Symplectic groupoid: geometry, networks, and noncommutativity

In the first part of the talk, I will describe the symplectic groupoid: a set of pairs $(B,\mathbb{A})$ with $\mathbb{A}$ unipotent upper-triangular matrices and $B\in G{L}_n$ being such that the matrix $\tilde{\mathbb{A}}=B\mathbb{A}{B}^{\text{T}}$ is itself unipotent upper triangular. Since works of J.Nelson, T.Regge, B.Dubrovin, and M. Ugaglia it was known that entries of $\mathbb{A}$ can be identified with geodesic functions on a Riemann surface with holes; these entries then enjoy a closed Poisson algebra (reflection equation) expressible in the $r$-matrix form. In our recent work with M. Shapiro, we solved the symplectic groupoid in terms of planar networks; we used this solution to construct a complete set of geodesic functions for a closed Riemann surface.

In the second part, I will present preliminary results on generalizing the above construction to the case of noncommutative symplectic groupoid subject to a Van der Bergh double Poisson bracket. Based on a forthcoming joint paper with I. Bobrova and M. Shapiro.