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

Abstract

The affinization morphism for the stack $\mathfrak{M}({\Pi }_Q)$ 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 $\mathfrak{M}({\Pi }_Q)$, 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 $\mathfrak{M}({\Pi }_Q)$ is isomorphic to the universal enveloping algebra of an associated BPS Lie algebra ${\mathfrak{g}}_{{\Pi }_Q}$. 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 subspace of canonical generators for ${\mathfrak{g}}_{{\Pi }_Q}$.

arXiv commentv6 - 48 pages, lots of improvements in presentation thanks to referee comments; v5 - corrected some signs, and a reference; v4 - added results on grading by dimension of supports; 51 pages. v3 - minor edits; 49 pages. v2 - minor edits; 45 pages, all comments welcome

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Cited by in corpus (3)

• Quantum toroidal algebras and solvable structures in gauge/string theory
• Cohomological Hall algebras and perverse coherent sheaves on toric Calabi-Yau 3-folds
• Purity and 2-Calabi-Yau categories