Rigid analytic quantum groups and quantum Arens-Michael envelopes

TL;DR

This paper develops a non-archimedean analytic framework for quantum groups, introducing rigid analytification and Arens-Michael envelopes, and establishes categorical equivalences with classical structures.

Contribution

It introduces rigid analytification of quantum groups and Arens-Michael envelopes over non-archimedean fields, and connects their representation categories to classical categories.

Findings

Quantum groups are shown to be topological Hopf and Fréchet-Stein algebras.

An analogue of category O is constructed for the quantum Arens-Michael envelope.

Equivalence between the new and classical category O is established.

Abstract

We introduce a rigid analytification of the quantized coordinate algebra of a semisimple algebraic group and a quantized Arens-Michael envelope of the enveloping algebra of the corresponding Lie algebra, working over a non-archimedean field and when $q$ is not a root of unity. We show that these analytic quantum groups are topological Hopf algebras and Fr\'echet-Stein algebras. We then introduce an analogue of the BGG category $\mathcal{O}$ for the quantum Arens-Michael envelope and show that it is equivalent to the category $\mathcal{O}$ for the corresponding quantized enveloping algebra.