翻訳と辞書 Words near each other ・ Weak convergence (Hilbert space) ・ Weak derivative ・ Weak dimension ・ Weak duality ・ Weak entity ・ Weak equivalence ・ Weak equivalence (homotopy theory) ・ Weak evolutionarily stable strategy ・ Weak focusing ・ Weak form and strong form ・ Weak formulation ・ Weak generative capacity ・ Weak gravitational lensing ・ Weak Hausdorff space ・ Weak heap ・ Weak Hopf algebra ・ Weak hypercharge ・ Weak in the Presence of Beauty ・ Weak in the Presence of Beauty (album) ・ Weak in the Presence of Beauty (song) ・ Weak inflection ・ Weak interaction ・ Weak interpretability ・ Weak inverse ・ Weak isospin ・ Weak key ・ Weak localization ・ Weak measurement ・ Weak Moments of the Shadows ・ Weak n-category Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Weak Hopf algebra ： ウィキペディア英語版 Weak Hopf algebra In mathematics, weak bialgebras are a generalization of bialgebras that are both algebras and coalgebras but for which the compatibility conditions between the two structures have been "weakened". In the same spirit, weak Hopf algebras are weak bialgebras together with a linear map S satisfying specific conditions; they are generalizations of Hopf algebras. These objects were introduced by Böhm, Nill and Szlachányi. The first motivations for studying them came from quantum field theory and operator algebras.〔Böhm, Nill, Szlachányi. p. 387〕 Weak Hopf algebras have quite interesting representation theory; in particular modules over a semisimple finite weak Hopf algebra is a fusion category (which is a monoidal category with extra properties). It was also shown by Etingof, Nikshych and Ostrik that any fusion category is equivalent to a category of modules over a weak Hopf algebra.〔Etingof, Nikshych and Ostrik, Cor. 2.22〕 ==Definition== A weak bialgebra $(H,\; \backslash mu,\; \backslash eta,\; \backslash Delta,\; \backslash varepsilon)$ over a field $k$ is a vector space $H$ such that * $(H,\; \backslash mu,\; \backslash eta)$ forms an associative algebra with multiplication $\backslash mu\; :\; H\; \backslash otimes\; H\; \backslash rightarrow\; H$ and unit $\backslash eta\; :\; k\; \backslash rightarrow\; H$, * $(H,\; \backslash Delta,\; \backslash varepsilon)$ forms a coassociative coalgebra with comultiplication $\backslash Delta\; :\; H\; \backslash rightarrow\; H\; \backslash otimes\; H$ and counit $\backslash varepsilon\; :\; H\; \backslash rightarrow\; k$, for which the following compatibility conditions hold : # Multiplicativity of the Comultiplication : #: $\backslash Delta\; \backslash circ\; \backslash mu\; =\; (\backslash mu\; \backslash otimes\; \backslash mu)\; \backslash circ\; (\backslash mathrm\_H\; \backslash otimes\; \backslash sigma\_\; \backslash otimes\; \backslash mathrm\_H)\; \backslash circ\; (\backslash Delta\; \backslash otimes\; \backslash Delta)$, # Weak Multiplicativity of the Counit : #: $\backslash varepsilon\; \backslash circ\; \backslash mu\; \backslash circ\; (\backslash mu\; \backslash otimes\; \backslash mathrm\_H)\; =\; (\backslash varepsilon\; \backslash otimes\; \backslash varepsilon\; )\; \backslash circ\; (\backslash mu\; \backslash otimes\; \backslash mu)\; \backslash circ\; (\backslash mathrm\_H\; \backslash otimes\; \backslash Delta\; \backslash otimes\; \backslash mathrm\_H)=\; (\backslash varepsilon\; \backslash otimes\; \backslash varepsilon\; )\; \backslash circ\; (\backslash mu\; \backslash otimes\; \backslash mu)\; \backslash circ\; (\backslash mathrm\_H\; \backslash otimes\; \backslash Delta^\; \backslash