U}_{T}^{-n}\mathcal{H} = {\bigoplus }_{-n}^{m}{U}_{T}^{i}V \oplus {U}_{T}^{m + 1}\mathcal{H}\left( {n, m > 0}\right) \) it follows that \( {L}^{2}\left( m\right) = {\bigoplus }_{n = 0}^{\infty }{U}_{T}^{n}V \oplus C \) where \( C \) is the subspace of constants. It
- suffices to show \( V \) is infinite-dimensional since if \( \left\{ {{f}_{1},{f}_{2},{f}_{3},\ldots }\right\} \) is a basis for \( V \), then \( \left\{ {{f}_{0} \equiv 1,{U}_{\Gamma }^{n}{f}_{j} : n \in Z, j > 0}\right\} \) is a basis for \( {L}^{2}\left( m\right) \) .
(c) We now show \( V \) is infinite-dimensional. Since \( T\mathcal{K} \neq \mathcal{K} \) (we are assuming \( \mathcal{B} \neq \mathcal{A} + 1 \) ) we know \( V \neq \{ 0\} \) . Let \( g \in V, g \neq 0 \) and then \( G = \) \( \{ x : g\left( x\right) \neq 0\} \) satisfies \( m\left( G\right) > 0 \) . Since \( g \) is \( \mathcal{K} \) -measurable we have \( G \in \mathcal{K} \) and using (a) we know \( {\chi }_{G}\mathcal{H} = \left\{ {{\chi }_{G}f : f \in \mathcal{H}}\right\} \) is infinite-dimensional. Also \( {\chi }_{G}\mathcal{H} = {V}^{\prime } \oplus {\chi }_{G}{U}_{T}\mathcal{H} \) where \( {V}^{\prime } \subset V \) so either \( {V}^{\prime } \) is infinite-dimensional (and hence \( V \) is) or \( {\chi }_{G}{U}_{T}\mathcal{H} \) is infinite-dimensional. In this second case there is a linearly independent sequence of functions \( \left\{ {{\chi }_{G}{U}_{T}{f}_{n}}\right\} \) where the \( {f}_{n} \) are bounded functions in \( \mathcal{H} \) . Then \( \left\{ {g{U}_{T}{f}_{n}}\right\} \) are linearly independent in \( \mathcal{H} \) . It suffices to show these functions are in \( V \) . But if \( f \in \mathcal{H} \) then
\[
\left( {g{U}_{T}{f}_{n},{U}_{T}f}\right) = \left( {g,{U}_{T}\left( {f{\bar{f}}_{n}}\right) }\right) = 0
\]
so \( g{U}_{T}{f}_{n} \in V \) .
Corollary 4.33.1 A Kolmogorov automorphism is strong-mixing.
Proof. By Theorem 2.12.
Kolmogorov automorphisms are connected to entropy theory by the following result (half of which was proved by Pinsker).
Theorem 4.34 (Rohlin and Sinai, see Rohlin [3]). Let \( \left( {X,\mathcal{B}, m}\right) \) be a Lebesgue space and let \( T : X \rightarrow X \) be an invertible measure-prescrving transformation. Then \( T \) is a Kolmogorov automorphism iff \( h\left( {T,\mathcal{A}}\right) > 0 \) for all finite \( \mathcal{A} \neq \mathcal{N} \) .
## Remarks
(1) One says that \( T \) has completely positive entropy wh
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1077_(GTM235)Compact Lie Groups
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Definition 2.30
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Definition 2.30. If \( H \) is a Lie subgroup of a Lie group \( G \) and \( V \) is a representation of \( G \), write \( {\left. V\right| }_{H} \) for the representation of \( H \) on \( V \) given by restricting the action of \( G \) to \( H \) .
For the remainder of this section, view \( O\left( {n - 1}\right) \) as a Lie subgroup of \( O\left( n\right) \) via the embedding \( g \rightarrow \left( \begin{array}{ll} 1 & 0 \\ 0 & g \end{array}\right) \) .
Lemma 2.31.
\[
{\left. {\mathcal{H}}_{m}\left( {\mathbb{R}}^{n}\right) \right| }_{O\left( {n - 1}\right) } \cong {\mathcal{H}}_{m}\left( {\mathbb{R}}^{n - 1}\right) \oplus {\mathcal{H}}_{m - 1}\left( {\mathbb{R}}^{n - 1}\right) \oplus \cdots \oplus {\mathcal{H}}_{0}\left( {\mathbb{R}}^{n - 1}\right) .
\]
Proof. Any \( p \in {V}_{m}\left( {\mathbb{R}}^{n}\right) \) may be uniquely written as \( p = \mathop{\sum }\limits_{{k = 0}}^{m}{x}_{1}^{k}{p}_{k} \) with \( {p}_{k} \in \) \( {V}_{m - k}\left( {\mathbb{R}}^{n - 1}\right) \) where \( {\mathbb{R}}^{n} \) is viewed as \( \mathbb{R} \times {\mathbb{R}}^{n - 1} \) . Since \( O\left( {n - 1}\right) \) acts trivially on \( {x}_{1}^{k} \) ,
(2.32)
\[
{\left. {V}_{m}\left( {\mathbb{R}}^{n}\right) \right| }_{O\left( {n - 1}\right) } \cong {\bigoplus }_{k = 0}^{m}{V}_{m - k}\left( {\mathbb{R}}^{n - 1}\right) .
\]
Applying Equation 2.28 first (restricted to \( O\left( {n - 1}\right) \) ) and then Equation 2.32, we get
\[
{\left. {\left. {V}_{m}\left( {\mathbb{R}}^{n}\right) \right| }_{O\left( {n - 1}\right) } \cong {\mathcal{H}}_{m}\left( {\mathbb{R}}^{n}\right) \right| }_{O\left( {n - 1}\right) } \oplus {\bigoplus }_{k = 0}^{m - 2}{V}_{m - 2 - k}\left( {\mathbb{R}}^{n - 1}\right) .
\]
Applying Equation 2.32 first and then Equation 2.28 yields
\[
{\left. {V}_{m}\left( {\mathbb{R}}^{n}\right) \right| }_{O\left( {n - 1}\right) } \cong {\bigoplus }_{k = 0}^{m}\left\lbrack {{\mathcal{H}}_{m - k}\left( {\mathbb{R}}^{n - 1}\right) \oplus {V}_{m - 2 - k}\left( {\mathbb{R}}^{n - 1}\right) }\right\rbrack
\]
\[
= \left\lbrack {{\bigoplus }_{k = 0}^{m}{\mathcal{H}}_{m - k}\left( {\mathbb{R}}^{n - 1}\right) }\right\rbrack \oplus \left\lbrack {{\bigoplus }_{k = 0}^{m - 2}{V}_{m - 2 - k}\left( {\mathbb{R}}^{n - 1}\right) }\right\rbrack .
\]
The proof is now finished by Lemma 2.29.
Theorem 2.33. \( {\mathcal{H}}_{m}\left( {\mathbb{R}}^{n}\right) \) is an irreducible \( O\left( n\right) \) -module and, in fact, is irreducible under \( \operatorname{SO}\left( n\right) \) for \( n \geq 3 \) .
Proof. See Exercise 2.31 for the case of \( n = 2 \) . In this proof assume \( n \geq 3 \) .
\( {\left. {\mathcal{H}}_{m}\left( {\mathbb{R}}^{n}\right) \right| }_{{SO}\left( {n - 1}\right) } \) contains, up to scalar multiplication, a unique \( {SO}\left( {n - 1}\right) \) - invariant function: If \( f \in {\mathcal{H}}_{m}\left( {\mathbb{R}}^{n}\right) \) is nonzero and \( {SO}\left( n\right) \) -invariant, then it is constant on each sphere in \( {\mathbb{R}}^{n} \) and thus a function of the radius. Homogeneity implies that \( f\left( x\right) = C{\left| x\right| }^{m} \) for some nonzero constant. It is trivial to check the condition that \( {\Delta f} = 0 \) now forces \( m = 0 \) . Thus only \( {\mathcal{H}}_{0}\left( {\mathbb{R}}^{n}\right) \) contains a nonzero \( {SO}\left( n\right) \) -invariant function. The desired result now follows from the previous observation and Lemma 2.31.
If \( V \) is a finite-dimensional \( {SO}\left( n\right) \) -invariant subspace of continuous functions on \( {S}^{n - 1} \), then \( V \) contains a nonzero \( {SO}\left( {n - 1}\right) \) -invariant function: Here the action of \( {SO}\left( n\right) \) on \( V \) is, as usual, given by \( \left( {gf}\right) \left( s\right) = f\left( {{g}^{-1}s}\right) \) . Since \( {SO}\left( n\right) \) acts transitively on \( {S}^{n - 1} \) and \( V \) is nonzero invariant, there exists \( f \in V \), so \( f\left( {1,0,\ldots ,0}\right) \neq 0 \) . Define \( \widetilde{f}\left( s\right) = {\int }_{{SO}\left( {n - 1}\right) }f\left( {gs}\right) {dg} \) . If \( \left\{ {f}_{i}\right\} \) is a basis of \( V \), then \( f\left( {gs}\right) = \left( {{g}^{-1}f}\right) \left( s\right) \) and so may be written as \( f\left( {gs}\right) = \mathop{\sum }\limits_{i}{c}_{i}\left( g\right) {f}_{i}\left( s\right) \) for some smooth functions \( {c}_{i} \) . By integrating, it follows that \( \widetilde{f} \in V \) . From the definition, it is clear that \( \widetilde{f} \) is \( {SO}\left( {n - 1}\right) \) - invariant. It is nonzero since \( \widetilde{f}\left( {1,0,\ldots ,0}\right) = f\left( {1,0,\ldots ,0}\right) \) .
