Overrings of primitive factors of quantum \(\mathfrak{sl}(2)\) (Q1901001)

This paper classifies completely prime Dixmier algebras over primitive quotients of the quantum enveloping algebra of \(\mathfrak {sl} (2)\); the corresponding classification over the ordinary enveloping algebra of \(\mathfrak {sl} (2)\) looks essentially the same and was done by \textit{S. P. Smith} [J. Pure Appl. Algebra 63, 207-218 (1990; Zbl 0849.17006)] and independently by the author (unpublished). In particular, as in the classical case, every completely prime Dixmier algebra has length at most two as a bimodule (modulo algebraic extensions of the base field) and the length two algebras have half-integral infinitesimal characters as bimodules. Everything is done by explicit calculation and thus one does not need quantized versions of standard structural results on enveloping algebras.