Okounkov's conjecture via BPS Lie algebras

Given an arbitrary finite quiver Q, Maulik and Okounkov defined a new Yangian-style quantum group. It is built from the FRT formalism and their construction of R matrices on the cohomology of Nakajima quiver varieties, via the stable envelopes that they also defined. Just as in the case of ordinary Yangians, there is a Lie algebra g_Q inside their new algebra, and the Yangian is a deformation of the current algebra of this Lie algebra.

Outside of extended ADE type, numerous basic features of g_Q have remained mysterious since the outset of the subject, for example, the dimensions of the graded pieces. A conjecture of Okounkov predicts that these dimensions are given by the coefficients of Kac’s polynomials, which count isomorphism classes of absolutely indecomposable Q-representations over finite fields. I will explain a recent proof, with Botta, of the result that the Maulik-Okounkov Lie algebra is isomorphic to the “BPS Lie algebra” associated to the tripled quiver with potential, defined in joint work with Meinhardt, following the work of Kontsevich and Soibelman on critical cohomological Hall algebras, and then completely described in joint work with Hennecart and Schlegel-Mejia. A corollary of these results is that Okounkov’s conjecture is true.