\( {\mathcal{H}}_{m}\left( {\mathbb{R}}^{n}\right) \) is an irreducible \( {SO}\left( n\right) \) -module: Suppose \( {\mathcal{H}}_{m}\left( {\mathbb{R}}^{n}\right) = {V}_{1} \oplus {V}_{2} \) for proper \( {SO}\left( n\right) \) -invariant subspaces. By homogeneity, restricting functions in \( {V}_{i} \) from \( {\mathbb{R}}^{n} \) to \( {S}^{n - 1} \) is injective. Hence, both \( {V}_{1} \) and \( {V}_{2} \) contain independent \( {SO}\left( {n - 1}\right) \) - invariant functions. But this contradicts the fact that \( {\mathcal{H}}_{m}\left( {\mathbb{R}}^{n}\right) \) has only one independent \( {SO}\left( {n - 1}\right) \) -invariant function.
A relatively small dose of functional analysis (Exercise 3.14) can be used to further show that \( {L}^{2}\left( {S}^{n - 1}\right) = {\left. {\bigoplus }_{m = 0}^{\infty }{\mathcal{H}}_{m}\left( {\mathbb{R}}^{n}\right) \right| }_{{S}^{n - 1}} \) (Hilbert space direct sum) and that \( {\left. {\mathcal{H}}_{m}\left( {\mathbb{R}}^{n}\right) \right| }_{{S}^{n - 1}} \) is the eigenspace of the Laplacian on \( {S}^{n - 1} \) with eigenvalue \( - m\left( {n + m - 2}\right) \) .
## 2.3.3 Spin and Half-Spin Representations
The spin representation \( S = \bigwedge W \) of \( {\operatorname{Spin}}_{n}\left( \mathbb{R}\right) \) for \( n \) odd and the half-spin representations \( {S}^{ \pm } = \mathop{\bigwedge }\limits^{ \pm }W \) for \( n \) even were constructed in \( §{2.1.2.4} \), where \( W \) is a maximal isotropic subspace of \( {\mathbb{C}}^{n} \) . This section shows that these representations are irreducible.
For \( n \) even with \( n = {2m} \), let \( W = \left\{ {\left( {{z}_{1},\ldots ,{z}_{m}, i{z}_{1},\ldots, i{z}_{m}}\right) \mid {z}_{k} \in \mathbb{C}}\right\} \) and \( {W}^{\prime } = \left\{ {\left( {{z}_{1},\ldots ,{z}_{m}, - i{z}_{1},\ldots , - i{z}_{m}}\right) \mid {z}_{k} \in \mathbb{C}}\right\} \) . Identify \( W \) with \( {\mathbb{C}}^{m} \) by projecting onto the first \( m \) coordinates. For \( x = \left( {{x}_{1},\ldots ,{x}_{m}}\right) \) and \( y = \left( {{y}_{1},\ldots ,{y}_{m}}\right) \) in \( {\mathbb{R}}^{m} \), let \( \left( {x, y}\right) = \left( {{x}_{1},\ldots ,{x}_{m},{y}_{1},\ldots ,{y}_{m}}\right) \in {\mathbb{R}}^{n} \) . In particular, \( \left( {x, y}\right) = \frac{1}{2}(x - {iy}, i(x - \) \( {iy})) + \frac{1}{2}\left( {x + {iy}, - i\left( {x + {iy}}\right) }\right) \) . Using Definition 2.7, the identification of \( {\mathbb{C}}^{m} \) with \( W \) , and noting \( \left( {\left( {a, - {ia}}\right) ,\left( {b,{ib}}\right) }\right) = 2\left( {a, b}\right) \), the spin action of \( {\operatorname{Spin}}_{2m}\left( \mathbb{R}\right) \) on \( \mathop{\bigwedge }\limits^{ \pm }{\mathbb{C}}^{m} \cong \) \( {S}^{ \pm } \) is induced by having \( \left( {x, y}\right) \) act as
(2.34)
\[
\frac{1}{2}\epsilon \left( {x - {iy}}\right) - {2\iota }\left( {x + {iy}}\right)
\]
For \( n \) odd with \( n = {2m} + 1 \), take \( W = \left\{ {\left( {{z}_{1},\ldots ,{z}_{m}, i{z}_{1},\ldots, i{z}_{m},0}\right) \mid {z}_{k} \in \mathbb{C}}\right\} \) , \( {W}^{\prime } = \left\{ {\left( {{z}_{1},\ldots ,{z}_{m}, - i{z}_{1},\ldots , - i{z}_{m},0}\right) \mid {z}_{k} \in \mathbb{C}}\right\} \), and \( {e}_{0} = \left( {0,\ldots ,0,1}\right) \) . As above, identify \( W \) with \( {\mathbb{C}}^{m} \) by projecting onto the first \( m \) coordinates. For \( x = \left( {{x}_{1},\ldots ,{x}_{m}}\right) \) and \( y = \left( {{y}_{1},\ldots ,{y}_{m}}\right) \) in \( {\mathbb{R}}^{m} \) and \( u \in \mathbb{R} \), let \( \left( {x, y, u}\right) = \) \( \left( {{x}_{1},\ldots ,{x}_{m},{y}_{1},\ldots ,{y}_{m}, u}\right) \in {\mathbb{R}}^{n} \) . In particular, \( \left( {x, y, u}\right) = \frac{1}{2}\left( {x - {iy}, i\left( {x - {iy}}\right) ,0}\right) + \) \( \frac{1}{2}\left( {x + {iy}, - i\left( {x + {iy}}\right) ,0}\right) + \left( {0,0, u}\right) \) . Using Definition 2.7 and the identification of \( {\mathbb{C}}^{m} \) with \( W \), the spin action of \( {\operatorname{Spin}}_{{2m} + 1}\left( \mathbb{R}\right) \) on \( \bigwedge {\mathbb{C}}^{m} \cong S \) is induced by having \( \left( {x, y, u}\right) \) act as
(2.35)
\[
\frac{1}{2}\epsilon \left( {x - {iy}}\right) - {2\iota }\left( {x + {iy}}\right) + {\left( -1\right) }^{\deg }{m}_{iu}.
\]
Theorem 2.36. For \( n \) even, the half-spin representations \( {S}^{ \pm } \) of \( {\operatorname{Spin}}_{n}\left( \mathbb{R}\right) \) are irreducible. For \( n \) odd, the spin representation \( S \) of \( {\operatorname{Spin}}_{n}\left( \mathbb{R}\right) \) is irreducible.
Proof. Using the standard basis \( {\left\{ {e}_{j}\right\} }_{j = 1}^{n} \), calculate
\[
\left( {{e}_{j} \pm i{e}_{j + m}}\right) \left( {{e}_{k} \pm i{e}_{k + m}}\right) = {e}_{j}{e}_{k} \pm i\left( {{e}_{j}{e}_{k + m} + {e}_{j + m}{e}_{k}}\right) - {e}_{j + m}{e}_{k + m}
\]
for \( 1 \leq j, k \leq m \) . Since \( {e}_{j}{e}_{k},{e}_{j}{e}_{k + m},{e}_{j + m}{e}_{k} \), and \( {e}_{j + m}{e}_{k + m} \) lie in \( {\operatorname{Spin}}_{n}\left( \mathbb{R}\right) \), Equations 2.34 and 2.35 imply that the operators \( \epsilon \left( {e}_{j}\right) \epsilon \left( {e}_{k}\right) \) and \( \iota \left( {e}_{j}\right) \iota \left( {e}_{k}\right) \) on \( \bigwedge {\mathbb{C}}^{m} \) are achieved by linear combinations of the action of elements of \( {\operator
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1068_(GTM227)Combinatorial Commutative Algebra
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Definition 7.8
|
Definition 7.8 A subset \( T \subseteq Q \) is called an ideal of \( Q \) if \( Q + T \subseteq T \) . A subsemigroup \( F \) of \( Q \) is called a face if the complement \( Q \smallsetminus F \) is an ideal of \( Q \) . The affine semigroup \( Q \) is pointed if its only unit is \( \mathbf{0} \), where a unit is an element \( \mathbf{a} \in Q \) whose additive inverse \( - \mathbf{a} \) also lies in \( Q \) .
By definition, then, \( F \) is a face precisely when each pair of elements \( \mathbf{a},\mathbf{b} \in Q \) satisfies
\[
\mathbf{a} + \mathbf{b} \in F\; \Leftrightarrow \;\mathbf{a} \in F\text{ and }\mathbf{b} \in F.
\]
(7.2)
The unique smallest face of \( Q \) is its group \( Q \cap \left( {-Q}\right) \) of units.
The \( {\mathbb{N}}^{n} \) -graded algebra we did over the polynomial ring \( \mathbb{k}\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) in Part I generalizes to the \( {\mathbb{Z}}^{d} \) -graded algebra of affine semigroups rings.
Definition 7.9 A monomial in the semigroup ring \( \mathbb{k}\left\lbrack Q\right\rbrack \) is an element of the form \( {\mathbf{t}}^{\mathbf{a}} \) for \( \mathbf{a} \in Q \) . An ideal \( I \subseteq \mathbb{k}\left\lbrack Q\right\rbrack \) is a monomial ideal if it is generated by monomials.
For any subset \( T \) of \( Q \), we write \( \mathbb{k}\{ T\} \) for the \( \mathbb{k} \) -linear span of the monomials \( {\mathbf{t}}^{\mathbf{a}} \) with \( \mathbf{a} \in T \) . Thus \( I \) is a monomial ideal if and only if \( I = \) \( \mathbb{k}\{ T\} \) for some ideal \( T \) of \( Q \), or equivalently, if \( I \) is homogeneous with respect to the tautological \( A \) -grading on \( \mathbb{k}\left\lbrack Q\right\rbrack \), which is defined by \( \deg \left( {\mathbf{t}}^{\mathbf{a}}\right) = \mathbf{a} \) . For any subset \( F \) of \( Q \) we abbreviate \( {P}_{F} = \mathbb{k}\{ Q \smallsetminus F\} \) .
