Kac's conjecture and the algebra of BPS states

Let Q be an affine quiver and let $\mathfrak{n}$ be the positive part of the affine Lie algebra associated to Q. We provide a construction of $\mathfrak{n}$ using the semistable irreducible components in the Lusztig nilpotent variety associated to Q. This confirms a conjecture of Frenkel, Malkin, and Vybornov on defining the so-called algebra of BPS states on the minimal resolution of a Kleinian singularity. Using the results of Crawley-Boevey and Van den Bergh, we show that our construction is closely connected to Kac's constant term conjecture in the case of an affine quiver.