Quiver Heisenberg algebras: a cubic analogue of preprojective algebras

Abstract: In this paper we study a certain class of central extensions of preprojective algebras of quivers under the name quiver Heisenberg algebras (QHA). There are several classes of algebras introduced before by different researchers from different view points, which have the QHA as a special case. While these have mainly been studied in characteristic zero, we also study the case of positive characteristic. Our results show that the QHA is closely related to the representation theory of the corresponding path algebra in a similar way to the preprojective algebra. Among other things, one of our main results is that the QHA provides an exact sequence of bimodules over the path algebra of a quiver, which can be called the universal Auslander-Reiten sequence. Moreover, we show that the QHA provides minimal left and right approximations with respect to the powers of the radical functor. Consequently, we obtain a description of the QHA as a module over the path algebra, which in the Dynkin case, gives a categorification (as well as a generalization to the positive characteristic case) of the dimension formula by Etingof-Rains.