All issues concerning primality and primary decomposition are compatible with the \( A \) -grading on \( \mathbb{k}\left\lbrack Q\right\rbrack \) . For instance, to test whether a homogeneous ideal \( I \) is prime or primary, it suffices to check homogeneous polynomials in the definition of "prime" or "primary". Also, the associated primes of an \( A \) -homogeneous ideal are automatically \( A \) -homogeneous. This follows from [Eis95, Exercise 3.5], because the grading group \( A \) can be totally ordered; see Proposition 8.11.
Lemma 7.10 A subset \( F \) of \( Q \) is a face if and only if \( {P}_{F} \) is a prime ideal.
Proof. The subspace \( {P}_{F} \) is an ideal if and only if the implication " \( \Rightarrow \) " holds in (7.2). Assuming that this is the case, the implication " \( \Leftarrow \) " says that \( {\mathbf{t}}^{\mathbf{a} + \mathbf{b}} \in {P}_{F} \) implies \( {\mathbf{t}}^{\mathbf{a}} \in {P}_{F} \) or \( {\mathbf{t}}^{\mathbf{b}} \in {P}_{F} \) . The latter condition is equivalent to \( {P}_{F} \) being a prime ideal, by the above remark about the \( A \) -grading.
Definition 7.11 If \( F \) is a face of \( Q \), then the localization of \( Q \) along \( F \) is the semigroup \( Q - F = Q + \mathbb{Z}F \) consisting of all differences \( \mathbf{a} - \mathbf{b} \) with \( \mathbf{a} \in Q \) and \( \mathbf{b} \in F \) . The quotient semigroup \( Q/F \) is the image of \( Q \) in the group \( {\mathbb{Z}}^{d}/\mathbb{Z}F \) .
The map \( Q \rightarrow Q/F \) always factors through the localization \( Q \rightarrow Q - F \) , and the quotient semigroup \( Q/F = \left( {Q - F}\right) /\mathbb{Z}F \) is always pointed.
The terms "face" and "pointed" refer to the relationship between affine semigroups and cones, whose polyhedral geometric definitions we recall. A (polyhedral) cone in \( {\mathbb{R}}^{d} \) is the intersection of finitely many closed linear half-spaces in \( {\mathbb{R}}^{d} \), each of whose bounding hyperplanes contains the origin. We write \( \dim \left( C\right) \) for the dimension of the linear span of \( C \) . Every polyhedral cone \( C \) is finitely generated: there exist \( {\mathbf{c}}_{1},\ldots ,{\mathbf{c}}_{r} \in {\mathbb{R}}^{d} \) with
\[
C = \left\{ {{\lambda }_{1}{\mathbf{c}}_{1} + \cdots + {\lambda }_{r}{\mathbf{c}}_{r} \mid {\lambda }_{1},\ldots ,{\lambda }_{r} \in {\mathbb{R}}_{ \geq 0}}\right\} .
\]
We call the cone \( C \) rational if \( {\mathbf{c}}_{1},\ldots ,{\mathbf{c}}_{r} \) can be chosen to have rational coordinates, and we say that \( C \) is simplicial if \( r = \dim \left( C\right) \) generators suffice. A face of a cone \( C \) is a subset of the form \( H \cap C \) in which \( H \) is the bounding hyperplane of a closed half-space \( {H}_{ \geq 0} \) that contains \( C \) . The unique smallest face of \( C \) is the lineality space \( C \cap \left( {-C}\right) \) . We call the cone \( C \) pointed if \( C \cap \left( {-C}\right) = \{ \mathbf{0}\} \) .
Lemma 7.12 The map \( F \mapsto {\mathbb{R}}_{ \geq 0}F \) is a bijection from the set of faces of the semigroup \( Q \) to the set of faces of the cone \( {\mathbb{R}}_{ \geq 0}Q \) . In particular, the semigroup \( Q \) is pointed if and only if the associated cone \( {\mathbb{R}}_{ \geq 0}Q \) is pointed.
Proof. Let \( F \) be a subset of \( Q \) and consider the following linear system of equations and inequalities in an indeterminate vector \( \mathbf{w} \in {\mathbb{R}}^{d} \) :
\[
\mathbf{w} \cdot \mathbf{a} = 0\text{ for }\mathbf{a} \in F\;\text{ and }\;\mathbf{w} \cdot \mathbf{b} > 0\text{ for }\mathbf{b} \in Q \smallsetminus F.
\]
If this system has a solution \( \mathbf{w} \), then \( F \) is a face of \( Q \) by definition. If this system has no solution, then by Farkas' Lemma [Zie95, Proposition 1.7], there exists a linear combination \( \mathbf{a} \) of vectors in \( F \) that equals a positive linear combination of some vectors \( \mathbf{b} \in Q \smallsetminus F \) . The vector \( \mathbf{a} \) can be moved into \( F \) by adding a vector from \( F \), and hence we may assume a itself lies in \( F \) . Since \( F \) is a face, some vector \( \mathbf{b} \in Q \smallsetminus F \) lies in \( F \) as well, a contradiction.
The argument in the previous paragraph shows that a subset \( F \) of \( Q \) is a face if and only if it has the form \( F = H \cap Q \), where \( H \) is the bounding
Figure 7.1: The primes in \( \mathbb{k}\left\lbrack Q\right\rbrack \) for \( Q = \) the saturated cone over a square
hyperplane of a closed half-space \( {H}_{ \geq 0} \) containing \( Q \) . If \( F \) is a face of \( Q \), then \( {\mathbb{R}}_{ \geq 0}F \) is a face of \( C \), and conversely, if \( {F}^{\prime } \) is a face of \( C \), then \( {F}^{\prime } \cap Q \) is a face of \( Q \) . These two maps are inverses to each other, for if \( H \) is a hyperplane satisfying \( H \cap Q = F \), then \( F \subseteq {\mathbb{R}}_{ \geq 0}F \subseteq H \), whence \( Q \cap {\mathbb{R}}_{ \geq 0}F = F \) .
Lemma 7.12 implies that affine semigroups \( Q \) have only finitely many faces \( F \), so affine semigroup rings \( \mathbb{k}\left\lbrack Q\right\rbrack \) have only finitely many homogeneous prime ideals \( {P}_{F} \) . Computing this list of prime ideals is a valuable preprocessing step in dealing with a semigroup ring. This will be important in our study of injective modules and injective resolutions in Chapter 11.
Example 7.13 Every monomial ideal \( I \) in any affine semigroup ring \( \mathbb{k}\left\lbrack Q\right\rbrack \) is an intersection of monomial ideals \( {I}_{F} \), at most one for each face \( F \), with \( {I}_{F} \) primary to \( {P}_{F} \) . We will prove this in Corollary 11.5, which rests mainly on Proposition 8.11, where we indicate how to derive a more general statement from [Eis95, Exercise 3.5]. For now, we present a 3-dimensional example that also serves to illustrate the other concepts from this section.
Let \( Q \) be the subsemigroup of \( {\mathbb{Z}}^{3} \) generated by \( \left( {1,0,0}\right) ,\left( {1,1,0}\right) ,\left( {1,1,1}\right) \) , \( \left( {1,0,1}\right) \) . Its semigroup ring equals
\[
\mathbb{k}\left\lbrack Q\right\rbrack \cong \mathbb{k}\left\lbrack {a, b, c, d}\right\rbrack /\langle {ac} - {bd}\rangle .
\]
The cone \( {\mathbb{R}}_{ \geq 0}Q \) is the cone over a square and therefore pointed. It has nine faces: one of dimension 0 , four of dimension 1 , and four of dimension 2 . Hence there are precisely nine homogeneous prime ideals in \( \mathbb{k}\left\lbrack Q\right\rbrack \) . They are
codim 3 primes: \( \;{P}_{\mathcal{O}} = \langle a, b, c, d\rangle \)
codim 2 primes: \( {P}_{a} = \langle b, c, d\rangle ,{P}_{b} = \langle a, c, d\rangle ,{P}_{c} = \langle a, b, d\rangle ,{P}_{d} = \langle a, b, c\rangle \)
codim 1 primes: \( {P}_{ab} = \langle c, d\rangle ,{P}_{bc} = \langle d, a\rangle ,{P}_{cd} = \langle a, b\rangle ,{P}_{da} = \langle b, c\rangle \) .
Figure 7.2: Primary decomposition in a 2-dimensional semigroup
The faces of \( {\mathbb{R}}_{ \geq 0}Q \) are labeled in Fig. 7.1, where (for example) the ray labeled \( {ab} \) contains all of the monomials outside of \( {P}_{ab} \) .
Computing intersections of monomial ideals in affine semigroup rings is more complicated than in a polynomial ring. Certain bad behavior arises, such as the fact that the intersection of two principal ideals is generally not principal. For instance, in our example, for any \( i \in \mathbb{N} \) ,
\[
\left\langle {a}^{i}\right\rangle \cap \left\langle {d}^{i}\right\rangle = \left\langle {{a}^{i}{d}^{i},{a}^{i - 1}b{d}^{i},{a}^{i - 2}{b}^{2}{d}^{i},{a}^{i - 3}{b}^{3}{d}^{i},\ldots, a{b}^{i - 1}{d}^{i},{b}^{i}{d}^{i}}\right\rangle .
\]
An arbitrary principal monomial ideal here has
\[
\left\langle {{a}^{i}{b}^{j}{c}^{k}{d}^{l}}\right\rangle = \la
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1059_(GTM219)The Arithmetic of Hyperbolic 3-Manifolds
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Definition 0.7.1
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Definition 0.7.1 The field \( K \) is said to be complete at \( v \) if every Cauchy sequence in \( K \) converges to an element of \( K \) .
