Euler continuants in noncommutative quasi-Poisson geometry

TL;DR

This paper links Euler continuants to noncommutative quasi-Poisson geometry, showing how certain wild character varieties can be understood through explicit Hamiltonian double quasi-Poisson algebras related to specific quivers.

Contribution

It introduces a noncommutative quasi-Poisson framework for Sibuya varieties, generalizing previous algebraic structures and demonstrating their factorization via fusion methods.

Findings

Euler continuants are realized as noncommutative moment maps.

The Poisson structures are induced by explicit Hamiltonian double quasi-Poisson algebras.

The algebra for d7n quivers factorizes into n copies of the algebra for d71.

Abstract

It was established by Boalch that Euler continuants arise as Lie group valued moment maps for a class of wild character varieties described as moduli spaces of points on $\mathbb{P}^1$ by Sibuya. Furthermore, Boalch noticed that these varieties are multiplicative analogues of certain Nakajima quiver varieties originally introduced by Calabi, which are attached to the quiver $\Gamma_n$ on two vertices and $n$ equioriented arrows. In this article, we go a step further by unveiling that the Sibuya varieties can be understood using noncommutative quasi-Poisson geometry modeled on the quiver $\Gamma_n$. We prove that the Poisson structure carried by these varieties is induced, via the Kontsevich-Rosenberg principle, by an explicit Hamiltonian double quasi-Poisson algebra defined at the level of the quiver $\Gamma_n$ such that its noncommutative multiplicative moment map is given in terms of Euler continuants. This result generalises the Hamiltonian double quasi-Poisson algebra associated with the quiver $\Gamma_1$ by Van den Bergh. Moreover, using the method of fusion, we prove that the Hamiltonian double quasi-Poisson algebra attached to $\Gamma_n$ admits a factorisation in terms of $n$ copies of the algebra attached to $\Gamma_1$.