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Lemma 4.3.23. Let \( \xi \) be a linear isomorphism of \( \mathfrak{h} \) onto \( \widetilde{\mathfrak{h}} \) with the property that its dual \( {\xi }^{ * } \) maps \( \bar{\Delta } \) onto \( \Delta \) . Then \( \xi \) preserves the respective restrictions of the Cartan-Killing forms, and \[ {\bar{a}}_{\alpha ,\beta ... | Proof. Let \( \xi \) be a linear isomorphism of \( \mathfrak{l} \) ) onto \( \check{\mathfrak{h}} \) such that \( {\xi }^{ * }\bar{\Delta } = \Delta \) . Then, for any \( H,{H}^{\prime } \in \mathfrak{l} \) ), \[ \left\langle {{\xi H},\xi {H}^{\prime }}\right\rangle = \mathop{\sum }\limits_{{\alpha \in \Delta }}\bar{\a... | Yes |
Corollary 4.3.25. There exists an automorphism \( \varphi \) of \( \mathfrak{g} \) which coincides with -1 on \( \mathfrak{h} \) . Any such automorphism is involutive, and if \( \varphi \) is one such, \( \varphi \) maps \( {\mathfrak{g}}_{\alpha } \) onto \( {\mathfrak{g}}_{-\alpha } \) for all \( \alpha \in \Delta \)... | Proof. The existence of \( \varphi \) is immediate from the theorem. The fact that \( \varphi \) maps \( {g}_{\alpha } \) onto \( {g}_{-\alpha } \) is also straightforward. We now prove that \( {\varphi }^{2} = 1 \) . Select \( {Z}_{\alpha } \in {\mathfrak{g}}_{\alpha } \) such that \( \left\langle {{Z}_{\alpha },{Z}_{... | Yes |
Theorem 4.3.26. \( \mathrm{g} \) always admits a Weyl basis. If \( {H}_{i}\left( {1 \leq i \leq l}\right) \) and \( {Z}_{\alpha }\left( {\alpha \in \Delta }\right) \) are members of a Weyl basis, the corresponding constants \( {N}_{\alpha ,\beta } \) are real, and\n\n\[{\mathfrak{g}}_{0} = {\mathfrak{h}}_{\mathbf{R}} +... | Proof. Let \( P \) be a positive system of roots and \( \varphi \) an automorphism of \( \mathfrak{g} \) such that \( \varphi \mid \mathfrak{h} = - 1 \) . It is obvious that we can select, for each \( \alpha \in P \), a \( {Z}_{\alpha } \in {\mathfrak{g}}_{\alpha } \) such that \( \left\langle {{Z}_{\alpha },\varphi {Z... | Yes |
Two semisimple Lie algebras over \( \mathbf{C} \) are isomorphic if and only if the corresponding equivalence classes of Cartan matrices are identical. | Let \( \mathfrak{g},\mathfrak{h} \) be as usual. Let \( \overline{\mathfrak{g}} \) be a semisimple Lie algebra over \( \mathbf{C} \) with CSA \( \bar{b} \), and suppose that the two Lie algebras give rise to the same equivalence class of Cartan matrices. Then we can find simple systems \( S = \left\{ {\alpha }_{1}\righ... | Yes |
Lemma 4.5.1. Let \( S \) be a scheme with \( n \) elements. Then there cannot be more than \( n - 1 \) links from \( S \) . | Proof. Let \( S = \left\{ {{\alpha }_{1},\ldots ,{\alpha }_{n}}\right\} \), let \( P \) be the set of links from \( S \), and let \( p \) be the number of members of \( P \) . Define \( \alpha = \mathop{\sum }\limits_{{1 \leq i \leq n}}{\left| {\alpha }_{i}\right| }^{-1}{\alpha }_{i} \) . Clearly, \( \alpha \neq 0 \), ... | Yes |
Corollary 4.5.2. No scheme can contain a cycle. If \( S \) is a scheme, \( {S}_{1} \) is a connected subscheme \( \neq S \), and \( \beta \in S \smallsetminus {S}_{1} \), then \( \beta \) cannot be linked to more than one vertex of \( {S}_{1} \) . | Proof. Since a cycle with \( n \) elements has at least \( n \) links, the first assertion is clear. For the second, if \( \left\{ {{\alpha }_{i},\beta }\right\} \) are links \( \left( {{\alpha }_{1},{\alpha }_{2} \in {S}_{1},{\alpha }_{1} \neq {\alpha }_{2}}\right) \), we can find \( {\gamma }_{0} = {\alpha }_{1},{\ga... | Yes |
Corollary 4.5.3. The number of lines issuing from a given vertex in a scheme cannot exceed 3. | Proof. Let \( \alpha \) be a vertex of \( S,{\beta }_{1},\ldots ,{\beta }_{p} \) distinct vertices such that \( \left\{ {\alpha ,{\beta }_{i}}\right\} \) are all links, \( 1 \leq i \leq p \) . If \( \gamma \) is the orthogonal projection of \( \alpha \) on the linear span of the \( {\beta }_{i},\left( {\gamma ,\gamma }... | Yes |
Lemma 4.5.4. Let \( C = \left\{ {{\alpha }_{1},\ldots ,{\alpha }_{n + 1}}\right\} \left( {n \geq 1}\right) \) be a simple chain in a scheme \( S \), and let \( \alpha = {\alpha }_{1} + \cdots + {\alpha }_{n + 1} \) . Then \( {\left| \alpha \right| }^{2} = {\left| {\alpha }_{i}\right| }^{2}\left( {1 \leq i \leq n + 1}\r... | Proof. That \( {\left| \alpha \right| }^{2} = {\left| {\alpha }_{i}\right| }^{2}\left( {1 \leq i \leq n + 1}\right) \) follows at once from (4.5.2). Fix \( \beta \in S \smallsetminus C \), and choose \( i \) with \( 1 \leq i \leq n + 1 \) such that \( \left( {\beta ,{\alpha }_{j}}\right) = 0 \) for \( j \neq i \) . The... | Yes |
Lemma 4.5.5. Let \( C \) be a chain with at least one double link. Then the graph of \( C \) has the form\n\n\n\nwhere \( p, q \) are integers \( \geq 1 \), and either \( \min \left( {p, q}\right) = 1 \) or \( p = q ... | Proof. We observe first that \( C \) does not have more than one double link. If it did there would be a subscheme \( {C}^{\prime } \) of \( C \) whose graph has the form\n\n\n\n\( {}^{3} \) By Corollary 4.5.2, there... | Yes |
Theorem 4.5.7. Let \( S \) be a connected scheme of rank \( l \) . Then the graph of \( S \) has one of the following forms: | Proof. Let \( \Gamma \) be the graph of \( S \) . If \( l = 1,\Gamma = {A}_{1} \) . So we may assume that \( l \geq 2 \) . Suppose first that \( \Gamma \) has a triple link \( \{ \alpha ,\beta \} \) . By Corollary 4.5.3, neither of \( \alpha \) and \( \beta \) can be linked with any other vertex of \( S \) . Since \( S... | Yes |
Corollary 4.6.2. \( \dim {\mathfrak{g}}_{\pm {\alpha }_{i}} = 1\left( {1 \leq i \leq l}\right) \), and \( \dim {\mathfrak{g}}_{\pm \lambda } < \infty \left( {\lambda \in \Gamma }\right) \) . In particular, the dimension of \( \mathfrak{g} \) is at most countable. | Proof. \( {\mathfrak{n}}^{ + } \) is spanned by the \( {X}_{k} \) and the \( \left( {\operatorname{ad}{X}_{{j}_{1}}\cdots \operatorname{ad}{X}_{{j}_{v}}}\right) \left( {X}_{j}\right) (1 \leq \) \( \left. {{j}_{1},\ldots ,{j}_{v}, j \leq l, v \geq 1}\right) \), which belong respectively to \( {\mathfrak{g}}_{{\alpha }_{... | Yes |
Lemma 4.6.3 Define \( {}^{4} \), for \( 1 \leq i, j \leq l \) with \( i \neq j \) ,\n\n\[ \n{\theta }_{ij}^{ + } = {\left( \operatorname{ad}{X}_{i}\right) }^{-{a}_{ij} + 1}\left( {X}_{j}\right) \]\n\n\[ \n{\theta }_{ij}^{ - } = {\left( \operatorname{ad}{Y}_{i}\right) }^{-{a}_{ij} + 1}\left( {Y}_{j}\right) \]\n\nThen, f... | Proof. Since the proofs for \( {\theta }_{ij}^{ + } \) and \( {\theta }_{ij}^{ - } \) are similar, we prove only the relation \( \left\lbrack {{Y}_{k},{\theta }_{ij}^{ + }}\right\rbrack = 0 \) . If \( k \neq i \) and \( k \neq j \), this is immediate, since \( \left\lbrack {{Y}_{k},{X}_{i}}\right\rbrack = \) \( \left\l... | Yes |
