Frobenius Degenerations of Preprojective Algebras

In this paper, we study a preprojective algebra for quivers decorated with $k$-algebras and bimodules, which generalizes work of Gabriel for ordinary quivers, work of Dlab and Ringel for $k$-species, and recent work of de Thanhoffer de V\"olcsey and Presotto, which has recently appeared from a different perspective in work of K\"ulshammer. As for undecorated quivers, we show that its moduli space of representations recovers the Hamiltonian reduction of the cotangent bundle over the space of representations of the decorated quiver. These algebras yield degenerations of ordinary preprojective algebras, by folding the quiver and then degenerating the decorations. We prove that these degenerations are flat in the Dynkin case, and conjecture, based on computer results, that this extends to arbitrary decorated quivers.