Application of the reflection functor to integrable systems on quiver varieties

TL;DR

This paper reviews the use of the reflection functor in the context of integrable systems on quiver varieties, demonstrating its role in establishing symplectic isomorphisms and applications to generalized Calogero-Moser systems.

Contribution

It introduces the application of the reflection functor to integrable Hamiltonian systems on quiver varieties, including new insights into their symplectic structure and integrable generalizations.

Findings

Reflection functor induces symplectic isomorphisms between moduli spaces.

Application to spin Calogero-Moser systems and KP hierarchies.

Review of the case of cyclic quivers and their integrable systems.

Abstract

The theory of representations of quivers and of their preprojective algebras are reviewed. In particular, moduli spaces of representations of these algebras, quiver varieties and reflection functor are described. The proof that the bijection between moduli spaces induced by the reflection functor is an isomorphism of symplectic affine varieties is presented. Hamiltonian systems with complete flows on the quiver varieties are introduced. An application of the reflection functor to these systems is described. The results on the case of the cyclic quiver are reviewed and a role of the reflection functor in this case is discussed. In particular, "spin" integrable generalisations of the Calogero-Moser systems and their application to the generalised KP hierarchies are considered.