The prime spectrum of quantum SL3 and the Poisson prime spectrum of its semiclassical limit

Abstract

A bijection ψ is defined between the prime spectrum of quantum SL3 and the Poisson prime spectrum of SL3, and we verify that ψ and ψ−1 both preserve inclusions of primes, i.e. that ψ 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 SL3 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 O(SL3) and obtain explicit generating sets for all of the Poisson primitive ideals.