Reflection Functor in the Representation Theory of Preprojective Algebras for Quivers and Integrable Systems

A brief overview of the representation theory of quivers and the associated (deformed) preprojective algebras, as well as of the theories of moduli spaces of these algebras, quiver varieties and a reflection functor, is given. It is proven that a bijection between moduli spaces (in particular, between quiver varieties), which is induced by a reflection function, is the isomorphism of symplectic affine varieties. The Hamiltonian systems on quiver varieties are defined, and the application of a reflection functor to them is described. The review of [1], concerning the case of a cyclic quiver is given, and a role of the reflection functor in this case is clarified. The "spin" integrable generalizations of Calogero-Moser systems and their application to the KP hierarchy generalizations are described.