## EVENT DETAILS AND ABSTRACT

**Geometry/Physics Seminar**
**Title:** Okounkov's conjecture via BPS Lie algebras

**Speaker:** Ben Davison

**Speaker Info:** University of Edinburgh

**Brief Description:**

**Special Note**:

**Abstract:**

Let $Q$ be an arbitrary finite quiver. We use nonabelian stable envelopes to relate representations of the Maulik-Okounkov Lie algebra $\mathfrak{g}^{\text{MO}}_Q$ to representations of the BPS Lie algebra associated to the tripled quiver $\tilde{Q}$ with its canonical potential. We use this comparison to provide an isomorphism between the Maulik-Okounkov Lie algebra and the BPS Lie algebra. Via this isomorphism we prove Okounkov's conjecture, equating the graded dimensions of the Lie algebra $\mathfrak{g}^{\text{MO}}_Q$ with the coefficients of Kac polynomials. Via general results regarding cohomological Hall algebras in dimensions two and three we furthermore give a complete description of $\mathfrak{g}^{\text{MO}}_Q$ as a generalised Kac-Moody Lie algebra with Cartan datum given by intersection cohomology of singular Nakajima quiver varieties, and prove a conjecture of Maulik and Okounkov, stating that their Lie algebra is obtained from a Lie algebra defined over the rationals, by extension of scalars.

**Date:** Tuesday, April 30, 2024

**Time:** 3:00pm

**Where:** Lunt 103

**Contact Person:** Ezra Getzler

**Contact email:** getzler@northwestern.edu

**Contact Phone:**

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