Modules and comodules over nonarchimedean Hopf algebras

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dc.contributor.author Lyubinin, Anton
dc.date.accessioned 2010-08-13T16:57:08Z
dc.date.available 2010-08-13T16:57:08Z
dc.date.issued 2010-08-13T16:57:08Z
dc.identifier.uri http://hdl.handle.net/2097/4639
dc.description.abstract The purpose of this work is to study Hopf algebra analogs of constructions in the theory of p-adic representations of p-adic groups. We study Hopf algebras and comodules, whose underlying vector spaces are either Banach or compact inductive limits of such. This framework is unifying for the study of continuous and locally analytic representations of compact p-adic groups, affinoid and sigma-affinoid groups and their quantized analogs. We define the analog of Frechet-Stein structure for Hopf algebra (which play role of the function algebra), which we call CT-Stein structure. We prove that a compact type structure on a CT-Hopf algebra is CT-Stein if its dual is a nuclear Frechet-Stein structure on the dual NF-Hopf algebra. We show that for every compact p-adic group the algebra of locally analytic functions on that group is CT-Stein. We describe admissible representations in terms of comodules, which we call admissible comodules, and thus we prove that admissible locally analytic representations of compact p-adic groups are compact inductive limits of artinian locally analytic Banach space representations. We introduce quantized analogs of algebras Ur(sl2;K) from [7] thus giving an example of in fite-dimensional noncommutative and noncocommutative nonarchimedean Banach Hopf algebra. We prove that these algebras are Noetherian. We also introduce a quantum analog of U(sl2;K) and we prove that it is a (in fite-dimensional non-commutative and non-cocommutative) Frechet-Stein Hopf algebra. We study the cohomology theory of nonarchimedean comodules. In the case of modules and algebras this was done by Kohlhasse, following the framework of J.L. Taylor. We use an analog of the topological derived functor of Helemskii to develop a cohomology theory of non-archimedean comodules (this approach can be applied to modules too). The derived functor approach allows us to discuss a Grothendieck spectral sequence (GSS) in our context. We apply GSS theorem to prove generalized tensor identity and give an example, when this identity is nontrivial. en_US
dc.language.iso en_US en_US
dc.publisher Kansas State University en
dc.subject Nonarchimedean en_US
dc.title Modules and comodules over nonarchimedean Hopf algebras en_US
dc.type Dissertation en_US
dc.description.degree Doctor of Philosophy en_US
dc.description.level Doctoral en_US
dc.description.department Department of Mathematics en_US
dc.description.advisor Zongzhu Lin en_US
dc.subject.umi Mathematics (0405) en_US
dc.date.published 2010 en_US
dc.date.graduationmonth August en_US


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