On the noncommutative Poisson geometry of certain wild character varieties

TL;DR

This paper explores the noncommutative Poisson geometry of wild character varieties, proposing a conjecture that their Poisson structures originate from $H_0$-Poisson structures on associated fission algebras, and proves it in specific cases.

Contribution

It introduces a conjecture linking Poisson structures on colored multiplicative quiver varieties to fission algebras and confirms this in the interval and triangle cases.

Findings

Conjecture that Poisson structures are induced by $H_0$-Poisson structures on fission algebras.

Proven the conjecture for the interval case.

Proven the conjecture for the triangle case.

Abstract

To show that certain wild character varieties are multiplicative analogues of quiver varieties, Boalch introduced colored multiplicative quiver varieties. They form a class of (nondegenerate) Poisson varieties attached to colored quivers whose representation theory is controlled by fission algebras: noncommutative algebras generalizing the multiplicative preprojective algebras of Crawley-Boevey and Shaw. Previously, Van den Bergh exploited the Kontsevich-Rosenberg principle to prove that the natural Poisson structure of any (non-colored) multiplicative quiver variety is induced by an $H_0$-Poisson structure on the underlying multiplicative preprojective algebra; indeed, it turns out that this noncommutative structure comes from a Hamiltonian double quasi-Poisson algebra constructed from the quiver itself. In this article we conjecture that, via the Kontsevich-Rosenberg principle, the natural Poisson structure on each colored multiplicative quiver variety is induced by an $H_0$-Poisson structure on the underlying fission algebra which, in turn, is obtained from a Hamiltonian double quasi-Poisson algebra attached to the colored quiver. We study some consequences of this conjecture and we prove it in two significant cases: the interval and the triangle.