翻訳と辞書 Words near each other ・ Tanna infuscata ・ Tanna japonensis ・ Tanna karenkonis ・ Tanna ornatipennis ・ Tanna Peak ・ Tanna sayurie ・ Tanna sozanensis ・ Tanna taipinensis ・ Tanna Tunnel ・ Tanna viridis ・ Tanna, Germany ・ Tannadice Park ・ Tannadice, Angus ・ Tannahill ・ Tannaim ・ Tannaka–Krein duality ・ Tannakian category ・ Tannase ・ Tannat ・ Tannate ・ Tannay ・ Tannay, Ardennes ・ Tannay, Nièvre ・ Tannay, Switzerland ・ Tannbach (Saale) ・ Tannbach (TV series) ・ Tanne ・ Tanne, Saxony-Anhalt ・ Tanneberger ・ Tanned Legs 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 Tannaka–Krein duality ： ウィキペディア英語版 Tannaka–Krein duality In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations. It is a natural extension of Pontryagin duality, between compact and discrete commutative topological groups, to groups that are compact but noncommutative. The theory is named for two men, the Soviet mathematician Mark Grigorievich Krein, and the Japanese Tadao Tannaka. In contrast to the case of commutative groups considered by Lev Pontryagin, the notion dual to a noncommutative compact group is not a group, but a category Π(''G'') with some additional structures, formed by the finite-dimensional representations of ''G''. Duality theorems of Tannaka and Krein describe the converse passage from the category Π(''G'') back to the group ''G'', allowing one to recover the group from its category of representations. Moreover, they in effect completely characterize all categories that can arise from a group in this fashion. Alexander Grothendieck later showed that by a similar process, Tannaka duality can be extended to the case of algebraic groups: see tannakian category. Meanwhile, the original theory of Tannaka and Krein continued to be developed and refined by mathematical physicists. A generalization of Tannaka–Krein theory provides the natural framework for studying representations of quantum groups, and is currently being extended to quantum supergroups, quantum groupoids and their dual quantum algebroids. == The idea of Tannaka–Krein duality: category of representations of a group == In Pontryagin duality theory for locally compact commutative groups, the dual object to a group ''G'' is its character group $\backslash hat,$ which consists of its one-dimensional unitary representations. If we allow the group ''G'' to be noncommutative, the most direct analogue of the character group is the set of equivalence classes of irreducible unitary representations of ''G''. The analogue of the product of characters is the tensor product of representations. However, irreducible representations of ''G'' in general fail to form a group, because a tensor product of irreducible representations is not necessarily irreducible. It turns out that one needs to consider the set Π(''G'') of all finite-dimensional representations, and treat it as a monoidal category, where the product is the usual tensor product of representations, and the dual object is given by the operation of the contragredient representation. A representation of the category Π(''G'') is a monoidal natural transformation from the identity functor $id\_$ to itself. In other words, it is a non-zero function φ that associates with any $T\backslash in\; Ob\; \backslash Pi(G)$ an endomorphism of the space of ''T'' and satisfies the conditions of compatibility with tensor products, $\backslash phi(T\backslash otimes\; U)=\backslash phi(T)\backslash otimes\backslash phi(U)$, and with arbitrary intertwining operators ''f:'' ''T'' → ''U'', namely, $f\backslash circ\; \backslash phi(T)\; =\; \backslash phi(U)\; \backslash circ\; f$. The collection Γ(Π(''G'')) of all representations of the category Π(''G'') can be endowed with multiplication φψ(''T'') = φ(''T'') ψ(''T'') and topology, in which convergence is defined pointwise, i.e. a sequence $\backslash$ converges to some $\backslash phi$ if and only if $\backslash$ converges to $\backslash phi(T)$ for all $T\backslash in\; Ob\backslash Pi(G)$. It can be shown that the set Γ(Π(''G'')) thus becomes a compact (topological) group. 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「Tannaka–Krein duality」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.