For a number field \( k \), we have indicated how to obtain all valuations. The field \( k \) is not complete with respect to any of these valuations, but for each valuation \( v \), one can construct a field \( {k}_{v} \) in which \( k \) embeds, such that the valuation \( v \) extends to \( {k}_{v} \) and \( {k}_{v} \) is complete with respect to this extended valuation. These field are the completions of \( k \) .
For the moment, consider any field \( K \) with a valuation \( v \) . Let \( \mathcal{C} \) be the set of all Cauchy sequences in \( K \) and let \( \mathcal{N} \) be the subset of null sequences,(i.e., those that converge to 0 ). Under pointwise addition and multiplication, \( \mathcal{C} \) is a commutative ring with 1 and \( \mathcal{N} \) is an ideal of \( \mathcal{C} \) . For \( x \in K \), the mapping \( x \mapsto \{ x\} + \mathcal{N} \), where \( \{ x\} \) is the constant sequence, defines an embedding of \( K \) in the quotient \( \widehat{K} \mathrel{\text{:=}} \mathcal{C}/\mathcal{N} \) . It can be shown that \( \widehat{K} \) is a field. (See Exercise 0.7, No. 1).
If \( \left\{ {a}_{n}\right\} \in \mathcal{C} \), then \( \left\{ {v\left( {a}_{n}\right) }\right\} \) is a Cauchy sequence in \( \mathbb{R} \) and so it has a limit. It then follows, defining \( \widehat{v} \) on \( \widehat{K} \) by
\[
\widehat{v}\left( {\left\{ {a}_{n}\right\} + \mathcal{N}}\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}v\left( {a}_{n}\right)
\]
that \( \widehat{v} \) is well-defined. Note that \( {\left. \widehat{v}\right| }_{K} = v \) . With some effort, the following can then be proved:
Theorem 0.7.2 The field \( \widehat{K} \) is complete with respect to \( \widehat{v} \) . Furthermore, it is unique. More generally, if \( \sigma : K \rightarrow L \) is a field embedding, where \( L \) has a valuation \( {v}_{1} \) with \( {v}_{1}\left( {\sigma \left( x\right) }\right) = v\left( x\right) \) for each \( x \in K \), then there is a unique embedding \( \widehat{\sigma } : \widehat{K} \rightarrow \widehat{L} \) such that \( {\widehat{v}}_{1}\left( {\widehat{\sigma }\left( x\right) }\right) = \widehat{v}\left( x\right) \) for all \( x \in \widehat{K} \) and the following diagram commutes:
\[
K\overset{\sigma }{ \rightarrow }L
\]
\[
\widehat{K}\;\overset{\widehat{\sigma }}{ \rightarrow }\;\widehat{L}
\]
Definition 0.7.3 The field \( \widehat{K} \) is called the completion of \( K \) at the valuation \( v \) .
The above theorem justifies calling this field the completion, as it is unique up to a valuation-preserving isomorphism. Equivalent valuations on \( K \) determine the same field \( \widehat{K} \) and the valuations extend to equivalent valuations on \( \widehat{K} \) . Furthermore, non-Archimedean valuations extend to non-Archimedean valuations by Exercise 0.6, No. 1 and Archimedean valuations extend to Archimedean valuations.
For these Archimedean valuations, we have the following theorem of Os-trowski:
Theorem 0.7.4 Let \( K \) be a field with an Archimedean valuation. If \( K \) is complete, then \( K \) is isomorphic to \( \mathbb{R} \) or \( \mathbb{C} \) and the valuation is equivalent to the usual absolute value.
Thus consider again a number field \( k \) and the places on \( k \) ,(i.e., the equivalence classes of valuations on \( k \), as described in Theorem 0.6.6).
Definition 0.7.5 If \( v \) is a valuation on \( k \), let \( {k}_{v} \) denote the completion of \( k \) at \( v \) . If \( v \) corresponds to a prime ideal \( \mathcal{P} \), we will also write this as \( {k}_{\mathcal{P}} \) . We use, if necessary, \( {i}_{v} \) or \( {i}_{\mathcal{P}} \) to denote an embedding of \( k \) into \( {k}_{v} \) or \( {k}_{\mathcal{P}} \) .
If \( v \) is Archimedean, then \( {k}_{v} \cong \mathbb{R} \) or \( \mathbb{C} \), by Theorem 0.7.4. Furthermore, if \( v \) belongs to the place corresponding to the embedding \( \sigma \), there will be an embedding \( {i}_{v} \) such that \( \widehat{v}\left( {{i}_{v}\left( x\right) }\right) = \left| {\sigma \left( x\right) }\right| \) .
If \( v \) is non-Archimedean, then \( v \) belongs to a place corresponding to a prime ideal \( \mathcal{P} \) . The field \( {k}_{\mathcal{P}} \) is usually referred to as a \( \mathcal{P} \) -adic field. The valuation ring of \( {k}_{\mathcal{P}} \) with respect to the extended valuation \( {\widehat{v}}_{\mathcal{P}} \) is the ring of \( \mathcal{P} \) -adic integers and is denoted by \( {R}_{\mathcal{P}} \) . Recall that the valuation ring \( R\left( {v}_{\mathcal{P}}\right) \) of \( k \) with respect to \( {v}_{\mathcal{P}} \) is a discrete valuation ring whose unique maximal ideal is generated by an element \( \pi \in {R}_{k} \) . The same can be proved for the ring \( {R}_{\mathcal{P}} \) . More precisely, the following holds:
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, \( {R}_{\mathcal{P}}/{i}_{\mathcal{P}}\left( \pi \right) {R}_{\mathcal{P}} \cong R\left( {v}_{\mathcal{P}}\right) /{\pi R}\left( {v}_{\mathcal{P}}\right) \), the residue field.
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
\[
0 \neq {\widehat{v}}_{\mathcal{P}}\left( \alpha \right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}{v}_{\mathcal{P}}\left( {a}_{n}\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}{c}^{{n}_{\mathcal{P}}\left( {a}_{n}\right) }.
\]
However, the sequence \( \left\{ {{c}^{n} : n \in \mathbb{Z}}\right\} \) is a discrete sequence, so that \( {\widehat{v}}_{\mathcal{P}}\left( \alpha \right) = \) \( {c}^{{n}_{0}} \) for some \( {n}_{0} \) . Also \( {\widehat{v}}_{\mathcal{P}}\left( {{i}_{\mathcal{P}}\left( \pi \right) }\right) = {v}_{\mathcal{P}}\left( \pi \right) = c \) .
Notation Because we have given a number of constructions related to the prime ideal \( \mathcal{P} \) of \( {R}_{k} \), we re-emphasise for clarity the notation for each of these constructions. Thus \( {v}_{\mathcal{P}} \) is the valuation on the number field \( k \) and \( R\left( {v}_{\mathcal{P}}\right) \) and \( \mathcal{P}\left( {v}_{\mathcal{P}}\right) \) are the valuation ring and the unique maximal ideal, respectively, of the valuation on \( k \) . The completion of \( k \) at \( {v}_{\mathcal{P}} \) is the \( \mathcal{P} \) - adic field \( {k}_{\mathcal{P}} \) and the unique extension of the valuation \( {v}_{\mathcal{P}} \) on \( k \) to \( {k}_{\mathcal{P}} \) is denoted \( {\widehat{v}}_{\mathcal{P}} \) . Subsequently, we may drop the hat. An embedding of \( k \) in \( {k}_{\mathcal{P}} \) is denoted by \( {i}_{\mathcal{P}} \) . The valuation ring of \( \mathcal{P} \) -adic integers of \( {\widehat{v}}_{\mathcal{P}} \) in \( {k}_{\mathcal{P}} \) is denoted by \( {R}_{\mathcal{P}} \) . We will denote its unique maximal ideal by \( \widehat{\mathcal{P}} \) and note that \( \widehat{\mathcal{P}} = \pi {R}_{\mathcal{P}} \), where we have identified \( \pi \) and its image \( {i}_{\mathcal{P}}\left( \pi \right) \) .
Definition 0.7.7 Such an element \( \pi \) as described in the above theorem, is called a uniformiser in \( {k}_{\mathcal{P}} \) . Thus a uniformiser in \( {k}_{\mathcal{P}} \) is an element of \( {R}_{k} \) (or \( R\left( {v}_{\mathcal{P}}\right) \), or \( {R}_{\mathcal{P}} \) ) such that \( {v}_{\mathcal{P}}\left( \pi \right) \) generates the group \( {v}_{\mathcal{P}}\left( {k}^{ * }\right) = {\widehat{v}}_{\mathcal{P}}\left( {k}_{\mathcal{P}}^{ * }\right) \) .
We can use this to give an alternative description of the elements of the \( \mathcal{P} \) -adic field \( {k}_{\mathcal{P}} \) as power series. Let \( \left\{ {c}_{i}\right\} \) be a set of coset representatives of the ideal \( \widehat{\mathcal{P}} \) in \( {R}_{\mathcal{P}} \), which can be identified with a set of coset representatives for the residue field. This set will thus have \( N\left( \mathcal{P}\right) \) elements and is always chosen so that 0 represents the zero coset.
Theorem 0.7.8 Every element \( \alpha \neq 0 \) in \( {k}_{\mathcal{P}} \) has a unique expression in the form
\[
\alpha = {\pi }^{r}\left( {\mathop{\sum }\limits_{{n = 0}}^{\infty }{c}_{{i}_{n}}{\pi }^{n}}\right)
\]
\( \left( {0.27}\right) \)
where \( {c}_{{i}_{0}} \neq 0 \) .
(See Exercise 0.7, No. 3.)
