Darboux coordinates on the Poisson space of triangular bilinear forms

(Joint with L. Chekhov)

In this project we give a construction of a system of Darboux-type coordinates on the space of unipotent upper triangular bilinear forms equipped with the Poisson structure discovered by A. Bondal in 1995. Some special cases of  low dimensional symplectic leaves were identified earlier by L. Chekhov and M. Mazzocco with the Poisson algebras of hyperbolic length functions where Darboux-type coordinates are obtained by hyperbolic lengths of special system of loops.

Utilizing the  construction of Fock-Goncharov coordinates for flat SL_N connections on the disc with 3 marked points on the boundary, we compute Darboux-type coordinates for the maximal symplectic leaves.