The Arens-Michael envelopes of the Jordan 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 affine scheme, it constructs the ring of “holomorphic functions” on it when viewed as a noncommutative complex analytic space. In this paper, we explicitly compute the Arens-Michael envelopes of the Jordan plane and the quantum enveloping algebra $U_q(\mathfrak{sl}(2))$ of $\mathfrak{sl}(2)$ for $|q|=1$.