The finite prime \( \mathcal{P} \) of a number field \( k \) gives rise to a complete field \( {k}_{\mathcal{P}} \) . If \( \mathcal{Q} \) is a prime in a finite extension \( \ell \mid k \) which lies over \( \mathcal{P} \), then \( {v}_{\mathcal{Q}} \mid k \) is readily shown to be equivalent to \( {v}_{\mathcal{P}} \) . Thus there is an embedding \( \widehat{i} : {k}_{\mathcal{P}} \rightarrow {\ell }_{\mathcal{Q}} \) by Theorem 0.7.2. Furthermore, the image in \( {\ell }_{\mathcal{Q}} \) of a basis for \( \ell \mid k \) will span \( {\ell }_{\mathcal{Q}} \) over \( {k}_{\mathcal{P}} \) . Thus \( {\ell }_{\mathcal{Q}} \mid {k}_{\mathcal{P}} \) is a finite extension.
In these circumstances, we have the following uniqueness result:
Theorem 0.7.9 Let \( K \) be a field which is complete with respect to a non-Archimedean valuation \( v \) whose valuation ring \( R \) is a discrete valuation ring. Let \( L \) be a finite extension of \( K \) of degree \( n \) . Then there is a unique extension \( {v}^{\prime } \) of \( v \) to \( L \) such that \( L \) is complete with respect to \( {v}^{\prime } \) and \( {v}^{\prime } \) is determined for all \( y \in L \) by
\[
{v}^{\prime }\left( y\ri
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1063_(GTM222)Lie Groups, Lie Algebras, and Representations
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Definition 1.1
|
Definition 1.1. The general linear group over the real numbers, denoted \( \mathrm{{GL}}\left( {n;\mathbb{R}}\right) \), is the group of all \( n \times n \) invertible matrices with real entries. The general linear group over the complex numbers, denoted \( \mathrm{{GL}}\left( {n;\mathbb{C}}\right) \), is the group of all \( n \times n \) invertible matrices with complex entries.
Definition 1.2. Let \( {M}_{n}\left( \mathbb{C}\right) \) denote the space of all \( n \times n \) matrices with complex entries.
We may identify \( {M}_{n}\left( \mathbb{C}\right) \) with \( {\mathbb{C}}^{{n}^{2}} \) and use the standard notion of convergence in \( {\mathbb{C}}^{{n}^{2}} \) . Explicitly, this means the following.
Definition 1.3. Let \( {A}_{m} \) be a sequence of complex matrices in \( {M}_{n}\left( \mathbb{C}\right) \) . We say that \( {A}_{m} \) converges to a matrix \( A \) if each entry of \( {A}_{m} \) converges (as \( m \rightarrow \infty \) ) to the corresponding entry of \( A \) (i.e., if \( {\left( {A}_{m}\right) }_{jk} \) converges to \( {A}_{jk} \) for all \( 1 \leq j, k \leq n \) ).
We now consider subgroups of \( \mathrm{{GL}}\left( {n;\mathbb{C}}\right) \), that is, subsets \( G \) of \( \mathrm{{GL}}\left( {n;\mathbb{C}}\right) \) such that the identity matrix is in \( G \) and such that for all \( A \) and \( B \) in \( G \), the matrices \( {AB} \) and \( {A}^{-1} \) are also in \( G \) .
Definition 1.4. A matrix Lie group is a subgroup \( G \) of \( \mathrm{{GL}}\left( {n;\mathbb{C}}\right) \) with the following property: If \( {A}_{m} \) is any sequence of matrices in \( G \), and \( {A}_{m} \) converges to some matrix \( A \), then either \( A \) is in \( G \) or \( A \) is not invertible.
The condition on \( G \) amounts to saying that \( G \) is a closed subset of \( \mathrm{{GL}}\left( {n;\mathbb{C}}\right) \) . (This does not necessarily mean that \( G \) is closed in \( {M}_{n}\left( \mathbb{C}\right) \) .) Thus, Definition 1.4 is equivalent to saying that a matrix Lie group is a closed subgroup of \( \mathrm{{GL}}\left( {n;\mathbb{C}}\right) \) . Throughout the book, all topological properties of a matrix Lie group \( G \) will be considered with respect to the topology \( G \) inherits as a subset of \( {M}_{n}\left( \mathbb{C}\right) \cong {\mathbb{C}}^{{n}^{2}} \) .
The condition that \( G \) be a closed subgroup, as opposed to merely a subgroup, should be regarded as a technicality, in that most of the interesting subgroups of \( \mathrm{{GL}}\left( {n;\mathbb{C}}\right) \) have this property. Most of the matrix Lie groups \( G \) we will consider have the stronger property that if \( {A}_{m} \) is any sequence of matrices in \( G \), and \( {A}_{m} \) converges to some matrix \( A \), then \( A \in G \) (i.e., that \( G \) is closed in \( {M}_{n}\left( \mathbb{C}\right) \) ).
An example of a subgroup of \( \mathrm{{GL}}\left( {n;\mathbb{C}}\right) \) which is not closed (and hence is not a matrix Lie group) is the set of all \( n \times n \) invertible matrices with rational entries. This set is, in fact, a subgroup of \( \mathrm{{GL}}\left( {n;\mathbb{C}}\right) \), but not a closed subgroup. That is, one can
Fig. 1.1 A small portion of the group \( G \) inside \( \bar{G} \) (left) and a larger portion (right)
(easily) have a sequence of invertible matrices with rational entries converging to an invertible matrix with some irrational entries. (In fact, every real invertible matrix is the limit of some sequence of invertible matrices with rational entries.)
Another example of a group of matrices which is not a matrix Lie group is the following subgroup of \( \mathrm{{GL}}\left( {2;\mathbb{C}}\right) \) . Let \( a \) be an irrational real number and let
\[
G = \left\{ {\left. \left( \begin{matrix} {e}^{it} & 0 \\ 0 & {e}^{ita} \end{matrix}\right) \right| \;t \in \mathbb{R}}\right\}
\]
(1.1)
Clearly, \( G \) is a subgroup of \( \mathrm{{GL}}\left( {2;\mathbb{C}}\right) \) . According to Exercise 10, the closure of \( G \) is the group
\[
\bar{G} = \left\{ {\left. \left( \begin{matrix} {e}^{i\theta } & 0 \\ 0 & {e}^{i\phi } \end{matrix}\right) \right| \;\theta ,\phi \in \mathbb{R}}\right\} .
\]
The group \( G \) inside \( \bar{G} \) is known as an "irrational line in a torus"; see Figure 1.1.
## 1.2 Examples
Mastering the subject of Lie groups involves not only learning the general theory but also familiarizing oneself with examples. In this section, we introduce some of the most important examples of (matrix) Lie groups. Among these are the classical groups, consisting of the general and special linear groups, the unitary and orthogonal groups, and the symplectic groups. The classical groups, and their associated Lie algebras, will be key examples in Parts II and III of the book.
## 1.2.1 General and Special Linear Groups
The general linear groups (over \( \mathbb{R} \) or \( \mathbb{C} \) ) are themselves matrix Lie groups. Of course, \( \mathrm{{GL}}\left( {n;\mathbb{C}}\right) \) is a subgroup of itself. Furthermore, if \( {A}_{m} \) is a sequence of matrices in \( \mathrm{{GL}}\left( {n;\mathbb{C}}\right) \) and \( {A}_{m} \) converges to \( A \), then by the definition of \( \mathrm{{GL}}\left( {n;\mathbb{C}}\right) \) , either \( A \) is in \( \mathrm{{GL}}\left( {n;\mathbb{C}}\right) \), or \( A \) is not invertible.
Moreover, \( \mathrm{{GL}}\left( {n;\mathbb{R}}\right) \) is a subgroup of \( \mathrm{{GL}}\left( {n;\mathbb{C}}\right) \), and if \( {A}_{m} \in \mathrm{{GL}}\left( {n;\mathbb{R}}\right) \) and \( {A}_{m} \) converges to \( A \), then the entries of \( A \) are real. Thus, either \( A \) is not invertible or \( A \in \mathrm{{GL}}\left( {n;\mathbb{R}}\right) \) .
The special linear group (over \( \mathbb{R} \) or \( \mathbb{C} \) ) is the group of \( n \times n \) invertible matrices (with real or complex entries) having determinant one. Both of these are subgroups of \( \mathrm{{GL}}\left( {n;\mathbb{C}}\right) \) . Furthermore, if \( {A}_{n} \) is a sequence of matrices with determinant one and \( {A}_{n} \) converges to \( A \), then \( A \) also has determinant one, because the determinant is a continuous function. Thus, \( \mathrm{{SL}}\left( {n;\mathbb{R}}\right) \) and \( \mathrm{{SL}}\left( {n;\mathbb{C}}\right) \) are matrix Lie groups.
## 1.2.2 Unitary and Orthogonal Groups
An \( n \times n \) complex matrix \( A \) is said to be unitary if the column vectors of \( A \) are orthonormal, that is, if
\[
\mathop{\sum }\limits_{{l = 1}}^{n}\overline{{A}_{lj}}{A}_{lk} = {\delta }_{jk}
\]
(1.2)
We may rewrite (1.2) as
\[
\mathop{\sum }\limits_{{l = 1}}^{n}{\left( {A}^{ * }\right) }_{jl}{A}_{lk} = {\delta }_{jk}
\]
(1.3)
where \( {\delta }_{jk} \) is the Kronecker delta, equal to 1 if \( j = k \) and equal to zero if \( j \neq k \) . Here \( {A}^{ * } \) is the adjoint of \( A \), defined by
\[
{\left( {A}^{ * }\right) }_{jk} = \overline{{A}_{kj}}
\]
Equation (1.3) says that \( {A}^{ * }A = I \) ; thus, we see that \( A \) is unitary if and only if \( {A}^{ * } = {A}^{-1} \) . In particular, every unitary matrix is invertible.
