Darboux coordinates for symplectic groupoid and cluster algebras

Abstract: The talk is based on Arxiv:2003:07499, joint work with Misha Shapiro. I will start with a short elementary excursion into cluster algebras---a fascinating branch of modern algebra introduced by Fomin and Zelvinsky in 2000's---and describe planar directed networks. We then concentrate on another interesting algebraic object---the $\mathcal A_n$ groupoid of upper-triangular matrices, which has had many appearances in studies of algebras of monodromies of $SL_2$ Fuchsian systems and in geometry, including the celebrated Goldman bracket. I will show how we can use Fock--Goncharov higher Teichm\"uller space variables to derive canonical (Darboux) coordinate representation for entries of general symplectic leaves of the $\mathcal A_n$ groupoid and, in a more general setting, of higher-dimensional symplectic leaves for algebras governed by the quantum reflection equation with the trigonometric $R$-matrix. For the groupoid of upper-triangular matrices, we represent braid-group transformations via sequences of cluster mutations in the special $\mathbb A_n$-quiver. Time permitting, I will also describe a generalization of this construction to affine Lie-Poisson algebras and to quantum loop algebras (Arxiv:2012:10982).

HEP - theorymathematical physicsalgebraic geometrycombinatoricsexactly solvable and integrable systems

Audience: researchers in the discipline

IBS-CGP Mathematical Physics Seminar

Series comments: Registration is required at cgp.ibs.re.kr/activities/talkregistration