Global BPS Lie algebra and tautological generation

The moduli stack and space of sheaves on quasi-projective symplectic surfaces (cotangent bundle of a curve, K3 or Abelian surface) are of high interest in algebraic geometry and enumerative geometry. Their cohomology encodes various enumerative invariants. The cohomology of the stack has the advantage of carrying extra structures: the cohomological Hall algebra structure. In this talk, I will explain how to construct a bialgebra structure on the cohomology of the moduli stack, and how to use it to define a global BPS Lie algebra without requiring the existence of a good moduli space. Then, I will explain how to generalize Markman's tautological generation theorem for smooth moduli spaces of sheaves on symplectic surfaces to non-smooth moduli spaces. These can be obtained from non-primitive Mukai vectors. This is joint work with Ben Davison, Tasuki Kinjo, Olivier Schiffmann and Eric Vasserot.