BPS Algebras and Generalised Kac-Moody Algebras from 2-Calabi-Yau Categories
The categories of representations of preprojective algebras of quivers, Higgs sheaves on a smooth projective curve, and sheaves on quasiprojective symplectic (e.g. K3 or Abelian) surfaces all give rise to 2-Calabi-Yau categories. One is interested in the topology of the moduli spaces of semistable objects in these categories and how it relates to the topology of the moduli stacks. The tool to achieve this goal is the cohomological Hall algebra, an algebra structure on the Borel-Moore homology of the stack of objects in the category. One can extract a smaller associative algebra using some perverse filtration, the BPS algebra. We proved that it is the enveloping algebra of some generalised Kac-Moody Lie algebra with explicit Cartan datum. In several cases, it is known how to obtain the Lie algebra geometrically by considering the 3-Calabi-Yau completion of the category and the corresponding perverse filtration. Our structural results have several nice consequences related to nonabelian Hodge theory, the cohomology of Nakajima quiver varieties, and Kac polynomials.