Tuesday, July 20, 2021

David Fernández (Bielefeld), Euler continuants in noncommutative quasi-Poisson geometry

Abstract:
Since their introduction in 1764 by Euler to study the numerators and denominators of continued fractions, Euler continuants have become essential in mathematics, appearing in diverse branches of mathematics. In representation theory, Boalch used Euler continuants to study certain wild character varieties, which can be described as quiver varieties attached to the quiver Γ_n with two vertices and n equioriented arrows. Interestingly, they carry a (nondegenerate) Poisson structure.
As a part of our program to understand wild character varieties in terms of Van den Bergh's noncommutative Poisson geometry, in this talk I shall explain how the abovementioned Poisson structure can be explained using an explicit Hamiltonian double quasi-Poisson structure on the path algebra of Γ_n, which is such that the Euler continuants are multiplicative moment maps. In addition, we will discuss its relationship with the celebrated Flaschka-Newell bracket. This is a joint work with Maxime Fairon (Glasgow).