Let Q be a finite acyclic quiver. In my talk I will discuss several algebras associated to Q and how they are related. As a starting point we’ll consider the path algebra of Q and how its representation theory is reflected in homological properties of the preprojective algebra of Q. One immediate connection is that the preprojective algebra is graded and its degree zero part is the path algebra.
Next we turn to the quiver Heisenberg algebra of Q. This algebra is a particular case of the central extensions of preprojective algebras introduced by Etingof-Rains. It has many similar properties to the preprojective algebra. Finally, we will consider a certain double cover of the quiver Heisenberg algebra, more precisely its second quasi-Veronese algebra. This algebra is also graded and turns out to be a higher preprojective algebra of its degree zero part B. The algebra B has many similarities with the original path algebra. It has global dimension 2 and is 2-hereditary algebra in the sense of Iyama’s higher dimensional Auslander-Reiten theory.
This talk is based on ongoing joint work with Hiroyuki Minamoto.