The Prime Spectrum of Quantum $SL_3$ and the Poisson-prime Spectrum of its Semi-classical Limit

TL;DR

This paper establishes a homeomorphism between the prime spectrum of quantum $SL_3$ and the Poisson prime spectrum of its classical limit, using a Poisson analogue of a known quantum algebra framework.

Contribution

It develops a Poisson analogue of Brown and Goodearl's framework and verifies a homeomorphism between quantum and Poisson spectra for $SL_3$, including explicit primitive ideals.

Findings

The bijection $bla$ is a homeomorphism between spectra.

Explicit generating sets for all Poisson primitive ideals are obtained.

The quantum and Poisson spectra have identical topological pictures for $SL_3$.

Abstract

A bijection $\psi$ is defined between the prime spectrum of quantum $SL_3$ and the Poisson prime spectrum of $SL_3$, and we verify that $\psi$ and $\psi^{-1}$ both preserve inclusions of primes, i.e. that $\psi$ is in fact a homeomorphism between these two spaces. This is accomplished by developing a Poisson analogue of Brown and Goodearl's framework for describing the Zariski topology of spectra of quantum algebras, and then verifying directly that in the case of $SL_3$ these give rise to identical pictures on both the quantum and Poisson sides. As part of this analysis, we study the Poisson primitive spectrum of $\mathcal{O}(SL_3)$ and obtain explicit generating sets for all of the Poisson primitive ideals.