Symplectic groupoid of unipotent nxn upper-triangular matrices is formed by pairs $(B, A)$ where $B$ is a nondegenerate nxn matrix, $A$ is a unipotent upper-triangular nxn matrix, and $BAB^t$ 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. This is a joint project with L. Chekhov.
The talk will be via Zoom at:
https://utoronto.zoom.us/j/99576627828
Passcode: 448487