Prime spectra of ambiskew polynomial rings

TL;DR

This paper characterizes the prime spectra of ambiskew polynomial rings over algebraically closed fields, linking their structure to well-known algebras like $U(sl_2)$ and its quantum version, with applications to quantum tori.

Contribution

It provides criteria for when the prime spectrum of an ambiskew polynomial ring resembles that of key quantum algebras, extending understanding of their prime ideal structure.

Findings

Prime spectrum characterized by specific prime ideals

Criteria established for spectra similar to $U(sl_2)$ and $U_q(sl_2)$

Applications to ambiskew rings over quantum tori

Abstract

We determine criteria for the prime spectrum of an ambiskew polynomial algebra $R$ over an algebraically closed field $K$ to be akin to those of two of the principal examples of such an algebra, namely the universal enveloping algebra $U(sl_2)$ (in characteristic $0$) and its quantization $U_q(sl_2)$ (when $q$ is not a root of unity). More precisely, we aim to determine when the prime spectrum of $R$ consists of $0$, the ideals $(z-\lambda)R$ for some central element $z$ of $R$ and all $\lambda\in K$, and, for some positive integer $d$ and each positive integer $m$, $d$ height two prime ideals with Goldie rank $m$. New applications are to certain ambiskew polynomial rings over coordinate rings of quantum tori which arise, as localizations of connected quantized Weyl algebras.