BPS Lie algebras and action on the cohomology of Nakajima quiver varieties

Kac--Moody Lie algebras form a family of Lie algebras constructed from a graph, extending the family of semisimple Lie algebras. Their representation theory is well understood, and a crucial role is played by the highest-weight modules. A geometric realisation of these modules was given by Nakajima in the 1990s using (a small part of) the cohomology of the now celebrated Nakajima quiver varieties. These geometric realisations provide powerful tools and, in particular, canonical bases. In my talk, I will explain how to decompose the full cohomology of Nakajima quiver varieties as a module over some generalised Kac--Moody Lie algebra. This answers a question of Nakajima and provides a new understanding of the cohomology of these varieties. Similar strategies may be used to decompose the cohomology of moduli spaces of framed sheaves on K3 or Abelian surfaces.