The adjoint operation on matrices satisfies \( {\left( AB\right) }^{ * } = {B}^{ * }{A}^{ * } \) . From this, we can see that if \( A \) and \( B \) are unitary, then
\[
{\left( AB\right) }^{ * }\left( {AB}\right) = {B}^{ * }{A}^{ * }{AB} = {B}^{-1}{A}^{-1}{AB} = I,
\]
showing that \( {AB} \) is also unitary. Furthermore, since \( {\left( A{A}^{-1}\right) }^{ * } = {I}^{ * } = I \), we see that \( {\left( {A}^{-1}\right) }^{ * }{A}^{ * } = I \), which shows that \( {\left( {A}^{-1}\right) }^{ * } = {\left( {A}^{ * }\right) }^{-1} \) . Thus, if \( A \) is unitary, we have
\[
{\left( {A}^{-1}\right) }^{ * }{A}^{-1} = {\left( {A}^{ * }\right) }^{-1}{A}^{-1} = {\left( A{A}^{ * }\right) }^{-1} = I,
\]
showing that \( {A}^{-1} \) is again unitary.
Thus, the collection of unitary matrices is a subgroup of \( \mathrm{{GL}}\left( {n;\mathbb{C}}\right) \) . We call this group the unitary group and we denote it by \( \mathrm{U}\left( n\right) \) . We may also define the special unitary group \( \mathrm{{SU}}\left( n\right) \), the subgroup of \( \mathrm{U}\left( n\right) \) consisting of unitary matrices with determinant 1 . It is easy to check that both \( \mathrm{U}\left( n\right) \) and \( \mathrm{{SU}}\left( n\right) \) are closed subgroups of \( \mathrm{{GL}}\left( {n;\mathbb{C}}\right) \) and thus matrix Lie groups.
Meanwhile, let \( \langle \cdot , \cdot \rangle \) denote the standard inner product on \( {\mathbb{C}}^{n} \), given by
\[
\langle x, y\rangle = \mathop{\sum }\limits_{j}\overline{{x}_{j}}{y}_{j}
\]
(Note that we put the conjugate on the first factor in the inner product.) By Proposition A.8, we have
\[
\langle x,{Ay}\rangle = \left\langle {{A}^{ * }x, y}\right\rangle
\]
for all \( x, y \in {\mathbb{C}}^{n} \) . Thus,
\[
\langle {Ax},{Ay}\rangle = \left\langle {{A}^{ * }{Ax}, y}\right\rangle
\]
from which we can see that if \( A \) is unitary, then \( A \) preserves the inner product on \( {\mathbb{C}}^{n} \), that is,
\[
\langle {Ax},{Ay}\rangle = \langle x, y\rangle
\]
for all \( x \) and \( y \) . Conversely, if \( A \) preserves the inner product, we must have \( \left\langle {{A}^{ * }{Ax}, y}\right\rangle = \langle x, y\rangle \) for all \( x, y \) . It is not hard to see that this condition holds only if \( {A}^{ * }A = I \) . Thus, an equivalent characterization of unitarity is that \( A \) is unitary if and only if \( A \) preserves the standard inner product on \( {\mathbb{C}}^{n} \) .
Finally, for any matrix \( A \), we have that \( \det {A}^{ * } = \overline{\det A} \) . Thus, if \( A \) is unitary, we have
\[
\det \left( {{A}^{ * }A}\right) = {\left| \det A\right| }^{2} = \det I = 1.
\]
Hence, for all unitary matrices \( A \), we have \( \left| {\det A}\right| = 1 \)
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1112_(GTM267)Quantum Theory for Mathematicians
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Definition 16.2
|
Definition 16.2 If \( {G}_{1} \) and \( {G}_{2} \) are matrix Lie groups, then a Lie group homomorphism of \( {G}_{1} \) to \( {G}_{2} \) is a continuous group homomorphism of \( {G}_{1} \) into \( {G}_{2} \) . A Lie group homomorphism is called a Lie group isomorphism if it is one-to-one and onto with continuous inverse. Two matrix Lie groups are called isomorphic if there exists a Lie group isomorphism between them.
Example 16.3 The real general linear group, denoted \( \mathrm{{GL}}\left( {n,\mathbb{R}}\right) \), is the group of invertible \( n \times n \) matrices with real entries. The groups \( \mathrm{{SL}}\left( {n,\mathbb{C}}\right) \) and \( \operatorname{SL}\left( {n,\mathbb{R}}\right) \) are, respectively, the groups of complex and real matrices with determinant 1. They are called the special linear groups.
Example 16.4 An \( n \times n \) matrix \( U \in {M}_{n}\left( \mathbb{C}\right) \) is said to be unitary if \( {U}^{ * }U = U{U}^{ * } = I \) . A matrix \( U \) is unitary if and only if
\[
\langle {Uv},{Uw}\rangle = \langle v, w\rangle
\]
for all \( v, w \in {\mathbb{C}}^{n} \) . The group of unitary matrices is denoted \( \mathrm{U}\left( n\right) \) and called the \( \left( {n \times n}\right) \) unitary group. The special unitary group, denoted \( \mathrm{{SU}}\left( n\right) \) , is the subgroup of \( \mathrm{U}\left( n\right) \) consisting of unitary matrices with determinant 1 .
The condition \( {\left( {U}^{ * }U\right) }_{jk} = {\delta }_{jk} \) is equivalent to the condition that the columns of \( U \) form an orthonormal set in \( {\mathbb{C}}^{n} \), as can be seen by direct computation. Geometrically, the condition \( {U}^{ * }U = I \) is equivalent to the condition that \( \left\langle {U{v}_{1}, U{v}_{2}}\right\rangle = \left\langle {{v}_{1},{v}_{2}}\right\rangle \) for all \( {v}_{1},{v}_{2} \in {\mathbb{C}}^{n} \), i.e., that \( U \) preserves the inner product on \( {\mathbb{C}}^{n} \) . By taking the determinant of the condition \( {U}^{ * }U = I \), we see that \( \left| {\det U}\right| = 1 \) for all \( U \in \mathrm{U}\left( n\right) \) .
In this, the finite-dimensional case, the condition \( {U}^{ * }U = I \) implies that \( {U}^{ * } \) is the inverse of \( U \) and thus that \( U{U}^{ * } = I \) . This result does not hold in the infinite-dimensional case.
Example 16.5 An \( n \times n \) real matrix \( R \in {M}_{n}\left( \mathbb{R}\right) \) is said to be orthogonal if \( {R}^{tr}R = R{R}^{tr} = I \) . A matrix \( R \) is orthogonal if and only if
\[
\langle {Rv},{Rw}\rangle = \langle v, w\rangle
\]
for all \( v, w \in {\mathbb{R}}^{n} \) . The group of orthogonal matrices is denoted \( \mathrm{O}\left( n\right) \) and is called the \( \left( {n \times n}\right) \) orthogonal group. The special orthogonal group, denoted \( \mathrm{{SO}}\left( n\right) \), is the subgroup of \( \mathrm{O}\left( n\right) \) consisting of orthogonal matrices with determinant 1.
As in the unitary case, the condition \( {R}^{tr}R = I \) implies that \( R{R}^{tr} = I \) and that the columns of \( R \) form an orthonormal set in \( {\mathbb{R}}^{n} \) . Geometrically, a real matrix \( R \) is in \( \mathrm{O}\left( n\right) \) if and only if \( \left\langle {R{v}_{1}, R{v}_{2}}\right\rangle = \left\langle {{v}_{1},{v}_{2}}\right\rangle \) for all \( {v}_{1},{v}_{2} \in {\mathbb{R}}^{n} \), i.e., if and only if \( R \) preserves the inner product on \( {\mathbb{R}}^{n} \) . By taking the determinant of the condition \( {R}^{tr}R = I \) we see that \( \det R = \pm 1 \) for all \( R \in \mathrm{O}\left( n\right) \) .
It is easy to verify that all the groups in Examples 16.3, 16.4, and 16.5 are, indeed, subgroups of \( \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \) and that they are closed.
Definition 16.6 A matrix Lie group \( G \) is connected if for all \( A, B \in G \) there is a continuous path \( A : \left\lbrack {0,1}\right\rbrack \rightarrow {M}_{n}\left( \mathbb{C}\right) \) such that \( A\left( 0\right) = A \) and \( A\left( 1\right) = B \) and such that \( A\left( t\right) \) lies in \( G \) for all \( t \) . A matrix Lie group \( G \) is simply connected if it is connected and every continuous loop in \( G \) can be shrunk continuously to a point in \( G \) . A matrix Lie group \( G \) is compact if it is compact as a subset of \( {M}_{n}\left( \mathbb{C}\right) \cong {\mathbb{R}}^{2{n}^{2}} \) .
By the Heine-Borel theorem (e.g., Proposition 0.26 of [12]), a matrix Lie group \( G \) is compact if and only if it is a closed and bounded subset of \( {M}_{n}\left( \mathbb{C}\right) \) . The condition we are calling "connected" is, more properly, the condition of being path connected. We will see, however, that each matrix Lie group is an embedded real submanifold of \( {M}_{n}\left( \mathbb{C}\right) \) and is, therefore, locally path connected. For matrix Lie groups, then, connectedness and path connectedness are equivalent.
To prove that a matrix Lie group \( G \) is connected, it suffices to prove that for all \( A \in G \), there is a continuous path in \( G \) connecting \( A \) to \( I \) . After all, if both \( A \) and \( B \) can be connected to \( I \), then they can be connected to each other.
Example 16.7 The groups \( \mathrm{O}\left( n\right) ,\mathrm{{SO}}\left( n\right) ,\mathrm{U}\left( n\right) \), and \( \mathrm{{SU}}\left( n\right) \) are compact.
Proof. The conditions defining these groups are obtained by setting certain continuous functions equal to a constant. The group \( \mathrm{{SU}}\left( n\right) \), for example, is defined by setting \( {\left( {U}^{ * }U\right) }_{jk} = {\delta }_{jk} \) for each \( j \) and \( k \) and by setting \( \det U = 1 \) . These groups are thus closed not just as subsets of \( \mathrm{{GL}}\left( {n;\mathbb{C}}\right) \) but also as subsets of \( {M}_{n}\left( \mathbb{C}\right) \) . Furthermore, each of these groups has the property that each column of any matrix in the group is a unit vector. Thus, each group is a bounded subset of \( {M}_{n}\left( \mathbb{C}\right) \) .