Lemma 4.6.4. Let \( \pi \) be a representation of \( \mathfrak{g} \) in a vector space \( V \) . Then the \( {V}_{\lambda } \) for distinct \( \lambda \in {\mathfrak{h}}^{ * } \) are linearly independent; and for any subspace \( U \) invariant under all \( \pi \left( H\right) \left( {H \in \mathfrak{h}}\right) \) , \[ ... | Proof. Let \( U \) be a subspace invariant under \( \pi \left( H\right) \) for all \( H \in \mathfrak{h} \) . To prove (4.6.9), it is enough to check the inclusion \( U \cap \mathop{\sum }\limits_{{\lambda \in {\mathfrak{h}}^{ * }}}{V}_{\lambda } \subseteq \mathop{\sum }\limits_{{\lambda \in {\mathfrak{h}}^{ * }}}\left... | Yes |
Lemma 4.6.5. Let \( \pi \) be a representation of \( \mathfrak{g} \) in a vector space \( V \) . Suppose \( v \in V \) is a nonzero vector such that\n\n\[ \begin{cases} \text{ (i) } & v \in {V}_{\lambda }\text{ for some }\lambda \in {\mathfrak{h}}^{ * } \\ \text{ (ii) } & \pi \left( {X}_{i}\right) v = 0,1 \leq i \leq l... | Proof. Let \( v \) be as above. We have already observed that \( \pi \) is a representation with weights. (i) of (4.6.12) is obvious, since the \( {X}_{i} \) generate \( {\mathfrak{n}}^{ + } \) .\n\nLet us write\n\n\[ {v}_{{j}_{1},\ldots ,{j}_{v}} = \pi \left( {{Y}_{{j}_{1}}\cdots {Y}_{{j}_{v}}}\right) v\;\left( {v \ge... | Yes |
Lemma 4.6.9. Let \( \pi \) be a representation of \( \mathfrak{g} \) in a vector space \( V \). Suppose \( \dim V < \infty \). Then \( \pi \) is a representation with weights, all its weights are integral, and for any \( \mu \in {\mathfrak{h}}^{ * } \), \[ \dim {V}_{\mu } = \dim {V}_{s\mu }\;\left( {s \in \mathfrak{w}}... | Proof. Let \( {\mathfrak{a}}_{i} \) be the linear span of \( {H}_{i},{X}_{i},{Y}_{i}\left( {1 \leq i \leq l}\right) \). Applying Corollary 4.2.3 to \( \pi \mid {a}_{i} \), we see that \( \pi \left( {H}_{i}\right) \) is a semisimple endomorphism with integral eigenvalues, while \( \pi \left( {X}_{i}\right) \) and \( \pi... | Yes |
Theorem 4.6.11. Let \( \mathfrak{g} \) be a Lie algebra over \( \mathbf{C} \) (possibly infinite-dimensional) generated by 31 linearly independent elements \( {H}_{i},{X}_{i},{Y}_{i}\left( {1 \leq i \leq l}\right) \) satisfying the commutation rules (4.6.1). Let \( \mathfrak{D} \) be the set of all dominant integral li... | Proof. For \( \lambda \in \mathfrak{D},\dim \left( {\mathfrak{G}/{\mathfrak{M}}_{\lambda }}\right) < \infty \) by Lemma 4.6.10, so \( {\pi }_{\lambda } \) is finite-dimensional. Theorem 4.6.11 is now an immediate consequence of the work so far. | No |
Lemma 4.7.2. Let \( \lambda \in {\mathfrak{D}}_{P} \), and let \( \pi \) be a finite-dimensional representation of \( \mathfrak{g} \) such that (i) \( \lambda \) is the highest weight of \( \pi \), and (ii) there is a vector of weight \( \lambda \) which is cyclic for \( \pi \) . Then \( \pi \) is equivalent to \( {\pi... | Proof. It is enough to prove that \( \pi \) is irreducible. We use the notation and results of \( \$ {4.6} \) . Let \( V \) be the vector space on which \( \pi \) acts, \( v \) a nonzero vector in \( {V}_{\lambda } \) which is cyclic for \( \pi \) . Since \( \lambda + {\alpha }_{i} \) is not a weight of \( \pi ,\pi \le... | Yes |
Theorem 4.7.3. Let \( \lambda \in {\mathfrak{D}}_{P} \), and let \( {\mathfrak{M}}_{\lambda } \) be as in \( §{4.6} \) . Write \( {\lambda }_{i} = \lambda \left( {H}_{i}\right) \) , \( 1 \leq i \leq l \), and use the notation of \( §{4.6} \) . Then\n\n(4.7.2)\n\n\[{\mathfrak{M}}_{\lambda } = \mathop{\sum }\limits_{{1 \... | Proof. In view of (4.7.1) and Lemma 4.6.10, it is clear that the right side of (4.7.2) is a left ideal of \( \mathfrak{G} \) of positive finite codimension. Let \( {\mathfrak{M}}_{\lambda }^{0} \) denote this left ideal. Then the representation of \( \mathfrak{G} \) induced in \( \mathfrak{G}/{\mathfrak{M}}_{\lambda }^... | Yes |
Lemma 4.7.4. \( {}^{6} \) Let \( \lambda \) be an integral linear function of \( \mathfrak{h} \) . Then there is a unique element of \( {\mathfrak{D}}_{P} \) in the orbit \( \mathfrak{w} \cdot \lambda \) . If \( \lambda \in {\mathfrak{D}}_{P}, s \cdot \lambda \preccurlyeq \lambda \) for all \( s \in \mathfrak{w} \) . | Proof. Let \( \lambda \) be an integral element of \( {\mathfrak{h}}^{ * } \) . Let \( \mu \) be an element of the orbit \( O = \mathfrak{w} \cdot \lambda \) that is maximal with respect to \( \prec \) . Since \( {s}_{{\alpha }_{i}}\mu = \mu - \mu \left( {H}_{i}\right) {\alpha }_{i} \) , it is clear from the maximality... | Yes |
Theorem 4.8.3. Let \( A = {\left( {a}_{ij}\right) }_{1 \leq i, j \leq l} \) be a Cartan matrix of rank \( l \geq 1 \) . Then there is a semisimple Lie algebra \( \overline{\mathfrak{g}} \) over \( \mathbf{C} \) with CSA \( \widetilde{\mathfrak{h}} \) and elements \( {\bar{H}}_{i},{\bar{X}}_{i},{\bar{Y}}_{i}\left( {1 \l... | Proof. We define \( \overline{\mathfrak{g}} = \mathfrak{g}/\mathfrak{q} \), where \( \mathfrak{q} \) is as in the previous lemma. Let \( {\bar{H}}_{i},{\bar{X}}_{i} \), and \( {\bar{Y}}_{i} \) be the respective images of \( {H}_{i},{X}_{i} \), and \( {Y}_{i} \) in \( \overline{\mathrm{g}}\left( {1 \leq i \leq l}\right)... | Yes |
Theorem 4.9.3. There exist 1 homogeneous algebraically independent elements \( {p}_{1},\ldots ,{p}_{l} \) of positive degree such that \( I = \mathbf{C}\left\lbrack {{p}_{1},\ldots ,{p}_{l}}\right\rbrack \) . | As an example, consider the case \( \mathfrak{g} = \mathfrak{{sl}}\left( {l + 1,\mathbf{C}}\right) \) . Let \( \mathfrak{h} \) be the CSA of all diagonal matrices in \( \mathfrak{g} \) . For any \( X \in \mathfrak{g} \) let \( c\left( {X : T}\right) \) be its characteristic polynomial, \( T \) being an indeterminate. T... | Yes |
Lemma 4.10.2. Let \( \lambda \in {\mathfrak{h}}^{ * } \), and let \( \pi \) be a representation of \( \mathfrak{g} \) in a vector space \( V \) such that (i) \( \lambda \) is the highest weight of \( \pi \), and (ii) there is a nonzero vector \( v \) in \( {V}_{\lambda } \) which is cyclic for \( \pi \) . Then if \( \m... | Proof. By Lemma 4.6.5, \( \dim {V}_{\lambda } = 1 \) . Obviously, \( \pi \left( X\right) v = 0 \) for \( X \in {\mathfrak{g}}_{\alpha } \) , \( \alpha \in P \) . So \( \pi \left( z\right) v = {\beta }_{P}\left( z\right) \left( \lambda \right) v \) for \( z \in \mathcal{Z} \) . Write \( U = \{ u : u \in V,\pi \left( z\r... | Yes |
Lemma 4.11.1. Let notation be as above. Then \( C \) is finitely generated. In particular, if \( C \) is infinite, there exist nontrivial homomorphisms of \( C \) into \( {\mathbf{R}}^{ + } \) . | Proof. One can choose a compact set \( D \) such that \( G = C{D}^{0} \), where \( {D}^{0} = \) interior \( D \) . By enlarging \( D \), we may assume that \( 1 \in {D}^{0} \) and \( D = {D}^{-1} \) . Since \( D \cdot {D}^{-1} \) is compact and \( \subseteq G = { \cup }_{c \in C}c{D}^{0} \), we can find \( {c}_{1},\ldo... | Yes |
