Poisson geometry and Azumaya loci of cluster algebras

TL;DR

This paper explores the Poisson geometric structure and Azumaya loci of cluster algebras, providing explicit descriptions of symplectic leaves and quantum Azumaya loci without restrictive assumptions.

Contribution

It proves the existence of a Zariski open orbit of symplectic leaves in upper cluster algebras and describes the Azumaya loci of their quantizations under certain conditions.

Findings

Spectrum of upper cluster algebras has a Zariski open orbit of symplectic leaves.

Explicit description of the symplectic leaf orbit in the spectrum.

Identification of the Azumaya loci of quantum cluster algebra quantizations.

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.