Euler continuants in noncommutative quasi-Poisson geometry

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 ${\Gamma }_1$.

51 pages, 1 figure. v3: Subsection 5.6 added + minor changes. Accepted version

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Cited by in corpus (1)

• On the noncommutative Poisson geometry of certain wild character varieties