The Arens-Michael envelopes of the Jordanian Plane and $U_q(\mathfrak{sl}(2))$

Abstract

The Arens-Michael functor in noncommutative geometry is an analogue of the analytification functor in algebraic geometry: out of the ring of "algebraic functions" on a noncommutative space it constructs the ring of "holomorphic functions" on it. In this paper, we explicitly compute the Arens-Michael envelopes of the Jordanian plane and the quantum enveloping algebra Uq(sl(2)) of sl(2) for |q| = 1.