Poisson geometry and Azumaya loci of cluster algebras (Q6608012)

Cluster algebras were defined by \textit{S. Fomin} and \textit{A. Zelevinsky} [J. Am. Math. Soc. 15, No. 2, 497--529 (2002; Zbl 1021.16017)], and since then, they have played prominent role in many areas of mathematics and mathematical physics. The upper cluster algebras \(\mathsf{U}\) with their Gekhtman--Shapiro--Vainshtein (GSV) Poisson brackets and their root of unity quantizations \(\mathsf{U}_{\varepsilon}\) are the main types of objects in the theory of cluster algebras. On the Poisson side, the authors prove that 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, the authors describe the fully Azumaya loci of the quantizations \(\mathsf{U}_{\varepsilon}\) under the assumption that \(\mathsf{A}_{\varepsilon}=\mathsf{U}_{\varepsilon}\) and \(\mathsf{U}_{\varepsilon}\) is a finitely generated algebra. All results allow frozen variables to be either inverted or not.