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 $\mathsf{U}$ with their Gekhtman-Shapiro-Vainshtein Poisson brackets and their root of unity quantizations ${\mathsf{U}}_{\epsilon }$. On the Poisson side, we prove that (without any assumptions) the spectrum of every finitely generated upper cluster algebra $\mathsf{U}$ 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 ${\mathsf{U}}_{\epsilon }$ under the assumption that ${\mathsf{A}}_{\epsilon }={\mathsf{U}}_{\epsilon }$ and ${\mathsf{U}}_{\epsilon }$ is a finitely generated algebra. All results allow frozen variables to be either inverted or not.

34 pages, v2: minor additions and corrections

Cited by in corpus (1)

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