Lemma 4.11.2. Let \( \varphi \) be a homomorphism of \( C \) into \( {\mathbf{R}}^{ + } \) . Then there is a continuous function \( h \) on \( G \) with positive values such that \( h \mid C = \varphi \) and \( h\left( {xc}\right) = \) \( h\left( x\right) \varphi \left( c\right) \) for all \( x \in G, c \in C \) . | Proof. Select a compact set \( D = {D}^{-1} \) such that \( G = {CD} \), and let \( g \) be a continuous function on \( G \) with compact support such that (i) \( g \geq 0 \), and (ii) \( g\left( x\right) = 1 \) for \( x \in D \) . Let\n\n\[ \n{h}_{1}\left( x\right) = \mathop{\sum }\limits_{{c \in C}}g\left( {xc}\right... | Yes |
Lemma 4.11.3. Let \( \bar{H} \) be a continuous real-valued function on \( \bar{G} \times \bar{G} \) such that \( \bar{H}\left( {\overline{1},\overline{1}}\right) = 0 \) and, for all \( \bar{x},\bar{y},\bar{z} \in G \) ,\n\n(4.11.1)\n\n\[ \bar{H}\left( {\bar{x}\bar{y},\bar{z}}\right) + \bar{H}\left( {\bar{x},\bar{y}}\r... | Proof. Let\n\n\[ \bar{a}\left( \bar{x}\right) = - {\int }_{G}\bar{H}\left( {\bar{x},\bar{y}}\right) d\bar{y}\;\left( {\bar{x} \in \bar{G}}\right) .\n\]\n\nIt is then a trivial verification, based on the biinvariant nature of \( d\bar{x} \), that \( \bar{a} \) has the required properties. | Yes |
Lemma 4.11.4. Let \( \varphi \) be a homomorphism of \( C \) into \( {\mathbf{R}}^{ + } \) . Then there exists a continuous homomorphism \( \chi \) of \( G \) into \( {\mathbf{R}}^{ + } \) such that \( \chi \mid C = \varphi \) . | Proof. Select a continuous function \( h \) on \( G \) with positive values such that the properties stated in Lemma 4.11.2 are satisfied. Define \( H \) by\n\n\[ H\left( {x, y}\right) = \log h\left( {xy}\right) - \log h\left( x\right) - \log h\left( y\right) \;\left( {x, y \in G}\right) . \]\n\nAny easy verification s... | Yes |
Theorem 4.11.5. Let \( G \) be a connected locally compact group satisfying the second axiom of countability, \( C \) a discrete central subgroup. Suppose that\n\n(a) \( G/C \) is compact\n\n(b) \( G \) has no nontrivial continuous homomorphisms into \( {\mathbf{R}}^{ + } \) .\n\nThen \( G \) is compact. This is, in pa... | Proof. If \( C \) is infinite, there is a nontrivial homomorphism of \( C \) into \( {\mathbf{R}}^{ + } \) , and this can be extended to a continuous homomorphism of \( G \) into \( {\mathbf{R}}^{ + } \) , contradicting (b). If we assume that the commutator subgroup of \( G \) is dense in \( G \), then any continuous h... | No |
Theorem 4.11.6. Let \( \bar{G} \) be a compact semisimple analytic group. Then its universal covering group is also compact. | Proof. Let \( G \) be the universal covering group of \( \bar{G} \) . We may then assume that \( \bar{G} = G/C \), where \( C \) is a discrete central subgroup of \( G \) . Clearly \( G \) is also semisimple. In order to prove that \( G \) is compact it is enough to prove that \( G \) satisfies condition (b) of the pre... | Yes |
Theorem 4.11.7. Let \( \mathfrak{g} \) be a Lie algebra over \( \mathbf{R}, G \) its adjoint group. Then the following statements are equivalent:\n\n(i) \( \mathfrak{g} \) is reductive and \( \mathfrak{{Dg}} \) is of compact type\n\n(ii) \( G \) is compact\n\n(iii) If \( X \in \mathfrak{g} \) , ad \( X \) is semisimple... | Proof. (i) \( \Rightarrow \) (ii) Let \( \mathfrak{c} = \) center \( \mathfrak{g},{\mathfrak{g}}_{1} = \mathfrak{D}\mathfrak{g} \) . Then \( {Y}^{\nu } = Y \) for \( Y \in \mathfrak{c} \) , \( y \in G \) . So if \( {G}_{1} \) is the adjoint group of \( {\mathfrak{g}}_{1}, y \mapsto y \mid {\mathfrak{g}}_{1} \) is an is... | Yes |
Theorem 4.11.9. Let \( \mathfrak{g} \) be a complex semisimple Lie algebra, \( G \) its adjoint group. Then \( \mathfrak{g} \) admits real forms of compact type. Any two such are conjugate via an element of \( G \) . | Proof. In view of the above lemma it is enough to prove that if \( {\mathfrak{u}}_{1} \) is a real form of compact type of \( \mathfrak{g} \), then there is \( x \in G \) such that \( {\mathfrak{u}}_{1}^{x} \) contains \( \mathfrak{b},\mathfrak{b} \) and \( \mathfrak{h} \) being as above. Let \( {\mathfrak{b}}_{1} \) b... | Yes |
Theorem 4.11.10. Let \( G \) be a complex semisimple analytic group, \( \mathfrak{g} \) its Lie algebra. Then \( G \) admits a compact real form. More precisely, if \( \mathfrak{u} \) is a compact type real form of \( \mathfrak{g} \) and \( U \) is the real analytic subgroup of \( G \) defined by \( \mathfrak{n} \), th... | Proof. If \( \mathfrak{u}, U \) are as in the statement above, \( U \) is compact by Theorem 4.11.7. The theorem follows easily from the previous theorem. | No |
Corollary 4.11.11. All finite-dimensional representations of a complex semisimple Lie algebra are semisimple. | Proof. Let \( \mathfrak{g} \) be a complex semisimple Lie algebra, and let \( G \) be a simply connected complex analytic group with Lie algebra \( \mathfrak{g} \) . Let \( \mathfrak{u}, U \) be as in the above theorem. Suppose \( \pi \) is a finite-dimensional representation of \( \mathfrak{g} \) . Then \( \pi \) can ... | Yes |
Lemma 4.11.12. Let \( U \) be a compact semisimple analytic group, \( \mathfrak{u} \) its Lie algebra. If every representation of \( \mathfrak{u} \) is the differential of a representation of \( U \), then \( U \) is simply connected. | Proof. Let \( \widetilde{U} \) be the universal covering group of \( U \) with covering homomorphism \( \pi \) . By Weyl’s theorem, \( \widetilde{U} \) is compact. Let \( C \) be the kernel of \( \pi \) . We identify the Lie algebra of \( \widetilde{U} \) with \( \mathfrak{u} \), so that \( {d\pi } \) is the identity. ... | Yes |
Lemma 4.12.1. Let \( \mathfrak{g} \) be a semisimple Lie algebra of compact type over \( \mathbf{R} \) , \( G \) its adjoint group. Then a subalgebra of \( \mathfrak{g} \) is a CSA if and only if it is maximal abelian. Suppose \( \mathfrak{b} \) is a CSA and \( \mathfrak{z} \) is a subalgebra of \( \mathfrak{g} \) such... | Proof. Theorems 4.1.5 and 4.11.7 imply that a subalgebra of \( g \) is a CSA if and only if it is maximal abelian. If \( z \) is a subalgebra of \( g \) containing a CSA \( \mathfrak{b} \), it is obvious that \( \mathfrak{b} \) is a CSA of \( \mathfrak{z} \) ; in particular, \( \operatorname{rk}\left( \mathfrak{z}\righ... | Yes |
Theorem 4.12.3. Let \( \mathfrak{b} \subseteq \mathfrak{g} \) be a CSA, \( B \) the corresponding analytic subgroup of \( G \) . Then \( B \) is a maximal torus of \( G \) . Every maximal torus can be obtained in this way. If \( {B}_{1},{B}_{2} \) are two maximal tori of \( G \), there is an \( x \in G \) such that \( ... | Proof. \( \mathfrak{b} \) is its own centralizer in \( \mathfrak{g} \) . So if \( A \) is the centralizer of \( \mathfrak{b} \) in \( G,\mathfrak{b} \) is the subalgebra defined by the closed subgroup \( A \) . This shows that \( B \) is the component of the identity of \( A.B \) is therefore closed. Since it is compac... | Yes |
