2022/09/23 by Greg Muller, Bach Nguyen, Muller, Greg +5 · 2 citations
Mathematics · #13F60 #16G30 #17B37 #53D17 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Cluster (spacecraft) #Cluster algebra #Combinatorics #FOS: Mathematics #Lie algebra #Mathematics #Physics #Poisson algebra #Poisson bracket #Poisson distribution #Poisson manifold #Pure mathematics #Quantum #Quantum Algebra (math.QA) #Quantum mechanics #Representation Theory (math.RT) #Symplectic Geometry (math.SG) #Symplectic geometry
paper · pdf · doi:10.48550/arxiv.2209.11622
openalex publication_date 2022/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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 Uε. 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 Uε under the assumption that \boldsymbol\mathsf Aε = \boldsymbol\mathsf Uε and \boldsymbol\mathsf Uε is a finitely generated algebra. All results allow frozen variables to be either inverted or not.