Poisson geometry and Azumaya loci of cluster algebras

Abstract

There are two main types of objects in the theory of cluster algebras: the upper cluster algebras ${{\boldsymbol{\mathsf U}}}$ with their Gekhtman-Shapiro-Vainshtein Poisson brackets and their root of unity quantizations ${{\boldsymbol{\mathsf U}}}_\varepsilon$. On the Poisson side, we prove that (without any assumptions) the spectrum of every finitely generated upper cluster algebra ${{\boldsymbol{\mathsf U}}}$ with its GSV Poisson structure always has a Zariski open orbit of symplectic leaves and give an explicit description of it. On the quantum side, we describe the fully Azumaya loci of the quantizations ${{\boldsymbol{\mathsf U}}}_\varepsilon$ under the assumption that ${{\boldsymbol{\mathsf A}}}_\varepsilon = {{\boldsymbol{\mathsf U}}}_\varepsilon$ and ${{\boldsymbol{\mathsf U}}}_\varepsilon$ is a finitely generated algebra. All results allow frozen variables to be either inverted or not.