Example 16.8 The group \( \mathrm{U}\left( n\right) \) is connected.
Proof. If \( U \in {M}_{n}\left( \mathbb{C}\right) \) is unitary, then \( U \) has an orthonormal basis of eigenvectors with eigenvalues of absolute value 1 . Thus, there is another unitary matrix \( V \) (the change of basis matrix) such that
\[
U = V\left( \begin{matrix} {e}^{i{\theta }_{1}} & & & \\ & {e}^{i{\theta }_{2}} & & \\ & & \ddots & \\ & & & {e}^{i{\theta }_{n}} \end{matrix}\right) {V}^{-1},
\]
for some real numbers \( {\theta }_{1},{\theta }_{2},\ldots ,{\theta }_{n} \) . Thus, we can define a family \( U\left( t\right) \) of unitary matrices by setting
\[
U\left( t\right) = V\left( \begin{matrix} {e}^{{it}{\theta }_{1}} & & & \\ & {e}^{{it}{\theta }_{2}} & & \\ & & \ddots & \\ & & & {e}^{{it}{\theta }_{n}} \end{matrix}\right) {V}^{-1}.
\]
Then \( U\left( \cdot \right) \) is a continuous path lying in \( \mathrm{U}\left( n\right) \) with \( U\left( 0\right) = I \) and \( U\left( 1\right) = U \) . -
Example 16.9 The group \( \mathrm{{SU}}\left( 2\right) \) is simply connected.
Proof. We claim that
\[
\mathrm{{SU}}\left( 2\right) = \left\{ {\left. \left( \begin{matrix} \alpha & - \bar{\beta } \\ \beta & \bar{\alpha } \end{matrix}\right) \right| \;\alpha ,\beta \in \mathbb{C},{\left| \alpha \right| }^{2} + {\left| \beta \right| }^{2} = 1}\right\} .
\]
It is easy to see that each matrix of the indicated form is indeed unitary and has determinant 1 . On the other hand, if \( U \) is any element of \( \mathrm{{SU}}\left( 2\right) \), then the first column of \( U \) is a unit vector \( \left( {\alpha ,\beta }\right) \in {\mathbb{C}}^{2} \) . The second column of \( U \) must then be orthogonal to \( \left( {\alpha ,\beta }\right) \) . Since \( \left( {-\bar{\beta },\bar{\alpha }}\right) \) is orthogonal to \( \left( {\alpha ,\beta }\right) \) and \( {\mathbb{C}}^{2} \) is 2-dimensional, the second column of \( U \) must be a multiple of \( \left( {-\bar{\beta },\bar{\alpha }}\right) \) . But the only multiple that produces a matrix with determinant 1 is 1 .
We see, then, that \( \mathrm{{SU}}\left( 2\right) \) is, topologically, the unit sphere \( {S}^{3} \) inside \( {\mathbb{C}}^{2} \cong \) \( {\mathbb{R}}^{4} \) and is, therefore, simply connected.
## 16.3 Lie Algebras
We now introduce the general algebraic concept of a Lie algebra. Once this is done, we will show how to associate a real Lie algebra with an arbitrary matrix Lie group.
Definition 16.10 A Lie algebra over a field \( \mathbb{F} \) is a vector space \( \mathfrak{g} \) over \( \mathbb{F} \), together with a "bracket" map \( \left\lbrack {\cdot , \cdot }\right\rbrack : \mathfrak{g} \times \mathfrak{g} \rightarrow \mathfrak{g} \) having the following properties:
1. \( \left\lbrack {\cdot , \cdot }\right\rbrack \) is bilinear
2. \( \left\lbrack {Y, X}\right\rbrack = - \left\lbrack {X, Y}\right\rbrack \) for all \( X, Y \in \mathfrak{g} \)
3. \( \left\lbrack {X, X}\right\rbrack = 0 \) for all \( X \in \mathfrak{g} \)
4. For all \( X, Y, Z \in \mathfrak{g} \) we have the Jacobi identity
\[
\left\lbrack {X,\left\lbrack {Y, Z}\right\rbrack }\right\rbrack + \left\lbrack {Y,\left\lbrack {Z, X}\right\rbrack }\right\rbrack + \left\lbrack {Z,\left\lbrack {X, Y}\right\rbrack }\right\rbrack = 0.
\]
If the characteristic of \( \mathbb{F} \) is not equal to 2, then Property 3 is a consequence of Property 2. If \( \mathbb{F} = \mathbb{R} \), then we say that \( \mathfrak{g} \) is a real Lie algebra. An example of a real Lie algebra is the vector space \( {\mathbb{R}}^{3} \) with the bracket equal to the cross product. Properties 1, 2, and 3 are evident from the definition of the cross product, while the Jacobi identity is a known property of the cross product that can be verified by direct calculation.
A large class of Lie algebras may be obtained by the following procedure.
Example 16.11 Let \( \math
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1094_(GTM250)Modern Fourier Analysis
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Definition 7.5.4
|
Definition 7.5.4. We denote by \( {\mathcal{S}}_{ * }\left( {\mathbf{R}}^{d}\right) \) the space of all Schwartz functions \( \Psi \) on \( {\mathbf{R}}^{d} \) whose Fourier transforms are supported in an annulus of the form \( {c}_{1} < \left| \xi \right| < {c}_{2} \) , are nonvanishing in a smaller annulus \( {c}_{1}^{\prime } \leq \left| \xi \right| \leq {c}_{2}^{\prime } \), for some choice of constants \( 0 < {c}_{1} < {c}_{1}^{\prime } < {c}_{2}^{\prime } < {c}_{2} < \infty \) and satisfy for some nonzero constant \( b \)
\[
\mathop{\sum }\limits_{{j \in \mathbf{Z}}}\widehat{\Psi }\left( {{2}^{-j}\xi }\right) = b
\]
(7.5.25)
for all \( \xi \in {\mathbf{R}}^{d} \smallsetminus \{ 0\} \) .
Recall that the Sobolev \( {L}_{\gamma }^{r} \) norm of a function \( g \) is defined as the \( {L}^{r} \) norm of the function \( {\left( I - \Delta \right) }^{\gamma /2}\left( g\right) \) . The main result of this section is as follows.
Theorem 7.5.5. Let \( 1 < r \leq 2 \) . Suppose that \( \sigma \) is a bounded function on \( {\left( {\mathbf{R}}^{n}\right) }^{m} \smallsetminus \{ 0\} \) . Let \( \Psi \) be in \( {\mathcal{S}}_{ * }\left( {\left( {\mathbf{R}}^{n}\right) }^{m}\right) \) . Suppose that for some \( \gamma \) satisfying \( \frac{mn}{r} < \gamma \leq {mn} \) we have
\[
\mathop{\sup }\limits_{{k \in \mathbf{Z}}}{\begin{Vmatrix}{\sigma }^{k}\widehat{\Psi }\end{Vmatrix}}_{{L}_{\gamma }^{r}\left( {\left( {\mathbf{R}}^{n}\right) }^{m}\right) } = K < \infty ,
\]
(7.5.26)
where
\[
{\sigma }^{k}\left( {{\xi }_{1},\ldots ,{\xi }_{m}}\right) = \sigma \left( {{2}^{k}{\xi }_{1},\ldots ,{2}^{k}{\xi }_{m}}\right) .
\]
(7.5.27)
Then the m-linear operator \( {T}_{\sigma } \) is bounded from \( {L}^{{p}_{1}}\left( {\mathbf{R}}^{n}\right) \times \cdots \times {L}^{{p}_{m}}\left( {\mathbf{R}}^{n}\right) \) to \( {L}^{p}\left( {\mathbf{R}}^{n}\right) \) , whenever \( \frac{mn}{\gamma } < {p}_{j} < \infty \) for all \( j = 1,\ldots, m \), and \( p \) satisfies
\[
\frac{1}{p} = \frac{1}{{p}_{1}} + \cdots + \frac{1}{{p}_{m}}
\]
Before we prove this theorem we discuss some preliminary facts.
Definition 7.5.6. For \( s \in \mathbf{R} \), we introduce the weight
\[
{w}_{s}\left( x\right) = {\left( 1 + 4{\pi }^{2}{\left| x\right| }^{2}\right) }^{s/2}.
\]
For \( 1 \leq p < \infty \) the weighted Lebesgue space \( {L}^{p}\left( {w}_{s}\right) \) is defined as the set of all measurable functions \( f \) on \( {\mathbf{R}}^{n} \) such that
\[
\parallel f{\parallel }_{{L}^{p}\left( {w}_{s}\right) } = {\left( {\int }_{{\mathbf{R}}^{n}}{\left| f\left( x\right) \right| }^{p}{w}_{s}\left( x\right) dx\right) }^{1/p} < \infty .
\]
We note that for \( 1 < r \leq 2 \) one has
\[
\parallel \widehat{g}{\parallel }_{{L}^{{r}^{\prime }}\left( {w}_{s}\right) } = {\left( {\int }_{{\mathbf{R}}^{n}}{\left| \widehat{g}\right| }^{{r}^{\prime }}{w}_{s}d\xi \right) }^{\frac{1}{{r}^{\prime }}}
\]
\[
= {\left( {\int }_{{\mathbf{R}}^{n}}{\left| \widehat{g}{w}_{s/{r}^{\prime }}\right| }^{{r}^{\prime }}d\xi \right) }^{\frac{1}{{r}^{\prime }}}
\]
\[
= {\left( {\int }_{{\mathbf{R}}^{n}}{\left| {\left\lbrack {\left( I - \Delta \right) }^{\frac{s}{2{r}^{\prime }}}g\right\rbrack }^{ \frown }\right| }^{{r}^{\prime }}d\xi \right) }^{\frac{1}{{r}^{\prime }}}
\]
(7.5.28)
\[
\leq {\left( {\int }_{{\mathbf{R}}^{n}}{\left| {\left( I - \Delta \right) }^{\frac{s}{2{r}^{\prime }}}g\right| }^{r}dx\right) }^{\frac{1}{r}}
\]
\[
= \parallel g{\parallel }_{{L}_{s/{r}^{\prime }}^{r}}
\]
via the Hausdorff-Young inequality (Proposition 2.2.16 in [156]).
