2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/124885In this paper we develop Poisson geometry for non-commutative algebras. This generalizes the bi-symplectic geometry which was recently, and independently, introduced by Crawley-Boevey, Etingof and Ginzburg. Our (quasi-)Poisson brackets induce classical (quasi-)Poisson brackets on representation spaces. As an application we show that the moduli spaces of representations associated to the deformed multiplicative preprojective algebras recently introduced by Crawley-Boevey and Shaw carry a natural Poisson structure.Many misprints and inaccuracies (found by the referee) were correctedQuantum AlgebraRings and Algebras53D30text