Moduli stack of quiver representations Let $Q$ be a finite quiver. As far as I know there's a great amount of work concerning the so-called quiver varieties one can associate to it. Loosely speaking, these are obtained by taking GIT quotients of semistable representations of the (deformed) preprojective algebra one get from $Q$.
These varieties turn out to have a wide range of applications to representation theory and algebraic geometry.
Is there any work going on pursuing the "stacky" approach? Like taking already the stack of semistable representations or maybe even the full stack of representations of the preprojective algebra?
 A: Homology groups of moduli spaces of stable framed representations are equipped with representations of the full Kac-Moody Lie algebra. This construction was motivated by an earlier work of Lusztig on moduli stacks of representations of the preprojective algebra, where the upper subalgebra of the Kac-Moody Lie algebra was constructed. Strictly speaking, this is not precise, as Lusztig consider the space of constructible functions on the moduli stack.
A: There is a differential graded version of the preprojective algebra, called Ginzburg dg-algebra, that is a special case of the Calabi-Yau completion introduced by Keller.
The derived moduli stack of object (as defined by Toen-Vaquié) of a Calabi-Yau completion becomes a ($0$-shifted) symplectic derived stack. This is more generally true for the moduli of object of a finite type $2$-Calabi-Yau dg-category, as shown by Brav-Dyckerhoff, and also Toen in a little less general situation.
These ideas have been used in a recent paper of mine with Bozec and Scherotzke to introduce and study (apparently new) lagrangian subvarieties in the Hilbert scheme of points in the plane.
There are also two papers of Yeung on related topics, that are very interesting.
