  We introduce 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 flat (unlike that for the
preprojective algebra). 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 proofs are
based on the fact that our central extension for the weight \rho is the image
of the quantum Heisenberg algebra in the fusion category of representations of
quantum SL(2) under a tensor functor into bimodules over a semisimple algebra.
Finally, we explain how our algebras are connected to cyclotomic Hecke algebras
of complex reflection groups of rank 2, and in particular show that the
dimension of the latter for generic parameters is equal to the order of the
group, as conjectured by Broue, Malle, and Rouquier.
