Central extensions of preprojective algebras, the quantum Heisenberg algebra, and 2-dimensional complex reflection groups

Preprojective algebras of quivers were introduced in 1979 by Gelfand and Ponomarev [GP], because for quivers of finite ADE type, they are models for indecomposable representations (they contain each indecomposable exactly once). Twenty years later, these algebras and their deformed versions introduced in [CBH] (for arbitrary quivers) became a subject of intense interest, since their representation varieties, called quiver varieties, played an important role in geometric representation theory. Ironically, it is exactly for quivers of finite ADE type that preprojective algebras fail to have good properties—they are not Koszul and their deformed versions are not flat.