Ben Davison, University of Edinburgh

Abstract: 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 this conjecture. By proving that the Maulik-Okounkov Lie algebra g_Q is isomorphic to certain BPS Lie algebras, we prove Okounkov's conjecture, as well as essentially determining the isomorphism class of g_Q, thanks to recent joint work of myself with Hennecart and Schlegel Mejia.