Theorem 4.13.1. Let \( \widetilde{B} \) be the normalizer of \( B \) in \( G \) . For \( x \in \widetilde{B} \), let\n\n\[ s\left( x\right) = \operatorname{Ad}\left( x\right) \mid {\mathfrak{b}}_{c}. \]\n\nThen \( x \mapsto s\left( x\right) \) induces an isomorphism of \( \widetilde{B}/B \) onto the Weyl group \( \math... | Proof. Since \( \mathfrak{b} = {\left( -1\right) }^{1/2}\mathop{\sum }\limits_{{\alpha \in \Delta }}\mathbf{R} \cdot {H}_{\alpha } \), it is obvious that \( \mathfrak{w} \) leaves \( \mathfrak{b} \) invariant. Theorem 4.9.1 implies that \( s\left( x\right) \in \mathfrak{w} \) for \( x \in \widetilde{B} \) and that \( s... | Yes |
Lemma 4.13.2. We have\n\n\[ L\left( R\right) \subseteq L\left( G\right) \subseteq L \]\n\nand all three are isomorphic as additive groups to \( {\mathbf{Z}}^{l}\left( {l = \operatorname{rank}\mathfrak{g}}\right) .L\left( G\right) \) is the set of all those integral linear functions on \( {\mathfrak{b}}_{c} \) which occ... | Proof. Since the roots are the weights of the adjoint representation, we have \( L\left( R\right) \subseteq L\left( G\right) \) . By the Frobenius reciprocity theorem (cf. Weil [1]), every character of \( B \) occurs in the decomposition with respect to \( B \) of some representation of \( G \) . So \( L\left( G\right)... | Yes |
Corollary 4.13.3. Let \( \delta = \frac{1}{2}\mathop{\sum }\limits_{{\alpha \in P}}\alpha \) . Then \( {2\delta } \in L\left( G\right) \) . If \( \lambda \in L,{s\lambda } - \lambda \) \( \in L\left( R\right) \) for all \( s \in \mathfrak{w} \) . | Proof. Follows on taking Lemma 4.7.4 into account. | No |
Lemma 4.13.4. Let \( \mathcal{F} \) be the algebra of all finite linear combinations of the exponential \( {e}^{\lambda },\lambda \in L \) . Let\n\n\[ \n{\mathcal{D}}_{P}^{ + } = \left\{ {{\lambda }^{\prime } : {\lambda }^{\prime } \in {\mathcal{D}}_{P},{\lambda }^{\prime }\left( {H}_{\alpha }\right) > 0\text{ for all ... | Proof. The properties of the \( {g}_{\lambda } \) are obvious. Suppose that \( g = \mathop{\sum }\limits_{{\mu \in L}}{c}_{\mu }{e}^{\mu } \) is an element of \( \mathcal{F} \) such that \( {g}^{s} = \epsilon \left( s\right) g \) for all \( s \in \mathfrak{w} \) . Then \( {c}_{s\mu } = \epsilon \left( s\right) {c}_{\mu... | Yes |
Corollary 4.13.5. \( {s\delta } + \delta \in L\left( R\right) \) for all \( s \in \mathfrak{w} \), and\n\n(4.13.20)\n\n\[ \n{D}_{P}\left( b\right) = \mathop{\sum }\limits_{{s \in w}}\epsilon \left( s\right) {\xi }_{{s\delta } + \delta }\left( b\right) \;\left( {b \in B}\right) .\n\] | Proof. \( {s\delta } + \delta = {s\delta } - \delta + {2\delta } \in L\left( R\right) \) for \( s \in \mathfrak{w} \) . Now for \( H \in \mathfrak{b} \), \n\n\[ \n{D}_{P}\left( {\exp H}\right) = \mathop{\prod }\limits_{{\alpha \in P}}\left( {{e}^{\alpha \left( H\right) } - 1}\right) \n\] \n\n\[ \n= {e}^{\delta \left( H... | Yes |
Theorem 4.13.7. For any continuous function \( f \) on \( {G}^{\prime }, f \in {\mathfrak{L}}^{1}\left( G\right) \) if and only if \( {\varphi }_{\left| f\right| } \cdot \left( {D \mid {B}^{\prime }}\right) \in {\mathcal{L}}^{1}\left( B\right) \) . In this case,\n\n\[{\int }_{G}f\left( x\right) {dx} = {\left\lbrack \ma... | Proof. Since \( {\varphi }^{ * } \) is a covering map of \( {G}^{ * } \times {B}^{\prime } \) onto \( {G}^{\prime } \), there is an integer \( k \geq 1 \) such that above any element of \( {G}^{\prime } \) there are exactly \( k \) elements of \( {G}^{ * } \times {B}^{\prime } \) . From the standard theory of integrati... | Yes |
Corollary 4.13.8. Let \( f \) be a continuous function on \( {G}^{\prime } \) that is invariant under all the inner automorphisms of \( G \) . Then \( f \in {\mathfrak{L}}^{1}\left( G\right) \) if and only if \( \left( {Df}\right) \mid {B}^{\prime } \) \( \in {\mathfrak{L}}^{1}\left( B\right) \), and in this case, | \[ {\int }_{G}f\left( x\right) {dx} = {\left\lbrack \mathfrak{w}\right\rbrack }^{-1}{\int }_{B}f\left( b\right) D\left( b\right) {db}. \] | Yes |
Lemma 4.14.1. Let \( f \) be a continuous function on \( {B}^{\prime } \) invariant with respect to \( \widetilde{B} \) (or \( \mathfrak{w} \) ). Then there exists a unique continuous function \( F \) on \( {G}^{\prime } \) such that\n\n(i) \( F \) is invariant under all inner automorphisms of \( G \)\n\n(ii) \( F \mid... | Proof. We have \( {G}^{\prime } = {\left( {B}^{\prime }\right) }^{G} \) . So we have to set \( F\left( {b}^{x}\right) = f\left( b\right) \) for \( b \in {B}^{\prime } \) , \( x \in G \) . To see that \( F \) is well defined, let \( {b}_{1},{b}_{2} \in {B}^{\prime } \) and \( {x}_{1},{x}_{2} \in G \) be such that \( {b}... | Yes |
Lemma 4.14.2. For \( \lambda \in L\left( G\right) \cap {\mathfrak{D}}_{P} \), let \( {u}_{\lambda } \) be the function on \( B \) defined \( b{y}^{14} \)\n\n(4.14.1)\n\n\[ \n{u}_{\lambda } = \mathop{\sum }\limits_{{s \in w}}\epsilon \left( s\right) {\xi }_{{s\lambda } + {s\delta } + \delta } \n\]\n\nDefine \( {v}_{\lam... | Proof. The existence, uniqueness, and continuity of \( {F}_{\lambda } \) will follow from the previous lemma provided we show that \( {v}_{\lambda } \) is invariant under \( \mathfrak{w} \) . If \( t \in \mathfrak{w} \) , a simple calculation based on (4.14.1) and (4.13.20) reveals that\n\n(4.14.4)\n\n\[ \n{D}_{P}^{t} ... | Yes |
Theorem 4.14.3. Let \( G \) be a compact connected semisimple Lie group, \( \mathrm{g} \) its Lie algebra, \( \mathfrak{b} \) a CSA of \( \mathfrak{g} \), and \( B \) the associated maximal torus. Let \( P \) be a positive system of roots of \( \left( {{\mathfrak{g}}_{c},{\mathfrak{b}}_{c}}\right) \) . Then the irreduc... | Proof. Let \( X \) be the set of irreducible characters of \( G \) . By the Schur orthogonality relations, we have\n\n\[ {\int }_{G}\chi {\chi }^{\prime \text{ conj }}{dx} = {\delta }_{\chi {\chi }^{\prime }}\;\left( {\chi ,{\chi }^{\prime } \in X}\right) . \]\n\nLet \( \Delta \) be the function on \( \mathfrak{b} \) g... | Yes |
Theorem 4.14.4. Let \( G \) be a compact connected semisimple Lie group. Suppose \( G \) is simply connected. Then \( L\left( G\right) = L \), and in particular, \( \delta \in L\left( G\right) \). If \( \lambda \in {\mathfrak{D}}_{P} \), the character of the irreducible representation of \( G \) with highest weight \( ... | Proof. In view of the work of \( \$ {4.11} \) we may assume that \( G \) is a real form of a complex analytic simply connected semisimple Lie group. If \( \lambda \in {\mathfrak{D}}_{P} \), the representation \( {\pi }_{\lambda } \) of \( {g}_{c} \) with highest weight \( \lambda \) lifts to a complex analytic represen... | Yes |
Theorem 4.14.6. Let notation be as in Theorem 4.14.3. Then for \( \lambda \in \) \( L\left( G\right) \cap {\mathfrak{D}}_{P} \), the dimension of the corresponding representation \( {\pi }_{\lambda } \) of \( G \) is given by\n\n\[ \dim \left( {\pi }_{\lambda }\right) = \mathop{\prod }\limits_{{\alpha \in P}}\frac{\lan... | Proof. Let \( {\psi }_{\lambda }\left( b\right) = \operatorname{tr}{\pi }_{\lambda }\left( b\right), b \in B \) . Then \( \dim \left( {\pi }_{\lambda }\right) = {\psi }_{\lambda }\left( 1\right) \) . But the formula (4.14.4) becomes indeterminate if we substitute \( b = 1 \) . So we have to calculate its limit when \( ... | Yes |
