The Prime Spectrum of Quantum $S{L}_3$ and the Poisson-prime Spectrum of its Semi-classical Limit

Abstract

A bijection $\psi$ is defined between the prime spectrum of quantum $S{L}_3$ and the Poisson prime spectrum of $S{L}_3$, 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 $S{L}_3$ 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 $\mathcal{O}(S{L}_3)$ and obtain explicit generating sets for all of the Poisson primitive ideals.

Minor updates, clarifications, etc throughout the text. To appear in Transactions of the London Mathematical Society

References in corpus (1)

• Quantum Matrices by Paths

Cited by in corpus (2)

• The Orbit Method for Poisson Orders
• Automorphism groups of deformations and quantizations of Kleinian singularities