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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Chern–Weil homomorphism ： ウィキペディア英語版 Chern–Weil homomorphism In mathematics, the Chern–Weil homomorphism is a basic construction in the Chern–Weil theory that computes topological invariants of vector bundles and principal bundles on a smooth manifold ''M'' in terms of connections and curvature representing classes in the de Rham cohomology rings of ''M''. That is, the theory forms a bridge between the areas of algebraic topology and differential geometry. It was developed in the late 1940s by Shiing-Shen Chern and André Weil, in the wake of proofs of the generalized Gauss–Bonnet theorem. This theory was an important step in the theory of characteristic classes. Let ''G'' be a real or complex Lie group with Lie algebra $\backslash mathfrak\; g$; and let $\backslash mathbb(g)$ denote the algebra of $\backslash mathbb$-valued polynomials on $\backslash mathfrak\; g$ (exactly the same argument works if we used $\backslash mathbb$ instead of $\backslash mathbb$.) Let $\backslash mathbb(g)^G$ be the subalgebra of fixed points in $\backslash mathbb\; C(g)$ under the adjoint action of ''G''; that is, it consists of all polynomials ''f'' such that for any ''g'' in ''G'' and ''x'' in $\backslash mathfrak$, $f(\backslash operatorname\_g\; x)\; =\; f(x).$ The Chern–Weil homomorphism is a homomorphism of $\backslash mathbb\; C$-algebras :$\backslash mathbb\; C(g)^\; \backslash to\; H^$ *(M,\mathbb C) where on the right cohomology is de Rham cohomology. Such a homomorphism exists and is uniquely defined for every principal G-bundle ''P'' on ''M''. If ''G'' is compact, then under the homomorphism, the cohomology ring of the classifying space for ''G''-bundles ''BG'' is isomorphic to the algebra $\backslash mathbb\; C(g)^$ of invariant polynomials: :$H^$ *(BG, \mathbb) \cong \mathbb C(g )^. (The cohomology ring of ''BG'' can still be given in the de Rham sense: :$H^k(BG,\; \backslash mathbb)\; =\; \backslash varinjlim\; \backslash operatorname\; (d:\; \backslash Omega^k(B\_jG)\; \backslash to\; \backslash Omega^(B\_jG))/\backslash operatorname\; d.$ when $BG\; =\; \backslash varinjlim\; B\_jG$ and $B\_jG$ are manifolds.) For non-compact groups like SL(''n'',R), there may be cohomology classes that are not represented by invariant polynomials. ==Definition of the homomorphism== Choose any connection form ω in ''P'', and let Ω be the associated curvature 2-form; i.e., Ω = ''D''ω, the exterior covariant derivative of ω. If $f\backslash in\backslash mathbb\; C(g)^G$ is a homogeneous polynomial function of degree ''k''; i.e., $f(a\; x)\; =\; a^k\; f(x)$ for any complex number ''a'' and ''x'' in $\backslash mathfrak\; g$, then, viewing ''f'' as a symmetric multilinear functional on $\backslash prod\_1^k\; \backslash mathfrak$ (see the ring of polynomial functions), let :$f(\backslash Omega)$ be the (scalar-valued) 2''k''-form on ''P'' given by :$f(\backslash Omega)(v\_1,\backslash dots,v\_)=\backslash frac\backslash sum\_,v\_),\backslash dots,\backslash Omega(v\_,\; v\_))$ where ''v''''i'' are tangent vectors to ''P'', $\backslash epsilon\_\backslash sigma$ is the sign of the permutation $\backslash sigma$ in the symmetric group on 2''k'' numbers $\backslash mathfrak\; S\_$ (see Lie algebra-valued forms#Operations as well as Pfaffian). If, moreover, ''f'' is invariant; i.e., $f(\backslash operatorname\_g\; x)\; =\; f(x)$, then one can show that $f(\backslash Omega)$ is a closed form, it