Poisson structures on moduli spaces of representations

We show that a Poisson structure can be induced on the affine moduli space of (semisimple) representations of an associative algebra from a suitable Lie algebra structure on the zeroth Hochschild homology of the algebra. In particular this applies to necklace Lie algebra for path algebras of doubled quivers and preprojective algebras. We call such structures H0-Poisson structures, and show that they are well behaved for Azumaya algebras and under Morita equivalence.