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

TL;DR

This paper computes the Arens-Michael envelopes of the Jordanian plane and the quantum algebra U_q(sl(2)) for |q|=1, advancing the understanding of noncommutative analytic structures.

Contribution

It explicitly determines the Arens-Michael envelopes for these specific noncommutative algebras, providing new insights into their analytic properties.

Findings

Explicit formulas for the envelopes of the Jordanian plane.

Explicit formulas for the envelopes of U_q(sl(2)) at |q|=1.

Enhanced understanding of noncommutative holomorphic functions.

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 $U_q(\mathfrak{sl}(2))$ of $\mathfrak{sl}(2)$ for $|q|=1$.