Lemma 4.15.2. Let \( S \) be a simple system, \( P \) the positive system containing S. Then for \( \alpha \in S \) , (4.15.5) \[ {s}_{\alpha }\alpha = - \alpha ,\;{s}_{\alpha } \cdot \left( {P\smallsetminus \{ \alpha \} }\right) = P \smallsetminus \{ \alpha \} . \] | Proof. If \( \beta \in P \smallsetminus \{ \alpha \} \), then \( \exists \delta \neq \alpha \) in \( S \) and \( c > 0 \) such that \( \beta = {c\delta } + \) \( \mathop{\sum }\limits_{{\delta \neq \gamma \in S}}{c}_{\gamma }\gamma \) . Now \( {s}_{\alpha }\beta \) is of the form \( \beta - {a\alpha } \) for some const... | Yes |
Theorem 4.15.3. The correspondence \( C \mapsto P\left( C\right) \) is a bijection of the set of all chambers onto the set of all positive systems. The group \( \mathfrak{w} \) is transitive on the sets of positive systems, simple systems, and chambers. | Proof. Let \( P \) be a positive system, \( S \) the simple system contained in \( P \) . For any positive system \( Q \), let \( r\left( Q\right) = \left\lbrack {\left( {-Q}\right) \cap P}\right\rbrack \) . We prove by induction on \( r\left( Q\right) \) that \( Q \) is conjugate to \( P \) under \( \mathfrak{w} \) . ... | Yes |
Theorem 4.15.4. Let \( S \) be a simple system of roots. Then \( \mathfrak{w} \) is generated by the \( {s}_{\alpha }\left( {\alpha \in S}\right) \), and \( \Delta = \mathfrak{w} \cdot S \) . | Proof. Let \( P \) be the positive system containing \( S \) . For \( \beta \in P \) we define the order \( O\left( \beta \right) \) of \( \beta \) by\n\n(4.15.6)\n\n\[ O\left( \beta \right) = \mathop{\sum }\limits_{{\alpha \in S}}{c}_{\alpha }\;\left( {\beta = \mathop{\sum }\limits_{{\alpha \in S}}{c}_{\alpha } \cdot ... | Yes |
Lemma 4.15.6. Let \( \alpha \in S, t \in \mathfrak{w} \) . Then \( N\left( t\right) = N\left( {t}^{-1}\right) \) and\n\n\[ N\left( {t{s}_{\alpha }}\right) = N\left( t\right) \pm 1\;\text{ according as }\;{t\alpha } \gtrless 0 \]\n\n(4.15.8)\n\n\[ N\left( {{s}_{\alpha }t}\right) = N\left( t\right) \pm 1\;\text{ accordin... | Proof. \( N\left( t\right) = \left\lbrack {-P\left( t\right) }\right\rbrack = \left\lbrack {-t \cdot P\left( t\right) }\right\rbrack = N\left( {t}^{-1}\right) \), proving the first assertion. We now take up (4.15.8). By (4.15.5) it is easily seen that\n\n\[ P\left( {t{s}_{\alpha }}\right) = P\left( t\right) \cup \{ \al... | Yes |
Corollary 4.15.7. \( N\left( {t{t}^{\prime }}\right) \equiv N\left( t\right) + N\left( {t}^{\prime }\right) \left( {\;\operatorname{mod}\;2}\right) \) for \( t,{t}^{\prime } \in \mathfrak{w} \) . In particular, if \( t = {s}_{{\alpha }_{1}}\cdots {s}_{{\alpha }_{m}}\left( {{\alpha }_{i} \in S}\right), N\left( t\right) ... | Proof. From (4.15.8) we find \( N\left( {t{s}_{\alpha }}\right) \equiv N\left( t\right) + N\left( {s}_{\alpha }\right) \left( {\;\operatorname{mod}\;2}\right) \), for \( t \in \mathfrak{w} \) , \( \alpha \in S \) . This leads quickly to the first assertion. The second follows trivially from the first. | Yes |
Lemma 4.15.8. \( N\left( {t{t}^{\prime }}\right) \leq N\left( t\right) + N\left( {t}^{\prime }\right) \) for \( t,{t}^{\prime } \in \mathfrak{w} \) . In particular, if \( t = {s}_{{\alpha }_{1}}\cdots {s}_{{\alpha }_{m}}\left( {{\alpha }_{i} \in S}\right), N\left( t\right) \leq m. \) | Proof. It is easily seen that for \( t,{t}^{\prime } \in \mathfrak{w} \)\n\n\[ \text{(4.15.10)}P\left( {t{t}^{\prime }}\right) = \left\{ {\left( \left( {P \smallsetminus P\left( {t}^{\prime }\right) }\right) \right) \cap {t}^{\prime - 1}\left( {P\left( t\right) }\right) }\right\} \cup \left\{ {P{\left( t\right) }^{\pri... | Yes |
Lemma 4.15.9. Let \( t = {s}_{1}\cdots {s}_{n},{s}_{i} = {s}_{{\alpha }_{i}} \) where \( {\alpha }_{i} \in S\left( {1 \leq i \leq n}\right) \) . Then the following assertions are equivalent:\n\n(i) \( N\left( t\right) < n \) .\n\n(ii) for some \( j\left( {1 \leq j \leq n - 1}\right) ,{s}_{1}{s}_{2}\cdots {s}_{j}{\alpha... | Proof. If (iii) is true for some \( i, j \), then we get from (4.15.2) the relation \( {s}_{i} = {s}_{i + 1}\cdots {s}_{j}{s}_{j + 1}{\left( {s}_{i + 1}\cdots {s}_{j}\right) }^{-1} \), leading to (iv) (for the same \( i, j \) ). If (iv) is assumed,(v) follows (for the same \( i, j \) ) on replacing \( {s}_{i + 1}{s}_{i... | Yes |
Theorem 4.15.10. For \( t \in \mathfrak{w}, N\left( t\right) \) is the number of terms in any minimal expression of \( t \) as a product of reflections corresponding to simple roots. If \( t = {s}_{{\alpha }_{1}}\cdots {s}_{{\alpha }_{n}} \) is such a minimal expression, \( {s}_{{\alpha }_{1}}\cdots {s}_{{\alpha }_{j -... | Proof. Let \( t = {s}_{1}\cdots {s}_{n}\left( {{s}_{i} = {s}_{{\alpha }_{i}},{\alpha }_{i} \in S}\right) \) be a minimal expression of \( t \) . By Lemma 4.15.8, \( N\left( t\right) \leq n \) . If \( N\left( t\right) < n \) ,(v) of Lemma 4.15.9 would contradict the minimality of the expression for \( t \) . So \( N\lef... | Yes |
Corollary 4.15.11. There exists \( {s}_{0} \in \mathfrak{w} \) such that \( {s}_{0} \cdot P = - P \) . If \( {s}_{0} = {s}_{1} \) \( \cdots {s}_{n}\left( {{s}_{i} = {s}_{{\alpha }_{i}},{\alpha }_{i} \in P}\right) \) is a minimal expression for \( {s}_{0} \), then \( n = \left\lbrack P\right\rbrack \) and \( {s}_{1}\cdo... | Proof. Existence of \( {s}_{0} \) follows from Theorem 4.15.3, since \( - P \) is also a positive system. Clearly, \( N\left( {s}_{0}\right) = n \) and \( P\left( {s}_{0}^{-1}\right) = P \), so the second statement follows from the above theorem. | No |
Theorem 4.15.12. 1v acts simply transitively on the set of positive systems (resp. simple systems, chambers). | Proof. It is enough to consider the action of \( \mathfrak{w} \) on the positive systems. We must therefore prove that if \( t \in \mathfrak{w} \) and \( t \cdot P = P \), then \( t = 1 \) . If \( t \neq 1 \) , then the minimal expression for \( t \) contains at least one term, so \( N\left( t\right) \geq 1 \) by Theor... | Yes |
Theorem 4.15.13. Every element of \( V \) is conjugate to exactly one element of \( \mathrm{{Cl}}\left( \mathrm{C}\right) \) under \( \mathfrak{w} \) . | Proof. Let \( \lambda \in V \) . Choose a sequence \( \left\{ {\lambda }_{n}\right\} \) from \( {V}^{\prime } \) such that \( {\lambda }_{n} \rightarrow \lambda \) as \( n \rightarrow infty \) . Since there are only finitely many chambers, we may assume that all the \( {\lambda }_{n} \) belong to a fixed chamber \( {C}... | Yes |
