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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 Clifford bundle ： ウィキペディア英語版 Clifford bundle In mathematics, a Clifford bundle is an algebra bundle whose fibers have the structure of a Clifford algebra and whose local trivializations respect the algebra structure. There is a natural Clifford bundle associated to any (pseudo) Riemannian manifold ''M'' which is called the Clifford bundle of ''M''. ==General construction== Let ''V'' be a (real or complex) vector space together with a symmetric bilinear form <·,·>. The Clifford algebra ''Cℓ''(''V'') is a natural (unital associative) algebra generated by ''V'' subject only to the relation :$v^2\; =\; -\backslash langle\; v,v\backslash rangle$ for all ''v'' in ''V''.〔There is an arbitrary choice of sign in the definition of a Clifford algebra. In general, one can take ''v''2 = ±<''v'',''v''>. In differential geometry, it is common to use the (−) sign convention.〕 One can construct ''Cℓ''(''V'') as a quotient of the tensor algebra of ''V'' by the ideal generated by the above relation. Like other tensor operations, this construction can be carried out fiberwise on a smooth vector bundle. Let ''E'' be a smooth vector bundle over a smooth manifold ''M'', and let ''g'' be a smooth symmetric bilinear form on ''E''. The Clifford bundle of ''E'' is the fiber bundle whose fibers are the Clifford algebras generated by the fibers of ''E'': :$C\backslash ell(E)\; =\; \backslash coprod\_\; C\backslash ell(E\_x,g\_x)$ The topology of ''Cℓ''(''E'') is determined by that of ''E'' via an associated bundle construction. One is most often interested in the case where ''g'' is positive-definite or at least nondegenerate; that is, when (''E'', ''g'') is a Riemannian or pseudo-Riemannian vector bundle. For concreteness, suppose that (''E'', ''g'') is a Riemannian vector bundle. The Clifford bundle of ''E'' can be constructed as follows. Let ''Cℓ''''n''R be the Clifford algebra generated by R''n'' with the Euclidean metric. The standard action of the orthogonal group O(''n'') on R''n'' induces a graded automorphism of ''Cℓ''''n''R. The homomorphism :$\backslash rho\; :\; \backslash mathrm\; O(n)\; \backslash to\; \backslash mathrm(C\backslash ell\_n\backslash mathbb\; R)$ is determined by :$\backslash rho(A)(v\_1v\_2\backslash cdots\; v\_k)\; =\; (Av\_1)(Av\_2)\backslash cdots(Av\_k)$ where ''v''''i'' are all vectors in R''n''. The Clifford bundle of ''E'' is then given by :$C\backslash ell(E)\; =\; F(E)\; \backslash times\_\backslash rho\; C\backslash ell\_n\backslash mathbb\; R$ where ''F''(''E'') is the orthonormal frame bundle of ''E''. It is clear from this construction that the structure group of ''Cℓ''(''E'') is O(''n''). Since O(''n'') acts by graded automorphisms on ''Cℓ''''n''R it follows that ''Cℓ''(''E'') is a bundle of Z2-graded algebras over ''M''. The Clifford bundle ''Cℓ''(''E'') can then be decomposed into even and odd subbundles: :$C\backslash ell(E)\; =\; C\backslash ell^0(E)\; \backslash oplus\; C\backslash ell^1(E).$ If the vector bundle ''E'' is orientable then one can reduce the structure group of ''Cℓ''(''E'') from O(''n'') to SO(''n'') in the natural manner. 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「Clifford bundle」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.