  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.
