2018/07/10 by Songhao Li, Li, Songhao, Dylan Rupel +1 · 1 citation
Mathematics · #13F60 #53D17 #70S05 #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA) #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.1807.03450
openalex publication_date 2018/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We construct symplectic groupoids integrating log-canonical Poisson structures on cluster varieties of type A and X over both the real and complex numbers. Extensions of these groupoids to the completions of the cluster varieties where cluster variables are allowed to vanish are also considered. In the real case, we construct source-simply-connected groupoids for the cluster charts via the Poisson spray technique of Crainic and Mărcuţ. These groupoid charts and their analogues for the symplectic double and blow-up groupoids are glued by lifting the cluster mutations to groupoid comorphisms whose formulas are motivated by the Hamiltonian perspective of cluster mutations introduced by Fock and Goncharov.