# On cohomological Hall algebras of quivers : generators

## TL;DR

This paper investigates the structure of the cohomological Hall algebra associated with quivers, demonstrating its purity, computing its Poincare polynomials, and proposing a conjectural relation to Yangians, with implications for Kac polynomials.

## Contribution

It introduces a family of algebra generators for the cohomological Hall algebra and conjectures its equivalence to the Yangian of Maulik and Okounkov.

## Key findings

- Proves the purity of the cohomological Hall algebra.
- Computes Poincare polynomials in terms of Kac polynomials.
- Establishes a variant of Okounkov's conjecture related to Kac polynomials.

## Abstract

We study the cohomological Hall algebra Y of a lagrangian substack of the moduli stack of representations of the preprojective algebra of an arbitrary quiver Q, and their actions on the cohomology of Nakajima quiver varieties. We prove that Y is pure and we compute its Poincare polynomials in terms of (nilpotent) Kac polynomials. We also provide a family of algebra generators. We conjecture that Y is equal, after a suitable extension of scalars, to the Yangian introduced by Maulik and Okounkov. As a corollary, we prove a variant of Okounkov's conjecture, which is a generalization of the Kac conjecture relating the constant term of Kac polynomials to root multiplicities of Kac-Moody algebras.