Geometric Representation Theory Seminar | Maulik-Okounkov Lie algebras and BPS Lie algebras

Abstract：

The Maulik-Okounkov (MO) 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, and in particular from the theory of cohomological Hall algebras associated to 3 Calabi-Yau categories. In this talk, I will explain how to identify the MO Lie algebra of Q with the BPS Lie algebra of the tripled quiver Q̃ with its canonical cubic potential. To link these seemingly diverse words, I will review the theory of non-abelian stable envelopes and use them to relate representations of the MO Lie algebra to representations of the BPS Lie algebra. As a byproduct, I will present a proof of 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.