Separating Ore sets for prime ideals of quantum algebras

TL;DR

This paper proves the existence of separating Ore sets for all incident pairs of torus invariant prime ideals in key classes of quantum algebras, advancing the understanding of their spectral topology.

Contribution

It establishes the existence of separating Ore sets for the largest classes of quantum algebras fitting Brown and Goodearl's conjecture, specifically for quantized coordinate rings and quantum Schubert cell algebras.

Findings

Confirmed separating Ore sets for simple algebraic group coordinate rings

Established separating Ore sets for quantum Schubert cell algebras

Progressed towards explicit spectral topology description in quantum algebras

Abstract

Brown and Goodearl stated a conjecture that provides an explicit description of the topology of the spectra of quantum algebras. The conjecture takes on a more explicit form if there exist separating Ore sets for all incident pairs of torus invariant prime ideals of the given algebra. We prove that this is the case for the two largest classes of algebras of finite Gelfand-Kirillov dimension that fit the setting of the conjecture: the quantized coordinate rings of all simple algebraic groups and the quantum Schubert cell algebras for all symmetrizable Kac-Moody algebras.