Euler continuants in noncommutative quasi-Poisson geometry. (10th October 2022)

• Record Type: Journal Article
• Authors: Fairon, Maxime; Fernández, David
• Is Part Of: Forum of mathematics. Volume 10(2022)
• Publication Date: 2022-10-10
• Subjects: 16G20 -- 17B63 -- 14A22 -- 53D30 -- 53D20; Mathematics -- Periodicals; 510
• DOI: 10.1017/fms.2022.76
• Languages: English
• ISSNs: 2050-5094

Abstract

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 modelled 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 $\Gamma _1$ .