On Central Extensions of Preprojective Algebras

μiei), where μ = (μi) is a regular weight (for the root system attached to Q), and ei are the vertex idempotents in P . 1 It is shown in [ER] that the algebra A has nicer properties than the ordinary Gelfand-Ponomarev preprojective algebra A0 = A/(z) of Q; in particular, the deformed version A(λ) of A = A(0) is flat, while this is not the case for A0. The paper [ER] also shows that A is a Frobenius algebra, and computes the Hilbert series of A. Finally, [ER] links the algebra A with cyclotomic Hecke algebras of complex reflection groups of rank 2. The goal of this paper is to continue to study the rich structure of the algebra A. In particular, we show that for generic μ (and specifically for μ = ρ) the algebra A has a unique trace functional, and compute the structure of the center Z of A and the trace space A/[A,A]. Namely, it turns out that Z and A/[A,A] are dual to each other under the trace form, and the dimension of the homogeneous subspace (A/[A,A])[2p] equals the number of positive roots for Q of height p+ 1. We also show that the elements z( ∑