2020/07/07 by Davison, Ben · 1 citation
#81R10 #Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2007.03289
The affinization morphism for the stack \mathfrakM(ΠQ) of representations of a preprojective algebra ΠQ 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 \mathfrakM(ΠQ), 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 \mathfrakM(ΠQ) is isomorphic to the universal enveloping algebra of an associated BPS Lie algebra \mathfrakgΠQ. 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 ΠQ-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 ΠQ-modules provide "cuspidal cohomology" - a conjecturally complete subspace of canonical generators for \mathfrakgΠQ.