Geometric leaf of symplectic groupoid
E. Brodsky, P. Dangwal, S. Hamlin, L. Chekhov, M. Shapiro, S. Sottile, X. Lian, and Z. Zhan

TL;DR
This paper explores the geometric structure of a symplectic groupoid related to unipotent upper-triangular matrices, analyzing Hamiltonian reductions and cluster structures for specific cases connected to Teichmüller spaces.
Contribution
It describes the Hamiltonian reduction of the symplectic groupoid for n=5 and 6, revealing cluster structures on related Teichmüller spaces.
Findings
Hamiltonian reduction for n=5 and 6 cases.
Induced cluster structures on reduced spaces.
Recovery of known cluster structures on Teichmüller spaces.
Abstract
We consider the symplectic groupoid of pairs with real unipotent upper-triangular matrix and being such that is also a unipotent upper-triangular matrix. Fock and Chekhov defined a Poisson map of Teichm\"uller space {\mathcal T_{g,s} of genus surfaces with holes into the space of unipotent upper-triangular matrices whose image forms the \emph{geometric locus}. The elements of geometric locus satisfy \emph{rank condition}. We describe the Hamiltonian reduction of the Poisson cluster variety of symplectic groupoid by the rank condition for and . In both cases, we analyze the induced cluster structures on the results of Hamiltonian reduction and recover celebrated cluster structure on for and for .
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Geometric and Algebraic Topology · Advanced Algebra and Geometry
