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 .sl.2// of sl.2/ for jqj D 1.