Maulik-Okounkov Lie algebras and BPS Lie algebras

The Maulik-Okounkov Lie algebra associated to a quiver Q controls the R-matrix formalism developed by Maulik and Okounkov in the context of quantum cohomology of Nakajima quiver varieties. On the other hand, the BPS Lie algebra originates from cohomological DT theory, particularly from the theory of cohomological Hall algebras associated with 3 Calabi-Yau categories. In this talk, I will explain how to identify the MO Lie algebra of an arbitrary quiver with the (appropriate) BPS Lie algebra. The bridge to compare these seemingly diverse words is the theory of non-abelian stable envelopes, which is exploited to relate representations of the MO Lie algebra to representations of the BPS Lie algebra. In conclusion, I will apply this result to deduce Okounkov’s conjecture, equating the graded dimensions of the MO Lie algebra with the coefficients of Kac polynomials. This is joint work with Ben Davison.