descends to a unique form on ''M'' and that the de Rham cohomology class of the form is independent of ''ω''. First, that $f(\backslash Omega)$ is a closed form follows from the next two lemmas: :Lemma 1: The form $f(\backslash Omega)$ on ''P'' descends to a (unique) form $\backslash overline(\backslash Omega)$ on ''M''; i.e., there is a form on ''M'' that pulls-back to $f(\backslash Omega)$. :Lemma 2: If a form φ on ''P'' descends to a form on ''M'', then dφ = Dφ. Indeed, Bianchi's second identity says $D\; \backslash Omega\; =\; 0$ and, since ''D'' is a graded derivation, $D\; f(\backslash Omega)\; =\; 0.$ Finally, Lemma 1 says $f(\backslash Omega)$ satisfies the hypothesis of Lemma 2. To see Lemma 2, let $\backslash pi:\; P\; \backslash to\; M$ be the projection and ''h'' be the projection of $T\_u\; P$ onto the horizontal subspace. Then Lemma 2 is a consequence of the fact that $d\; \backslash pi(h\; v)\; =\; d\; \backslash pi(v)$ (the kernel of $d\; \backslash pi$ is precisely the vertical subspace.) As for Lemma 1, first note :$f(\backslash Omega)(d\; R\_g(v\_1),\; \backslash dots,\; d\; R\_g(v\_))\; =\; f(\backslash Omega)(v\_1,\; \backslash dots,\; v\_),\; \backslash ,\; R\_g(u)\; =\; ug;$ which is because $R\_g^$ * \Omega = \operatorname_(\Omega) by the formula: :$\backslash overline(\backslash Omega)(\backslash overline,\; \backslash dots,\; \backslash overline)$ where $v\_i$ are any lifts of $\backslash overline$: $d\; \backslash pi(v\_i)\; =\; \backslash overline\_i$. Next, we show that the de Rham cohomology class of $\backslash overline(\backslash Omega)$ on ''M'' is independent of a choice of connection.〔The argument for the independent of a choice of connection here is taken from: Akhil Mathew, Notes on Kodaira vanishing (). Kobayashi-Nomizu, the main reference, gives a more concrete argument.〕 Let $\backslash omega\_0,\; \backslash omega\_1$ be arbitrary connection forms on ''P'' and let $p:\; P\; \backslash times\; \backslash mathbb\; \backslash to\; P$ be the projection. Put :$\backslash omega\text{'}\; =\; t\; \backslash ,\; p^$ * \omega_1 + (1 - t) \, p^ * \omega_0 where ''t'' is a smooth function on $P\; \backslash times\; \backslash mathbb$ given by $(x,\; s)\; \backslash mapsto\; s$. Let $\backslash Omega\text{'},\; \backslash Omega\_0,\; \backslash Omega\_1$ be the curvature forms of $\backslash omega\text{'},\; \backslash omega\_0,\; \backslash omega\_1$. Let $i\_s:\; M\; \backslash to\; M\; \backslash times\; \backslash mathbb,\; \backslash ,\; x\; \backslash mapsto\; (x,\; s)$ be the inclusions. Then $i\_0$ is homotopic to $i\_1$. Thus, $i\_0^$ * \overline(\Omega') and $i\_1^$ * \overline(\Omega') belong to the same de Rham cohomology class by the homotopy invariance of de Rham cohomology. Finally, by naturality and by uniqueness of descending, :$i\_0^$ * \overline(\Omega') = \overline(\Omega_0) and the same for $\backslash Omega\_1$. Hence, $\backslash overline(\backslash Omega\_0),\; \backslash overline(\backslash Omega\_1)$ belong to the same cohomology class. The construction thus gives the linear map: (cf. Lemma 1) :$\backslash mathbb\; C(g)^\_k\; \backslash rightarrow\; H^(M,\backslash mathbb\; C),\; \backslash ,\; f\; \backslash mapsto\; \backslash left().$ In fact, one can check that the map thus obtained: :$\backslash mathbb\; C(g)^\; \backslash rightarrow\; H^$ *(M,\mathbb C) is an algebra homomorphism. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Chern–Weil homomorphism」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.