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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 Compact quantum group ： ウィキペディア英語版 Compact quantum group In mathematics, a compact quantum group is an abstract structure on a unital separable C *-algebra axiomatized from those that exist on the commutative C *-algebra of "continuous complex-valued functions" on a compact quantum group. The basic motivation for this theory comes from the following analogy. The space of complex-valued functions on a compact Hausdorff topological space forms a ''commutative'' C *-algebra. On the other hand, by the Gelfand Theorem, a commutative C *-algebra is isomorphic to the C *-algebra of continuous complex-valued functions on a compact Hausdorff topological space, and the topological space is uniquely determined by the C *-algebra up to homeomorphism. S. L. Woronowicz 〔Woronowicz, S.L. "Compact Matrix Pseudogrooups", Commun. Math. Phys. 111 (1987), 613-665〕 introduced the important concept of compact matrix quantum groups, which he initially called compact pseudogroups. Compact matrix quantum groups are abstract structures on which the "continuous functions" on the structure are given by elements of a C *-algebra. The geometry of a compact matrix quantum group is a special case of a noncommutative geometry. ==Formulation== For a compact topological group, , there exists a C *-algebra homomorphism :$\backslash Delta\; :\; C(G)\; \backslash to\; C(G)\; \backslash otimes\; C(G)$ where is the minimal C *-algebra tensor product — the completion of the algebraic tensor product of and ) — such that :$\backslash Delta(f)(x,y)\; =\; f(xy)$ for all $f\; \backslash in\; C(G)$, and for all $x,\; y\; \backslash in\; G$, where :$(f\; \backslash otimes\; g)(x,y)\; =\; f(x)\; g(y)$ for all $f,\; g\; \backslash in\; C(G)$ and all $x,\; y\; \backslash in\; G$. There also exists a linear multiplicative mapping :$\backslash kappa\; :\; C(G)\; \backslash to\; C(G)$, such that :$\backslash kappa(f)(x)\; =\; f(x^)$ for all $f\; \backslash in\; C(G)$ and all $x\; \backslash in\; G$. Strictly speaking, this does not make into a Hopf algebra, unless is finite. On the other hand, a finite-dimensional representation of can be used to generate a *-subalgebra of which is also a Hopf *-algebra. Specifically, if :$g\; \backslash mapsto\; (u\_(g))\_$ is an -dimensional representation of , then :$u\_\; \backslash in\; C(G)$ for all , and :$\backslash Delta(u\_)\; =\; \backslash sum\_k\; u\_\; \backslash otimes\; u\_$ for all . It follows that the *-algebra generated by $u\_$ for all and $\backslash kappa(u\_)$ for all is a Hopf *-algebra: the counit is determined by :$\backslash epsilon(u\_)\; =\; \backslash delta\_$ for all $i,\; j$ (where $\backslash delta\_$ is the Kronecker delta), the antipode is , and the unit is given by :$1\; =\; \backslash sum\_k\; u\_\; \backslash kappa(u\_)\; =\; \backslash sum\_k\; \backslash kappa(u\_)\; u\_.$ 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「Compact quantum group」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.