Michael Shapiro: Log-canonical coordinates for symplectic groupoid and cluster structure on the Teichmueller space of closed genus two surfaces

Abstract.

Symplectic groupoid of unipotent nxn upper-triangular matrices is formed by pairs \((B, A)\) where \(B\) is a nondegenerate \(n \times n\) matrix, \(A\) is a unipotent upper-triangular \(n \times n\) matrix, such that \(BAB^t\) is unipotent upper triangular. The symplectic groupoid is equipped with the natural symplectic form defined by Weinstein, which induces a Poisson bracket on the space of upper triangular unipotent matrices studied by Bondal, Dubrovin-Ugaglia, and others.

We compute the cluster structure compatible with the Poisson structure and discuss its connection with the Teichmueller space of genus \(g\) curves with one or two holes equipped with a Goldman Poisson bracket.

As an unexpected byproduct, we obtain a cluster structure on the Teichmueller space of closed genus two curves unknown earlier.