Abstract: I will report on joint work in progress with Sabin Cautis and Yukinobu
Toda. The Grothendieck group of the Hall category of a quiver is
isomorphic to a positive half of the quantum affine algebra corresponding
to the quiver. We show that two structures on the Hall category are
compatible: a perverse coherent t-structure and a filtration by window
categories.
In particular, we construct t-structures on the so-called quasi-BPS
categories, which generate Hall categories. We thus obtain a refinement of
the BPS cohomology of the quiver in terms of nilpotent orbits.
I will discuss examples of quasi-BPS categories for finite-type quivers,
where they may be compared with categories of representations of KLR
algebras, and for the Jordan quiver. As a corollary, we prove a conjecture
of Gorsky–Negut about generators of the category of coherent sheaves on the commuting stack.