On the noncommutative Poisson geometry of certain wild character varieties

TL;DR

The paper develops a noncommutative Poisson framework for Boalch’s colored multiplicative quiver varieties by embedding their Poisson geometry into Hamiltonian double quasi-Poisson algebras on Boalch/fission algebras. Central to the program is the Kontsevich–Rosenberg principle, which transfers noncommutative brackets to Poisson structures on representation schemes and yields $H_0$-Poisson structures on fission algebras, hence Poisson structures on colored MQVs. The authors formulate a conjecture asserting the existence of Hamiltonian double quasi-Poisson structures on Boalch algebras for any colored quiver and prove two significant instances: the monochromatic interval and the monochromatic triangle, using a detailed cascade of case analyses to verify quasi-Poisson axioms. These results provide a concrete noncommutative origin for the Poisson geometry of wild character varieties and indicate a path toward a full noncommutative classification via fusion of monochromatic components. The work suggests broader connections to fusion, pre-Calabi–Yau structures, and potential integrable systems arising from tautologically colored multiplicative quiver varieties.

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.

Figures (2)

• Figure 1: The quiver $\Upsilon$ contains two monochromatic pieces, whose colors are represented by the solid and dashed styles of the lines. The quiver $\Upsilon$ can be obtained from $\Upsilon_1$ and $\Upsilon_2$ by identifying the vertices $s$ and $s'$.
• Figure 2: The quivers $\Delta$, $\overline{\Delta}$, and $\widetilde{\Delta}$ used to introduce the algebra $\mathcal{B}(\Delta)$. In $\widetilde{\Delta}$, a loop based at the vertex $i$ represents $\gamma_i$, while a plain (resp. dashed) arrow from the vertex $i$ to the vertex $j$ represents $v_{ji}$ (resp. $w_{ji}$).

Theorems & Definitions (68)

• Definition 2.1: VdB1
• Definition 2.2: VdB1, Definition 5.1.1
• Remark 2.3
• Definition 2.4: VdB1, Definition 5.1.4
• Definition 2.5: CB11
• Proposition 2.6: VdB1, Proposition 5.1.5
• Theorem 2.7: VdB1, Theorem 7.12.2, Proposition 7.13.2; CB11, Theorem 4.5
• Definition 2.8: CBShaw
• Theorem 2.9: VdB1, Theorems 6.5.1 and 6.7.1, and Proposition 6.8.1
• Remark 2.10