Lemma 7.5.7. Let \( 1 < p < q < \infty \) . Let \( {R}_{0} > 0 \) . Then for every \( s \geq 0 \) there exists a constant \( C = C\left( {p, q, s, n,{R}_{0}}\right) > 0 \) such that for all functions \( g \) in \( {L}_{s}^{q} \) that are supported in a ball of radius \( {R}_{0} \) in \( {\mathbf{R}}^{n} \) we have
\[
\parallel g{\parallel }_{{L}_{s}^{p}\left( {\mathbf{R}}^{n}\right) } \leq C\parallel g{\parallel }_{{L}_{s}^{q}\left( {\mathbf{R}}^{n}\right) }.
\]
(7.5.29)
Proof. We fix a smooth and compactly supported function \( \varphi \) that is equal to one on the ball of radius \( {R}_{0} \) . It will suffice to prove that
\[
\parallel {\varphi g}{\parallel }_{{L}_{s}^{p}\left( {\mathbf{R}}^{n}\right) } \leq C\parallel g{\parallel }_{{L}_{s}^{q}\left( {\mathbf{R}}^{n}\right) }.
\]
(7.5.30)
Since Schwartz functions are dense in \( {L}_{s}^{q} \) (Exercise 2.2.4), it will suffice to prove (7.5.30) for a Schwartz function \( g \) . If \( g \) is a Schwartz function, then so are \( {\left( I - \Delta \right) }^{s/2}\left( g\right) \) and \( {\left( I - \Delta \right) }^{-s/2}\left( g\right) \) ; thus, we may write (7.5.30) equivalently as
\[
{\begin{Vmatrix}{\left( I - \Delta \right) }^{s/2}\left\lbrack \varphi {\left( I - \Delta \right) }^{-s/2}\left( \psi \right) \right\rbrack \end{Vmatrix}}_{{L}^{p}\left( {\mathbf{R}}^{n}\right) } \leq C\parallel \psi {\parallel }_{{L}^{q}\left( {\mathbf{R}}^{n}\right) },
\]
(7.5.31)
where \( \psi \) is a Schwartz function.
We fix an index \( r \) such that \( 1/p = 1/q + 1/r \) . We fix a Schwartz function \( h \) with \( {L}^{{p}^{\prime }} \) norm equal to one. For \( z \in \mathbf{C} \) we define an entire function
\[
G\left( z\right) = {\int }_{{\mathbf{R}}^{n}}{\left( I - \Delta \right) }^{z/2}\left\lbrack {\varphi {\left( I - \Delta \right) }^{-z/2}\left( \psi \right) }\right\rbrack \left( x\right) \overline{h\left( x\right) }{dx}
\]
\[
= {\int }_{{\mathbf{R}}^{n}}{\left( 1 + 4{\pi }^{2}{\left| \xi \right| }^{2}\right) }^{z/2}{\int }_{{\mathbf{R}}^{n}}\widehat{\varphi }\left( {\xi - \eta }\right) {\left( 1 + 4{\pi }^{2}{\left| \eta \right| }^{2}\right) }^{-z/2}\widehat{\psi }\left( \eta \right) {d\eta }\overline{\widehat{h}\left( \xi \right) }{d\xi }.
\]
We show that
\[
\left| {G\left( z\right) }\right| \leq C{\left( 1 + \left| \operatorname{Im}z\right| \right) }^{c}\parallel \psi {\parallel }_{{L}^{q}\left( {\mathbf{R}}^{n}\right) }
\]
(7.5.32)
where \( C, c \) are positive constants and \( z \) is a complex number that satisfies either \( \operatorname{Re}z = 0 \) or \( \operatorname{Re}z = 2\left\lbrack s\right\rbrack + 2 \) . Note that in view of the Mihlin multiplier theorem (Theorem 6.2.7 in [156]), we have that
\[
{\begin{Vmatrix}{\left( I - \Delta \right) }^{-{it}/2}\left( g\right) \end{Vmatrix}}_{{L}^{q}} \leq {C}_{n, q}{\left( 1 + \left| t\right| \right) }^{\left\lbrack {n/2}\right\rbrack + 1}\parallel g{\parallel }_{{L}^{q}}.
\]
When \( z = 0 + {it} \), using Hölder’s inequality we obtain
\[
\left| {G\left( {it}\right) }\right| \leq {\begin{Vmatrix}{\left( I - \Delta \right) }^{{it}/2}\left\lbrack \varphi {\left( I - \Delta \right) }^{-{it}/2}\left( \psi \right) \right\rbrack \end{Vmatrix}}_{{L}^{p}}\parallel h{\parallel }_{{L}^{{p}^{\prime }}}
\]
\[
\leq c{\left( 1 + \left| t\right| \right) }^{\left\lbrack \frac{n}{2}\right\rbrack + 1}{\begin{Vmatrix}\varphi {\left( I - \Delta \right) }^{-{it}/2}\left( \psi \right) \end{Vmatrix}}_{{L}^{p}}\parallel h{\parallel }_{{L}^{{p}^{\prime }}}
\]
\[
\leq c\parallel \varphi {\parallel }_{{L}^{r}}{\left( 1 + \left| t\right| \right) }^{\left\lbrack \frac{n}{2}\right\rbrack + 1}{\begin{Vmatrix}{\left( I - \Delta \right) }^{-{it}/2}\left( \psi \right) \end{Vmatrix}}_{{L}^{q}}
\]
\[
\leq {c}^{\prime }{\left( 1 + \left| t\right| \right) }^{2\left\lbrack \frac{n}{2}\right\rbrack + 2}\parallel \psi {\parallel }_{{L}^{q}}.
\]
\[
\text{Set}N = \left\lbrack s\right\rbrack + 1\text{. When}z = {it} + {2N}\text{, we have}
\]
\[
\left| {G\left( {{it} + {2N}}\right) }\right|
\]
\[
\leq {\begin{Vmatrix}{\left( I - \Delta \right) }^{{it}/2 + N}\left\lbrack \varphi {\left( I - \Delta \right) }^{-{it}/2 - N}\left( \psi \right) \right\rbrack \end{Vmatrix}}_{{L}^{p}}\parallel h{\parallel }_{{L}^{{p}^{\prime }}}
\]
\[
\leq c{\left( 1 + \left| t\right| \right) }^{\left\lbrack \frac{n}{2}\right\rbrack + 1}{\begin{Vmatrix}{\left( I - \Delta \right) }^{N}\left\lbrack \varphi {\left( I - \Delta \right) }^{-{it}/2 - N}\left( \psi \right) \right\rbrack \end{Vmatrix}}_{{L}^{p}}
\]
\[
\leq c{\left( 1 + \left| t\right| \right) }^{\left\lbrack \frac{n}{2}\right\rbrack + 1}\mathop{\sum }\limits_{{\left| \alpha \right| \leq {2N}}}{C}_{\alpha }{\begin{Vmatrix}{\partial }^{\alpha }\left\lbrack \varphi {\left( I - \Delta \right) }^{-{it}/2 - N}\left( \psi \right) \right\rbrack \end{Vmatrix}}_{{L}^{p}}
\]
\[
\leq c{\left( 1 + \left| t\right| \right) }^{\left\lbrack \frac{n}{2}\right\rbrack + 1}\mathop{\sum }\limits_{{\left| \beta \right| + \left| \gamma \right| \leq {2N}}}{C}_{\beta ,\gamma }{\begin{Vmatrix}{\partial }^{\beta }\varphi {\partial }^{\gamma }{\left( I - \Delta \right) }^{-{it}/2 - N}\left( \psi \right) \rbrack \end{Vmatrix}}_{{L}^{p}}
\]
\[
\leq c{\left( 1 + \left| t\right| \right) }^{\left\lbrack \frac{n}{2}\right\rbrack + 1}\mathop{\sum }\limits_{{\left| \beta \right| + \left| \gamma \right| \leq {2N}}}{C}_{\beta ,\gamma }{\begin{Vmatrix}{\partial }^{\beta }\varphi \end{Vmatrix}}_{{L}^{r}}{\begin{Vmatrix}{\left( I - \Delta \right) }^{-{it}/2}{\partial }^{\gamma }{\left( I - \Delta \right) }^{-N}\left( \psi \right) \rbrack \end{Vmatrix}}_{{L}^{q}}
\]
\[
\leq {c}^{\prime }{\left( 1 + \left| t\right| \right) }^{2\left\lbrack \frac{n}{2}\right\rbrack + 2}\mathop{\sum }\limits_{{\left| \beta \right| + \left| \gamma \right| \leq {2N}}}{C}_{\beta ,\gamma }{\begin{Vmatrix}{\partial }^{\beta }\varphi \end{Vmatrix}}_{{L}^{r}}{\begin{Vmatrix}\left. {\partial }^{\gamma }{\left( I - \Delta \right) }^{-N}\left( \psi \right) \right\rbrack \end{Vmatrix}}_{{L}^{q}}
\]
\[
\leq {c}^{\prime \prime }{\left( 1 + \left| t\right| \right) }^{2\left\lbrack \frac{n}{2}\right\rbrack + 2}\parallel \mathbf{\psi }{\parallel }_{{L}^{q}}
\]
since \( {\partial }^{\gamma }{\left( I - \Delta \right) }^{-N} \) is an \( {L}^{q} \) multiplier operator as long as \( \left| \gamma \right| \leq {2N} \) by another application of the Mihlin multi
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