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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 Quasitriangular Hopf algebra ： ウィキペディア英語版 Quasitriangular Hopf algebra In mathematics, a Hopf algebra, ''H'', is quasitriangular〔Montgomery & Schneider (2002), (p. 72 ).〕 if there exists an invertible element, ''R'', of $H\; \backslash otimes\; H$ such that : *$R\; \backslash \; \backslash Delta(x)R^\; =\; (T\; \backslash circ\; \backslash Delta)(x)\; \backslash \; R$ for all $x\; \backslash in\; H$, where $\backslash Delta$ is the coproduct on ''H'', and the linear map $T\; :\; H\; \backslash otimes\; H\; \backslash to\; H\; \backslash otimes\; H$ is given by $T(x\; \backslash otimes\; y)\; =\; y\; \backslash otimes\; x$, : *$(\backslash Delta\; \backslash otimes\; 1)(R)\; =\; R\_\; \backslash \; R\_$, : *$(1\; \backslash otimes\; \backslash Delta)(R)\; =\; R\_\; \backslash \; R\_$, where $R\_\; =\; \backslash phi\_(R)$, $R\_\; =\; \backslash phi\_(R)$, and $R\_\; =\; \backslash phi\_(R)$, where $\backslash phi\_\; :\; H\; \backslash otimes\; H\; \backslash to\; H\; \backslash otimes\; H\; \backslash otimes\; H$, $\backslash phi\_\; :\; H\; \backslash otimes\; H\; \backslash to\; H\; \backslash otimes\; H\; \backslash otimes\; H$, and $\backslash phi\_\; :\; H\; \backslash otimes\; H\; \backslash to\; H\; \backslash otimes\; H\; \backslash otimes\; H$, are algebra morphisms determined by :$\backslash phi\_(a\; \backslash otimes\; b)\; =\; a\; \backslash otimes\; b\; \backslash otimes\; 1,$ :$\backslash phi\_(a\; \backslash otimes\; b)\; =\; a\; \backslash otimes\; 1\; \backslash otimes\; b,$ :$\backslash phi\_(a\; \backslash otimes\; b)\; =\; 1\; \backslash otimes\; a\; \backslash otimes\; b.$ ''R'' is called the R-matrix. As a consequence of the properties of quasitriangularity, the R-matrix, ''R'', is a solution of the Yang-Baxter equation (and so a module ''V'' of ''H'' can be used to determine quasi-invariants of braids, knots and links). Also as a consequence of the properties of quasitriangularity, $(\backslash epsilon\; \backslash otimes\; 1)\; R\; =\; (1\; \backslash otimes\; \backslash epsilon)\; R\; =\; 1\; \backslash in\; H$; moreover $R^\; =\; (S\; \backslash otimes\; 1)(R)$, $R\; =\; (1\; \backslash otimes\; S)(R^)$, and $(S\; \backslash otimes\; S)(R)\; =\; R$. One may further show that the antipode ''S'' must be a linear isomorphism, and thus ''S2'' is an automorphism. In fact, ''S2'' is given by conjugating by an invertible element: $S^2(x)=\; u\; x\; u^$ where $u\; :=\; m\; (S\; \backslash otimes\; 1)R^$ (cf. Ribbon Hopf algebras). It is possible to construct a quasitriangular Hopf algebra from a Hopf algebra and its dual, using the Drinfeld quantum double construction. ==Twisting== The property of being a quasi-triangular Hopf algebra is preserved by twisting via an invertible element $F\; =\; \backslash sum\_i\; f^i\; \backslash otimes\; f\_i\; \backslash in\; \backslash mathcal$ such that $(\backslash varepsilon\; \backslash otimes\; id\; )F\; =\; (id\; \backslash otimes\; \backslash varepsilon)F\; =\; 1$ and satisfying the cocycle condition :$(F\; \backslash otimes\; 1)\; \backslash circ\; (\backslash Delta\; \backslash otimes\; id)\; F\; =\; (1\; \backslash otimes\; F)\; \backslash circ\; (id\; \backslash otimes\; \backslash Delta)\; F$ Furthermore, $u\; =\; \backslash sum\_i\; f^i\; S(f\_i)$ is invertible and the twisted antipode is given by $S\text{'}(a)\; =\; u\; S(a)u^$, with the twisted comultiplication, R-matrix and co-unit change according to those defined for the quasi-triangular quasi-Hopf algebra. Such a twist is known as an admissible (or Drinfeld) twist. 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「Quasitriangular Hopf algebra」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.