  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.
