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In mathematics, introduced an example of an infinite-dimensional Hopf algebra, and Sweedler's Hopf algebra ''H''4 is a certain 4-dimensional quotient of it that is neither commutative nor cocommutative. ==Definition== The following infinite dimensional Hopf algebra was introduced by . The Hopf algebra is generated as an algebra by three elements ''x'', ''g'', and ''g''−1. The coproduct Δ is given by :Δ(g) = ''g'' ⊗''g'', Δ(''x'') = 1⊗''x'' + ''x'' ⊗''g'' The antipode ''S'' is given by :''S''(''x'') = –''x'' ''g''−1, ''S''(''g'') = ''g''−1 The counit ε is given by :ε(''x'')=0, ε(''g'') = 1 Sweedler's 4-dimensional Hopf algebra ''H''4 is the quotient of this by the relations :''x''2 = 0, ''g''2 = 1, ''gx'' = –''xg'' so it has a basis 1, ''x'', ''g'', ''xg'' . Note that Montgomery describes a slight variant of this Hopf algebra using the opposite coproduct, i.e. the coproduct described above composed with the tensor flip on ''H''4⊗''H''4. Sweedler's 4-dimensional Hopf algebra is a quotient of the Pareigis Hopf algebra, which is in turn a quotient of the infinite dimensional Hopf algebra. 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「Sweedler's Hopf algebra」の詳細全文を読む スポンサード リンク
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