The BPS sheaves and quiver varieties

I will explain how to describe the BPS cohomology as a (positive part of a) generalized Kac-Moody Lie algebra and its action on the cohomology of Nakajima quiver varieties. This structure gives a decomposition of the cohomology of Nakajima quiver varieties as a direct sum of irreducible, lowest representations, generalizing simultaneously the action of the infinite Heisenberg Lie algebra on the cohomology of Hilbert schemes of points on the plane (Nakajima, Grojnowski) and the geometric realization of the lowest representations of Kac-Moody algebras (Nakajima). This is joint with Ben Davison and Sebastian Schlegel Mejia.