Symplectic groupoids for cluster manifolds

Abstract

We construct symplectic groupoids integrating log-canonical Poisson structures on cluster varieties of type $\mathcal{A}$ and $\mathcal{X}$ 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.

arXiv comment33 pages

References in corpus (7)

• The quantum dilogarithm and representations quantum cluster varieties
• On the integration of Poisson manifolds, Lie algebroids, and coisotropic submanifolds
• Structure of seeds in generalized cluster algebras
• Double Bruhat Cells in Kac-Moody Groups and Integrable Systems
• The quantization of the symplectic groupoid of the standard Podles sphere
• The Symplectization Functor
• Companion cluster algebras to a generalized cluster algebra

Cited by in corpus (1)

• Symplectic Groupoids for Poisson Integrators