Darboux coordinates for symplectic groupoid and cluster algebras

Abstract: The talk is based on Arxiv:2003:07499, joint work with Misha Shapiro. We use Fock--Goncharov higher Teichmüller space variables to derive Darboux coordinate representation for entries of general symplectic leaves of the $\mathcal A_n$ groupoid of upper-triangular matrices and, in a more general setting, of higher-dimensional symplectic leaves for algebras governed by the quantum reflection equation with the trigonometric $R$-matrix. This result can be generalized to any planar directed network on disc with separated sinks and sources. For the groupoid of upper-triangular matrices, we represent braid-group transformations via sequences of cluster mutations in the special $\mathbb A_n$-quiver. We prove the groupoid relations for quantum transport matrices and, as a byproduct, obtain quantum commutation relations having the Goldman bracket as their semiclassical limit. Time permitting, I will also describe a generalization of this construction to affine Lie-Poisson algebras and to quantum loop algebras (Arxiv:2012:10982).

mathematical physicscommutative algebraalgebraic geometrycombinatoricsquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the topic

Online Cluster Algebra Seminar (OCAS)