BPS Lie algebras and the less perverse filtration on the preprojective CoHA

Abstract

The affinization morphism for the stack $M\left({\Pi }_Q\right)$ of representations of a preprojective algebra ${\Pi }_Q$ is a local model for the morphism from the stack of objects in a general 2-Calabi–Yau category to the good moduli space. We show that the derived direct image of the dualizing complex along this morphism is pure, and admits a decomposition in the sense of the Beilinson–Bernstein–Deligne–Gabber decomposition theorem.

We introduce a new perverse filtration on the Borel–Moore homology of $M\left({\Pi }_Q\right)$, using this decomposition. We show that the zeroth piece of the resulting filtration on the cohomological Hall algebra built out of the Borel–Moore homology of $M\left({\Pi }_Q\right)$ is isomorphic to the universal enveloping algebra of an associated BPS Lie algebra $g_{{\Pi }_Q}$. This Lie algebra is defined via the Kontsevich–Soibelman theory of critical cohomological Hall algebras for 3-Calabi–Yau categories. We then lift this Lie algebra to a Lie algebra object in the category of perverse sheaves on the coarse moduli space of ${\Pi }_Q$-modules, and use this algebra structure to prove results about the summands appearing in the above decomposition theorem. In particular, we prove that the intersection cohomology of singular spaces of semistable ${\Pi }_Q$-modules provide “cuspidal cohomology” – a conjecturally complete canonical subspace of generators for $g_{{\Pi }_Q}$.