Corollary 4.15.14. If \( \lambda \in {Cl}\left( C\right) ,\lambda - {s\lambda } \) is a linear combination of elements of \( S \) with coefficients which are all \( \geq 0 \) . | Proof. Given \( \mu, v \in V \), we write \( \mu \geq v\left( {v \preccurlyeq \mu }\right) \) if \( \mu - v = \mathop{\sum }\limits_{{\gamma \in S}}{c}_{\gamma }\gamma \) , where the \( {c}_{\gamma } \) are all \( \geq 0. \preccurlyeq \) is a partial order in \( V \) . Let \( v \in \mathfrak{w} \cdot \lambda \) . Among... | Yes |
Lemma 4.15.15. Let \( \lambda \in V \) . Then \( \mathfrak{w}\left( \lambda \right) \) is the subgroup of \( \mathfrak{w} \) generated by those \( {s}_{\alpha }\left( {\alpha \in P}\right) \) for which \( \left( {\lambda ,\alpha }\right) = 0 \) . In particular, \( \mathfrak{w}\left( \lambda \right) \) is trivial if and... | Proof. Select \( {t}_{0} \in \mathfrak{w} \) such that \( \mu = {t}_{0}\lambda \in {Cl}\left( C\right) \) . Then \( \mathfrak{w}\left( \mu \right) = {t}_{0}\mathfrak{w}\left( \lambda \right) {t}_{0}^{-1} \) , and it is clearly sufficient to prove the lemma with \( \mu \) instead of \( \lambda \) . Let \( \bar{v}\left( ... | Yes |
Corollary 4.15.16. The \( {s}_{\alpha }\left( {\alpha \in P}\right) \) are the only reflections in \( \mathfrak{w} \) . | Proof. Suppose \( t \in \mathfrak{w} \) is a reflection which is not any one of the \( {s}_{\alpha } \) . Let \( L \) be the hyperplane of points fixed by \( t \) . Then no root \( \alpha \) is orthogonal to \( L \) . So we can find \( \lambda \in L \) such that \( \left( {\lambda ,\alpha }\right) \neq 0 \) for any \( ... | Yes |
Theorem 4.15.17. Let \( \Phi \) be any subset of \( {V}_{c} \) . Then \( \mathfrak{w}\left( \Phi \right) \) is generated by those \( {s}_{\alpha }\left( {\alpha \in P}\right) \) for which \( \alpha \) is orthogonal to \( \Phi \) . If \( \Phi \subseteq {Cl}\left( C\right) ,\mathfrak{w}\left( \Phi \right) \) is generated... | Proof. We may replace \( \mathbf{\Phi } \) by the set consisting of the real and imaginary parts \( {}^{17} \) of its members without changing \( \mathfrak{w}\left( \Phi \right) \) . We may therefore assume \( \Phi \subseteq V \) . Since we may replace \( \Phi \) by any basis of the \( \mathbf{R} \) -linear subspace of... | Yes |
Lemma 4.15.20. There are finite sets \( M \) of homogeneous elements of \( {I}^{ + } \) such that \( \mathcal{F} \) is the ideal generated by \( M \) . If \( L = \left\{ {{p}_{1},\ldots ,{p}_{m}}\right\} \) is one such, \( I = \) \( \mathbf{C}\left\lbrack {{p}_{1},\ldots ,{p}_{m}}\right\rbrack \) . | Proof. Let \( \mathfrak{M} \) be the collection of all finite sets of homogeneous elements in \( {I}^{ + } \) ; for \( M \in \mathfrak{M} \) let \( {\mathfrak{F}}_{M} \) be the ideal in \( \mathcal{P} \) generated by \( M \) . By the Hilbert basis theorem there is a finite set \( N \subseteq \mathcal{F} \) such that \(... | Yes |
Lemma 4.15.21. Let \( \mathfrak{w} \) be a frg. Let \( {p}_{1},\ldots ,{p}_{m} \in I \) be such that \( {p}_{1} \notin \) \( \mathop{\sum }\limits_{{2 \leq j \leq m}}I{p}_{j} \) . If \( {q}_{1},\ldots ,{q}_{m} \) are homogeneous elements of \( \mathcal{O} \) with\n\n(4.15.23)\n\n\[ \n{q}_{1}{p}_{1} + \cdots + {q}_{m}{p... | Proof. We argue by induction on \( \deg \left( {q}_{1}\right) \) . From (4.15.23) we get \( {\widetilde{q}}_{1}{p}_{1} + \) \( \cdots + {\bar{q}}_{m}{p}_{m} = 0 \) . If \( {q}_{1} \) is constant, this constant is 0 ; for otherwise \( {p}_{1} = \) \( - {q}_{1}^{-1}\mathop{\sum }\limits_{{2 \leq i \leq m}}{\bar{q}}_{i}{p... | Yes |
Theorem 4.15.23. Let \( \mathfrak{w} \) be a finite reflection subgroup of \( O\left( V\right) \) . Then there are \( l \) algebraically independent homogeneous elements \( {p}_{1},\ldots ,{p}_{l} \) of positive degree such that \( I = \mathbf{C}\left\lbrack {{p}_{1},\ldots ,{p}_{l}}\right\rbrack \) . | Proof. By Lemma 4.15.21 and 4.15.22, we can find homogeneous \( {p}_{i} \) \( \left( {1 \leq i \leq m}\right) \) which are algebraically independent and of positive degree such that \( I = \mathbf{C}\left\lbrack {{p}_{1},\ldots ,{p}_{m}}\right\rbrack \) . It only remains to prove that \( m = l \) . Clearly, \( m \leq l... | Yes |
Lemma 4.15.25. Let \( \mathfrak{w} \) be any finite subgroup of \( O\left( V\right) \) . Then the Poincaré series of \( I \) is given by \[ {P}_{I}\left( t\right) = {\left\lbrack \mathfrak{w}\right\rbrack }^{-1}\mathop{\sum }\limits_{{s \in \mathfrak{w}}}{\left( \det \left( 1 - ts\right) \right) }^{-1}. \] | Proof. For any linear automorphism \( L \) of \( {V}_{c} \) and any integer \( d \geq 0 \), let \( {L}^{\left( d\right) } \) be the corresponding induced linear transformation of \( {\mathcal{O}}_{d} \) . Then \[ {\left( \det \left( 1 - tL\right) \right) }^{-1} = \mathop{\sum }\limits_{{d = 0}}\operatorname{tr}\left( {... | Yes |
Theorem 4.15.26. Let \( \mathfrak{w} \) be a finite subgroup of \( O\left( V\right) \) . Suppose there are algebraically independent homogeneous elements \( {p}_{1},\ldots ,{p}_{m} \) of positive degree such that \( I = \mathbf{C}\left\lbrack {{p}_{1},\ldots ,{p}_{m}}\right\rbrack \) . Then \( m = l \), and \( \mathfra... | Proof. The proof that \( m = l \) is the same as in Chevalley’s theorem. Let \( {d}_{i} = \deg \left( {p}_{i}\right) \) . Then the Poincaré series of \( I \) is \( \mathop{\prod }\limits_{{1 \leq i \leq l}}{\left( 1 - {t}^{{d}_{i}}\right) }^{-1} \), so by (4.15.28) we have, writing \( w = \left\lbrack \mathfrak{w}\righ... | Yes |
Theorem 4.15.28. Let \( \mathfrak{w} \) be a finite reflection subgroup of \( O\left( V\right) \), and let notation be as above. Write \( w = \left\lbrack \mathfrak{w}\right\rbrack \) . Then \( \mathcal{O} \) is a free I-module of rank \( w \) . More precisely, let \( H \) be a graded subspace of \( \mathcal{O} \) such... | Proof. It is obvious that there are graded subspaces \( H \) of \( \vartheta \) such that \( \mathcal{P} = \mathfrak{F} + H \) is a direct sum. Choose and fix one such. Then the map \( p, u \mapsto {pu} \) ( \( p \in I, u \in H \) ) \ | No |
Proposition 1.1 Let \( R \) be a unital ring. The maximal right ring of quotients \( {Q}_{r}\left( R\right) \) satisfies the following properties:\n\n(i) \( R \) is a subring of \( {Q}_{r}\left( R\right) \) with the same 1,\n\n(ii) for any \( q \in {Q}_{r}\left( R\right) \) there exists a dense right ideal \( I \) of \... | Proof Since \( \left\lbrack {e, e}\right\rbrack = 0 \), we see from Lemma 2.4 that \( \phi \left( {e, e}\right) = {e\phi }\left( {e, e}\right) f \) . So \( \phi \left( {e, e}\right) \notin {Z}_{\sigma }\left( A\right) \) . By Lemma 2.1, we have\n\n\[ \phi \left( {e, e}\right) \left\lbrack {x, y}\right\rbrack = \left\lb... | No |
Corollary 3.2 Let \( R \) be a local ring. The following are equivalent:\n\n(1) \( R/J\left( R\right) \cong {\mathbb{Z}}_{2} \) .\n\n(2) \( {M}_{n}\left( R\right) \) is a GJ-clean ring for any \( n \geq 1 \) . | Proof \( \left( 1\right) \Rightarrow \left( 2\right) \) . Since \( R/J\left( R\right) \cong {\mathbb{Z}}_{2}, R \) is a J-clean ring. So the result follows by Proposition 3.5.\n\n\( \left( 2\right) \Rightarrow \left( 1\right) \) . In view of Theorem 3.1, \( {M}_{n}\left( {R/J\left( R\right) }\right) \) is nil clean. As... | Yes |
