published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

This thesis is divided into the following three parts. <b>A categorical reconstruction of crystals and quantum groups at</b> q = 0. The quantum co-ordinate algebra A_q(&gfr;) associated to a KacMoody Lie algebra g forms a Hopf algebra whose comodules are direct sums of finite dimensional irreducible U_q(&gfr;) modules. In Part I we investigate whether an analogous result is true when q = 0. We classify crystal bases as coalgebras over a comonadic functor on the category of pointed sets and encode the monoidal structure of crystals into a bicomonadic structure. In doing this we prove that there is no coalgebra in the category of pointed sets whose comodules are equivalent to crystal bases. We then construct a bialgebra over &Zopf; whose based comodules are equivalent to crystals, which we conjecture is linked to Lusztig's quantum group at v = ∞. <b>A Tannakian Reconstruction Theorem for IndBanach Spaces.</b> Classically, Tannaka-Krein duality allows us to reconstruct a (co)algebra from its category of representation. In Part II we present an approach that allows us to generalise this theory to the setting of Banach spaces. This leads to several interesting applications in the directions of analytic quantum groups, bounded cohomology and Galois descent. A large portion of Part II is dedicated to such examples. <b>On analytic analogues of quantum groups.</b> In Part III we present a new construction of analytic analogues of quantum groups over non-Archimedean fields and construct braided monoidal categories of their representations. We do this by constructing analytic Nichols algebras and use Majid's double-bosonisation construction to glue them together. We then go on to study the rigidity of these analytic quantum groups as algebra deformations of completed enveloping algebras through bounded cohomology. This provides the first steps towards a p-adic Drinfel'd-Kohno Theorem, which should relate this work to Furusho's p-adic Drinfel'd associators. Finally, we adapt these constructions to working over Archimedean fields.