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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 Group Hopf algebra ： ウィキペディア英語版 Group Hopf algebra In mathematics, the group Hopf algebra of a given group is a certain construct related to the symmetries of group actions. Deformations of group Hopf algebras are foundational in the theory of quantum groups. ==Definition== Let ''G'' be an arbitrary group and ''k'' a field. The ''group Hopf algebra'' of ''G'' over ''k'', denoted ''kG'' (or ''k''()), is as a set (and vector space) the free vector space on ''G'' over ''k''. As an algebra, its product is defined by linear extension of the group composition in ''G'', with multiplicative unit the identity in ''G''; this product is also known as convolution. Note that while the group algebra of a ''finite'' group can be identified with the space of functions on the group, for an infinite group these are different. The group algebra, consisting of ''finite'' sums, corresponds to functions on the group that vanish for cofinitely many points; topologically (using the discrete topology), these correspond to functions with compact support. However, the group algebra ''k''() and the space of function ''k''''G'' := Hom(''G'',''k'') are dual: given an element of the group algebra $x\; =\; \backslash sum\_\; a\_g\; g$ and a function on the group $f\backslash colon\; G\; \backslash to\; k,$ these pair to give an element of ''k'' via $(x,f)\; =\; \backslash sum\_\; a\_g\; f(g),$ which is a well-defined sum because it is finite. 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「Group 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.