广义弱双frobenius代数上等变循环上同调

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1、广义弱双Frobenius代数上的等变循环上同调摘要：本篇博士论文围绕弱Hopfl弋数、双Frobenius代数以及Hopf群-余代数展开一系列深入研究，主要表现在以下儿个方面：首先，我们构造了弱Hop玳数情形下弱模代数的等变循环上同调及弱量子Yetter-Drinfel’d代数的等变K一理论，接着讨论了等变上同调与弱Smash积代数的循环上同调的关系，进而构造了等变循环上同调和等变K一理论之间的Connes配对。其次，我们引入了弱双Frobenius代数的概念，然后讨论了弱双Frobenius代数的子结构，并证明了弱双Frob

2、enius代数的Radford公式，进一步考虑了弱双Frobenius代数与SzlachgmyiFrobe-nius双重代数的关系。再次，固定a∈7l-(Tr是一个群)，我们讨论了忘却函子F(。)：缎一c-÷阮。的可分性，并得到函子F(n)可分的充要条件。作为结果的应用，Doi—Hopf7r一模的Maschke型定理被证明。最后，我们引入了两个偏Doi—Hopf移$范畴间的导出函子，并证明导出函子具有右伴随函子。另一方面，我们引入了偏Dci—Hopf数据正规化积分概念，并刻画了『F规化积分存在的标准。关键词：弱Hop玳数；弱模代

3、数；可分函子；循环上同调；Radford公式；弱Smash积代数；双Frobenius代数：积分EquivariantCyclicCohomologyofGeneralizedWeakBi—FrobeniusAlgebrasAbstract：TheaimofPh．D．dissertationistocarryoutaseriesoffurtherresearchonweakHopfalgebra、bi—FrobeniusalgebraandHopf7r—coalgebrasfromthefollowingaspectsFirst

4、，wewillshowhowtoconstructtheequivariantcycliccohomologyforaweakmod—ulealgebraandequivariantK—theoryforaweakquantumYetter—Drinfel’dalgebrainweakHopfalgebrasetting，anddiscusstherelationshipbetweentheequivariantcycliccohomologyandthecycliccohomologyofthecorrespondingwea

5、ksmashproductalgebra，andestablishageneralizedConnes’pairingbetweensuchequivariantcycliccohomologytheoryandsuchequivariantK—theorySecond，weakbi—Frobeniusalgebrasareintroduced，andthesubstructuresarediscussed．Also，wemainlyshowRadford’SformulaforS4inthesettingofwbFalgebr

6、asanddiscusstherelationshipbetweenourwe'okbi—FrobeniusalgebrasandSzlach∈inyi’SFrobeniusdoublealgebras．Again．fixQ∈7r(rragroup)，wewilldiscusstheseparabilityoftheforgetfulfunctorF(Q)：磁一c-÷阮。，andgetthesufficientandnecessaryconditionsforF‘⋯．Astheapplicationofourresults．we

7、willprovetheMaschke—typeTheoremsforDoi—Hopf丌一modules．Finally，wewilldefineaninductionflmctorbetweenthetwocorrespondingcategoriesofpartialDoi—Hopfmodules，andprovethisfunctorhasarightadjoint．Also，wewillintroducetheconceptofnormalizedintegralofapartialDoi—Hopfdatum，andch

8、aracterizethecriterionfortheexistenceofthenormalizedintegral．Keywords：weakHopfalgebra；weakmodulealgebra；separablefunctor；cycliccoho