Example 3.1 Let \( {\mathbb{Z}}_{\left( 2\right) } \) be the localization of \( \mathbb{Z} \) at (2). Then, by Corollary \( {3.2},{M}_{n}\left( R\right) \) is GJ-clean for any \( n \geq 2 \), but it is neither nil clean nor J-clean. | Indeed, as \( J\left( {{M}_{n}\left( R\right) }\right) \) is not nil, by \( \left\lbrack {{10}\text{, Corollary 3.17}}\right\rbrack {M}_{n}\left( R\right) \) is not nil clean; since \( {M}_{n}\left( R\right) \) is never GUJ, \( {M}_{n}\left( R\right) \) is not J-clean by Theorem 2.1. | Yes |
Proposition 3.6 Let \( R \) be a ring. Then the power series ring \( R\left\lbrack \left\lbrack x\right\rbrack \right\rbrack \) is GJ-clean if and only if \( R \) is GJ-clean. | Proof Suppose that \( R \) is GJ-clean. Let \( \left( x\right) = {xR}\left\lbrack \left\lbrack x\right\rbrack \right\rbrack \) . Then \( \left( x\right) \) is an ideal of \( R\left\lbrack \left\lbrack x\right\rbrack \right\rbrack \) and \( \left( x\right) \subseteq J\left( {R\left\lbrack \left\lbrack x\right\rbrack \ri... | Yes |
Proposition 2.1 Let \( {M}_{r}^{n}\left( {n > 2}\right) \) be a nonminimal hypersurface in de Sitter space \( {\mathbb{S}}_{1}^{n + 1}\left( 1\right) \) . Assume that the shape operator of \( {M}_{r}^{n} \) has Form (I). Then \( {M}_{r}^{n} \) is \( \eta \) -biharmonic if and only if it is an open part of a generalized... | Proof It is easy to compute that \( {\mu }_{{A}_{n + 1}}\left( x\right) = {\left( x - \lambda \right) }^{2} \), hence \( {M}_{r}^{n} \) is an open part of a generalized umbilical hypersurface in \( {\mathbb{S}}_{1}^{n + 1}\left( 1\right) \) . Now, we construct a pseudo-Riemannian orthonormal basis \( \left\{ {{e}_{1},{... | Yes |
Theorem 2.1 Suppose that \( f : {\mathbb{R}}^{n} \rightarrow \overline{\mathbb{R}} \) and \( g : {\mathbb{R}}^{n} \rightarrow \overline{\mathbb{R}} \) are proper and lower semicontinuous, \( f \) is strongly convex with modulus \( \sigma > 0, g \) is weakly convex with modulus \( \omega > 0 \) and \( \omega < \sigma \)... | Proof (a) From (2.7) and the fact \( \alpha \in \left( {{2\omega },{2\sigma }}\right) \), we see that\n\n\[ 0 \leq \alpha \left( {\alpha - {2\omega }}\right) {\begin{Vmatrix}{y}_{e}^{k + 1}\end{Vmatrix}}^{2} + \alpha \left( {{2\sigma } - \alpha }\right) {\begin{Vmatrix}{x}_{e}^{k + 1}\end{Vmatrix}}^{2} \leq {\begin{Vma... | Yes |
Theorem 1.1 Let \( X \) be a cofibrant object and \( Y \) be a fibrant object of a model category \( \mathcal{C} \) . Then\n\n(1) Two maps \( X \rightarrow Y \) are left homotopic if and only if they are right homotopic.\n\n(2) The relation of left or right homotopy on \( {\operatorname{Hom}}_{\mathcal{C}}\left( {X, Y}... | Proof See [4, Section 4]. | No |
Theorem 1.2 Let \( F : \mathcal{C} \rightarrow \mathcal{D} \) be a functor from a model category \( \mathcal{C} \) to a category \( \mathcal{D} \) such that for any weak equivalence \( f \in \mathcal{C} \) its image \( F\left( f\right) \in \mathcal{D} \) is an isomorphism. Then there exists a unique functor \( G : \) h... | Proof See [4, Theorem 4.2]. | No |
Proposition 1.1 Let \( \mathcal{C} \) be a model category such that every object of \( \mathcal{C} \) is cofibrant. Then \( \mathcal{C} \) is left proper. Dually, if every object of \( \mathcal{C} \) is fibrant, then \( \mathcal{C} \) is right proper. | Proof See [15, Proposition A.2.4.2]. | No |
Proposition 1.2 Let \( Y \leftarrow X \rightarrow Z \) be a diagram in a left proper model category where \( X \rightarrow Y \) is a cofibration. Then \( Y{ \coprod }_{X}Z \) is weakly equivalent to \( Y{ \coprod }_{X}^{h}Z \) . | Proof See [15, Proposition A.2.4.4]. | No |
Theorem 1.4 Let \( X \) be a cofibrant object and \( Y \) be a fibrant object in a model category \( \mathcal{C} \). (1) For any object \( A \) there exists a simplicial set \( {\operatorname{Map}}_{l}\left( {A, Y}\right) \), such that \( {\pi }_{0}{\operatorname{Map}}_{l}\left( {A, Y}\right) \cong \) \( {\left\lbrack ... | Proof See [13, Section 5.4]. | No |
Theorem 2.1 Let \( \mathcal{C} \) be a compactly generated pointed closed model category with \( * \) denoting its initial-terminal object. Suppose that a set valued contravariant functor \( F \) on \( \mathcal{C} \) satisfies the following conditions:\n\n(1) \( F\left( *\right) = * \) ,\n\n(2) \( F \) takes weak equiv... | Proof This is \( \left\lbrack {{14}\text{, Theorem 19}}\right\rbrack \) . | No |
Proposition 4.1 The category Vect is anti-equivalent to pcVect, and the category pcDGVect is anti-equivalent to DGVect. | Proof Given a vector space \( V \), its \( k \) -linear dual \( {V}^{ * } \) is pseudocompact. Indeed, denoting by \( \left\{ {V}_{\alpha }\right\} \) the collection of finite-dimensional subspaces of \( V \), we have \( V = \mathop{\lim }\limits_{{ \rightarrow \alpha }}{V}_{\alpha } \) and therefore \( {V}^{ * } = \ma... | Yes |
Proposition 4.5 The functors Cobar : \( {\mathrm{{pcDGA}}}_{\mathrm{{loc}}}^{\mathrm{{op}}} \leftrightarrows \mathrm{{DGA}}/\mathrm{k} \) : Bar form an adjoint pair. | Proof We only need to notice that for \( A \in {\operatorname{pcDGA}}_{\text{loc }} \) and \( \mathfrak{g} \in \mathrm{{DGA}}/\mathrm{k} \) there are natural isomorphisms\n\n\[ \n{\operatorname{Hom}}_{\mathrm{{DGA}}/\mathrm{k}}\left( {\operatorname{Cobar}\left( A\right) ,\mathfrak{g}}\right) \cong \operatorname{MC}\lef... | Yes |
Theorem 4.4 The category \( {\mathrm{{pcDGA}}}_{\text{loc }} \) together with the classes of fibrations, cofibrations and weak equivalences is a model category. Moreover, the adjoint pair of functors (Cobar, Bar) is a Quillen equivalence between \( {\mathrm{{pcDGA}}}_{\text{loc }}^{\text{op }} \) and DGA/k. | Proof See [19]. | No |
Theorem 4.5 There are the following isomorphisms, natural in both variables:\n\n\[ \n{\left\lbrack \operatorname{Cobar}\left( A\right) ,\mathfrak{g}\right\rbrack }_{\mathrm{{DGA}}/\mathrm{k}} \cong \mathcal{M}\mathcal{C}\left( {\mathfrak{I}\left( \mathfrak{g}\right), A}\right) \cong {\left\lbrack \operatorname{Bar}\lef... | Proof The proof is the same as that of Theorem 4.3 with \( \operatorname{Harr}\left( A\right) \) and \( \mathrm{{CE}}\left( \mathfrak{g}\right) \) replaced by \( \operatorname{Cobar}\left( A\right) \) and \( \operatorname{Bar}\left( \mathfrak{g}\right) \) respectively. The only difference is that we choose the smaller ... | Yes |
Proposition 4.6 The category \( {\mathrm{{pcDGA}}}_{\text{loc }}^{\text{op }} \) is compactly generated. | Proof The argument is the same as in Proposition 4.4, using Theorem 4.4 in place of Theorem 4.2. | No |
