On analytic analogues of quantum groups

Abstract

In this paper 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.