Beijing Institute of Mathematical Sciences and Applications

Moduli spaces of quiver representations sit at a meeting point of algebra, geometry and representation theory: they include resolutions of Kleinian singularities, Hilbert schemes of points on surfaces, and the Calogero-Moser spaces. A remarkable feature of these spaces is that they carry canonical Poisson structures. The goal of this course is to explain where these structures originate and the answer turns out to be noncommutative: they are the shadows, taken along the trace map, of "double" Poisson brackets on the path algebra itself.

After a brief review of quiver representations, with the (deformed) preprojective algebra as the central object, we construct representation schemes and their quotients: Moment maps and quiver varieties, with the three examples above as running illustrations. The third part of the course is devoted to the symplectic geometry of these moduli: Hamiltonian reduction and quiver realization of the Calogero-Moser space. The final part introduces Van den Bergh's double Poisson geometry: double derivations, the necklace bracket, double moment maps, and noncommutative symplectic geometry in the sense of Crawley-Boevey--Etingof--Ginzburg, and shows how the classical Poisson structures on quiver moduli are induced by the canonical double bracket on the double quiver.

Connections to integrable systems (Calogero-Moser particle dynamics, rational solutions of the KP hierarchy) and to quantization (Cherednik algebras) will be presented as an outlook; the course is self-contained and does not assume apriori knowledge of this particular applications.