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 representatation 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. One of the goals of this paper is to partially correct this problem. We do so by introducing a central extension of the preprojective algebra of a finite Dynkin quiver (depending on a regular weight for the corresponding root system), whose natural deformed version is actually flat, although it ceases to be flat after factorization by the central element. We calculate the Hilbert polynomial of the central extension, and show that it is a Frobenius algebra. As a corollary, we obtain the Hilbert series of the usual deformed preprojective algebra in which the deformation parameters are variables, and show that this algebra is Gorenstein (although it is not a flat module over the ring of parameters). The main tool in the proofs is the fact that our central extension for the weight ρ is the image of the quantum Heisenberg algebra in the fusion category of representations of quantum SL2 under a tensor functor into Rbimodules (where R is the algebra of idempotents of the quiver). This is a generalization of the result of [MOV] which says that the usual preprojective algebra is the image of the quantum symmetric algebra under the same functor. We also construct Riemann-Hilbert homomorphisms from the cyclotomic Hecke algebras of certain 2-dimensional complex reflection groups to the “spherical” subalgebras of the deformed central extensions of preprojective algebras. This allows us to show that if all parameters of the cyclotomic Hecke algebra are equal to 1 (i.e. the generators are unipotent) then the