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

TL;DR

This paper explores the structure of Poisson ideals in cluster algebras, linking symplectic geometry to quantum algebra spectra, with applications to quantum matrices and general linear groups.

Contribution

It characterizes Poisson ideals and spectra in cluster algebras with compatible Poisson structures, advancing understanding of quantum coordinate rings.

Findings

Describes Poisson ideals and symplectic geometry in cluster algebras

Determines the spectrum of quantum cluster algebras

Analyzes the topology of spectra in 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.