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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 Hopf algebroid ： ウィキペディア英語版 Hopf algebroid In mathematics, in the theory of Hopf algebras, a Hopf algebroid is a generalisation of weak Hopf algebras and certain skew Hopf algebras: the concept was introduced by J.-H. Lu in 1996 as a result on work on groupoids in Poisson geometry (later shown equivalent in nontrivial way to a construction of Takeuchi from the 1970s and another by Xu around the year 2000). They may be loosely thought of as Hopf algebras over a noncommutative base ring, where weak Hopf algebras become Hopf algebras over a separable algebra. It is a theorem that a Hopf algebroid satisfying a finite projectivity condition over a separable algebra is a weak Hopf algebra, and conversely a weak Hopf algebra ''H'' is a Hopf algebroid over its separable subalgebra ''HL''. The antipode axioms have been changed by G. Böhm and K. Szlachanyi (J. Algebra) in 2004 for tensor categorical reasons and to accommodate examples associated to depth two Frobenius algebra extensions. ==Definition== A left Hopf algebroid (''H'', ''R'') is a left bialgebroid together with an antipode: the bialgebroid (''H'', ''R'') consists of a total algebra ''H'' and a base algebra ''R'' and two mappings, an algebra homomorphism ''s'': ''R'' → ''H'' called a source map, an algebra anti-homomorphism ''t'': ''R'' → ''H'' called a target map, such that the commutativity condition ''s''(''r''1) ''t''(''r''2) = ''t''(''r''2) ''s''(''r''1) is satisfied for all ''r''1, ''r''2 ∈ ''R''. The axioms resemble those of a Hopf algebra but are complicated by the possibility that ''R'' is a non-commutative algebra or its images under ''s'' and ''t'' are not in the center of ''H''. In particular a left bialgebroid (''H'', ''R'') has an ''R''-''R''-bimodule structure on ''H'' which prefers the left side as follows: ''r''1 ⋅ ''h'' ⋅ ''r''2 = ''s''(''r''1) ''t''(''r''2) ''h'' for all ''h'' in ''H'', ''r''1, ''r''2 ∈ ''R''. There is a coproduct Δ: ''H'' → ''H'' ⊗''R'' ''H'' and counit ε: ''H'' → ''R'' that make (''H'', ''R'', Δ, ε) an ''R''-coring (with axioms like that of a coalgebra such that all mappings are ''R''-''R''-bimodule homomorphisms and all tensors over ''R''). Additionally the bialgebroid (''H'', ''R'') must satisfy Δ(''ab'') = Δ(''a'')Δ(''b'') for all ''a'', ''b'' in ''H'', and a condition to make sure this last condition makes sense: every image point Δ(''a'') satisfies ''a''(1) ''t''(''r'') ⊗ ''a''(2) = ''a''(1) ⊗ ''a''(2) ''s''(''r'') for all ''r'' in ''R''. Also Δ(1) = 1 ⊗ 1. The counit is required to satisfy ε(1''H'') = 1''R'' and the condition ε(''ab'') = ε(''as''(ε(''b''))) = ε(''at''(ε(''b''))). The antipode ''S'': ''H'' → ''H'' is usually taken to be an algebra anti-automorphism satisfying conditions of exchanging the source and target maps and satisfying two axioms like Hopf algebra antipode axioms; see the references in Lu or in Böhm-Szlachanyi for a more example-category friendly, though somewhat more complicated, set of axioms for the antipode ''S''. The latter set of axioms depend on the axioms of a right bialgebroid as well, which are a straightforward switching of left to right, ''s'' with ''t'', of the axioms for a left bialgebroid given above. 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「Hopf algebroid」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.