Poisson Ideals in Cluster Algebras and the Spectra of Quantized Coordinate Rings

Abstract: We describe the Poisson ideals and attached symplectic geometry of a cluster algebra with compatible Poisson structure. We apply these results to determine the spectrum of a quantum cluster algebra. As an application, we describe the topology on the spectra of quantized coordinate rings such as quantum matrices and the quantized function algebra of the general linear group.