otimes\; \backslash mathrm\_H)$, # Weak Comultiplicativity of the Unit : #: $(\backslash Delta\; \backslash otimes\; \backslash mathrm\_H)\; \backslash circ\; \backslash Delta\; \backslash circ\; \backslash eta\; =\; (\backslash mathrm\_H\; \backslash otimes\; \backslash mu\; \backslash otimes\; \backslash mathrm\_H)\; \backslash circ\; (\backslash Delta\; \backslash otimes\; \backslash Delta)\; \backslash circ\; (\backslash eta\; \backslash otimes\; \backslash eta)\; =\; (\backslash mathrm\_H\; \backslash otimes\; \backslash mu^\; \backslash otimes\; \backslash mathrm\_H)\; \backslash circ\; (\backslash Delta\; \backslash otimes\; \backslash Delta)\; \backslash circ\; (\backslash eta\; \backslash otimes\; \backslash eta)$, where $\backslash sigma\_\; :\; V\; \backslash otimes\; W\; \backslash rightarrow\; W\; \backslash otimes\; V\; :\; v\; \backslash otimes\; w\; \backslash mapsto\; w\; \backslash otimes\; v$ flips the two tensor factors. Moreover $\backslash mu^\; =\; \backslash mu\; \backslash circ\; \backslash sigma\_$ is the opposite multiplication and $\backslash Delta^\; =\; \backslash sigma\_\; \backslash circ\; \backslash Delta$ is the opposite comultiplication. Note that we also implicitly use Mac Lane's coherence theorem for the monoidal category of vector spaces, identifying $(U\; \backslash otimes\; V)\; \backslash otimes\; W\; \backslash cong\; U\; \backslash otimes\; (V\; \backslash otimes\; W)$ as well as $V\; \backslash otimes\; k\; \backslash cong\; V\; \backslash cong\; k\; \backslash otimes\; V$. The definition is fairly self-explanatory, one sees that it is the compatibility between the algebra and coalgebra structures that is weaken. A weak Hopf algebra $(H,\; \backslash mu,\; \backslash eta,\; \backslash Delta,\; \backslash varepsilon,\; S)$ is a weak bialgebra $(H,\; \backslash mu,\; \backslash eta,\; \backslash Delta,\; \backslash varepsilon)$ with a linear map $S\; :\; H\; \backslash to\; H$, called the antipode, that satisfies: * $\backslash mu\; \backslash circ\; (\backslash mathrm\_H\; \backslash otimes\; S)\; \backslash circ\; \backslash Delta\; =\; (\backslash varepsilon\; \backslash otimes\; \backslash mathrm\_H)\; \backslash circ\; (\backslash mu\; \backslash otimes\; \backslash mathrm\_H)\; \backslash circ\; (\backslash mathrm\_H\; \backslash otimes\; \backslash sigma\_)\; \backslash circ\; (\backslash Delta\; \backslash otimes\; \backslash mathrm\_H)\; \backslash circ\; (\backslash eta\; \backslash otimes\; \backslash mathrm\_H)$, *$\backslash mu\; \backslash circ\; (S\; \backslash otimes\; \backslash mathrm\_H)\; \backslash circ\; \backslash Delta\; =\; (\backslash mathrm\_H\; \backslash otimes\; \backslash varepsilon)\; \backslash circ\; (\backslash mathrm\_H\; \backslash otimes\; \backslash mu)\; \backslash circ\; (\backslash sigma\_\; \backslash otimes\; \backslash mathrm\_H)\; \backslash circ\; (\backslash mathrm\_H\; \backslash otimes\; \backslash Delta)\; \backslash circ\; (\backslash mathrm\_H\; \backslash otimes\; \backslash eta)$, *$S\; =\; \backslash mu\; \backslash circ\; (\backslash mu\; \backslash otimes\; \backslash mathrm\_H)\; \backslash circ\; (S\; \backslash otimes\; \backslash mathrm\_H\; \backslash otimes\; S)\; \backslash circ\; (\backslash Delta\; \backslash otimes\; \backslash mathrm\_H)\; \backslash circ\; \backslash Delta$. 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「Weak 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. 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