Proposition 4.7 The following diagrams of model categories and Quillen functors between them is commutative in the sense that there is a functor isomorphism \( U \circ \operatorname{Harr} \cong \) Cobar \( \circ \) Ass and \( \mathrm{{CE}} \circ \mathrm{{Lie}} \cong \mathrm{{Ab}} \circ \mathrm{{Bar}} \) . | Proof Straightforward unravelling of the definitions. | No |
Theorem 5.1 Let \( \mathfrak{g} \) be a \( \mathrm{{dg}} \) Lie algebra.\n\n(1) If \( A \rightarrow B \) is a weak equivalence in \( {\operatorname{pcCDGA}}_{\text{loc }} \), then the induced map \( {\operatorname{Def}}_{\mathfrak{g}}\left( A\right) \rightarrow \) \( {\operatorname{Def}}_{\mathfrak{g}}\left( B\right) \... | Proof This follows from Theorem 4.3. | No |
Theorem 5.2 The set-valued functor \( {\operatorname{Def}}_{\mathfrak{g}} \) on ho(pcCDGA \( {}_{\text{loc }} \) ) is representable by the local pseudocompact commutative \( \mathrm{{dg}} \) algebra \( \mathrm{{CE}}\left( \mathfrak{g}\right) \) . Conversely, any functor on ho(pcCDGA \( {}_{\text{loc }} \) ) that is hom... | Proof By Theorem 4.3 we have \( {\operatorname{Def}}_{\mathfrak{g}}\left( A\right) = \mathcal{M}\mathcal{C}\left( {\mathfrak{g}, A}\right) \cong \left\lbrack {\mathrm{{CE}}\left( \mathfrak{g}\right), A}\right\rbrack \), which means that \( {\operatorname{Def}}_{\mathfrak{g}} \) is representable by \( \mathrm{{CE}}\left... | Yes |
Theorem 5.3 Let \( F \) be a set-valued functor on \( {\mathrm{{pcCDGA}}}_{\text{loc }} \) such that\n\n(1) \( F \) is homotopy invariant: it takes weak equivalences in \( {\mathrm{{pcCDGA}}}_{\text{loc }} \) to bijections of sets.\n\n(2) \( F \) is normalized: \( F\left( k\right) \) is a one-element set.\n\n(3) \( F \... | Proof This follows from Brown representability, Theorem 2.1, taking into account that the model category \( {\mathrm{{pcCDGA}}}_{\text{loc }}^{\text{op }} \) is compactly generated, cf. Proposition 4.4. | Yes |
Theorem 5.4 Let \( \mathfrak{g} \) be an augmented dg algebra.\n\n(1) If \( A \rightarrow B \) is a weak equivalence in \( {\operatorname{pcDGA}}_{\text{loc }} \), then the induced map \( {\operatorname{Def}}_{\mathfrak{g}}\left( A\right) \rightarrow {\operatorname{Def}}_{\mathfrak{g}}\left( B\right) \) is an isomorphi... | Proof This follows from Theorem 4.5. | No |
Theorem 5.5 The set-valued functor \( {\operatorname{Def}}_{\mathfrak{g}} \) on ho(pcDGA \( {}_{\text{loc }} \) ) is representable by the local pseudocompact \( \mathrm{{dg}} \) algebra \( \mathrm{{Bar}}\left( \mathfrak{g}\right) \) . Conversely, any functor on ho(pcDGA \( {}_{\text{loc }} \) ) that is homotopy represe... | Proof The proof is the same as that of Theorem 5.2, applying Theorem 4.5 instead of Theorem 4.3. | No |
Theorem 5.6 Let \( F \) be a set-valued functor on \( {\mathrm{{pcDGA}}}_{\text{loc }} \) such that:\n\n(1) \( F \) is homotopy invariant: it takes weak equivalences in \( {\mathrm{{pcDGA}}}_{\text{loc }} \) to bijections of sets;\n\n(2) \( F \) is normalized: \( F\left( k\right) \) is a one-element set.\n\n(3) \( F \)... | Proof This follows from Brown representability, Theorem 2.1, taking into account that the model category \( {\mathrm{{pcDGA}}}_{\text{loc }}^{\text{op }} \) is compactly generated, cf. Proposition 4.4. | Yes |
Theorem 5.7 Let \( \mathfrak{g} \) be a \( \mathrm{{dg}} \) algebra. Then the deformation functor \( {\operatorname{Def}}_{\mathfrak{g}} \) on \( {\mathrm{{pcDGA}}}_{\text{loc }} \) restricts to the deformation functor \( {\operatorname{Def}}_{\operatorname{Lie}\left( \mathfrak{g}\right) } \) on \( {\operatorname{pcCDG... | Proof We know by Theorem 5.5 that \( {\operatorname{Def}}_{\mathfrak{g}} \) is represented by a \( \mathrm{{dg}} \) algebra \( \operatorname{Bar}\left( \mathfrak{g}\right) \) . Then for \( \mathfrak{h} \in {\operatorname{pcCDGA}}_{\text{loc }} \) we have \( {\operatorname{Def}}_{\mathfrak{g}}\left( \mathfrak{h}\right) ... | Yes |
Lemma 0.2 The minimum genus of a minor of a graph \( G \) can never be larger than \( \gamma \left( G\right) \) . | Proof Let the graph \( G \) be embedded in a surface \( S \), then contracting an edge \( e \) of \( G \) on \( S \) can obtain an embedding of the contracted graph \( G/e \) on \( S \) . Moreover, edge deletion can never increase embedding genus. Thus, the lemma is obtained. | Yes |
Lemma 0.3 If an orientable surface \( S \) has the form as \( \left( {{AxByC}{x}^{ - }D{y}^{ - }E}\right) \), then \( g\left( S\right) \geq 1 \) , furthermore, the genus of \( S \) is \( p\left( { \geq 1}\right) \) if and only if \( {ADCBE} \) is with genus \( p - 1 \) . | Proof According to Transform 4, it is obvious. | No |
Lemma 1.2 If the apex-vertex of the near-wheel graph is the type-2 apex-vertex \( {v}_{\text{typ }}^{2} \) , then \( \gamma \left( {{W}_{n} \oplus {v}_{\text{typ }}^{2}}\right) = 1 \) . | Proof It is easy to find out that \( {K}_{3,3} \) is a minor of \( {W}_{n} \oplus {v}_{\text{typ }}^{2} \) . According to Lemma 0.2, we can get that \( \gamma \left( {{W}_{n} \oplus {v}_{\mathrm{{typ}}}^{2}}\right) \geq 1 \) . Let \( {v}_{1},{v}_{2},{v}_{3} \) be the three antennal-vertices of \( {W}_{n} \oplus {v}_{\m... | Yes |
Lemma 1.3 If the apex-vertex of the near-wheel graph is the type-3 apex-vertex \( {v}_{\text{typ }}^{3} \) , then \( \gamma \left( {{W}_{n} \oplus {v}_{\text{typ }}^{3}}\right) = 1 \) . | Proof It is not difficult to find out that \( {K}_{3,3} \) is a minor of \( {W}_{n} \oplus {v}_{\text{typ }}^{3} \) . According to Lemma 0.2, we can get that \( \gamma \left( {{W}_{n} \oplus {v}_{\text{typ }}^{3}}\right) \geq 1 \) .\n\nCase 1: The three antennal-vertices of \( {W}_{n} \oplus {v}_{\text{typ }}^{3} \) ar... | Yes |
Theorem 3 Let \( S\left( {{a}_{1},{a}_{2},\cdots ,{a}_{t},{b}_{1},{b}_{2},\cdots ,{b}_{t}}\right) \) be a star-like tree with \( \Delta \left( S\right) \leq n + 1 \) for an odd integer \( n \geq 2 \) and \( t \geq 3 \) . Then \( S▱{P}_{n} \) is not AP. | Proof To prove \( G = S▱{P}_{n} \) is not AP, we prove that \( G \) is not \( \left( {2,2,\cdots ,2}\right) \) -partitionable, or \( \left( {1,2,2,\cdots ,2}\right) \) -partitionable. That is, \( G \) does not have perfect matching or quasi-perfect matching.\n\nRecall that \( {S}^{g} \) denotes the \( g \) -th cope of ... | Yes |
Proposition 1.3 Every idempotent in a \( * \) -reversible ring is a projection. | Proof Let \( R \) be a \( * \) -reversible ring. Note that if \( e \in R \) is an idempotent, then \( e\left( {1 - e}\right) = 0 \) implies \( \left( {1 - e}\right) {e}^{ * } = 0 \) . This implies that \( {e}^{ * } = e{e}^{ * } = e \) . | No |
Proposition 1.4 Let \( R \) be a \( * \) -reflexive ring. Then any central idempotent of \( R \) is a projection. | Proof For any central idempotent \( e \in R, e\left( {1 - e}\right) = 0 \) implies \( {Re}\left( {1 - e}\right) = 0 \) . Then we have \( {eR}\left( {1 - e}\right) = 0 \) since \( e \) is central. Thus, \( \left( {1 - e}\right) R{e}^{ * } = 0 \) since \( R \) is \( * \) -reflexive. It follows that \( \left( {1 - e}\righ... | Yes |
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