On cohomological Hall algebras of quivers: Generators

Abstract

Abstract We study the cohomological Hall algebra Y ♭ {\operatorname{Y}\nolimits^{\flat}} of a Lagrangian substack Λ ♭ {\Lambda^{\flat}} 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 ♭ {\operatorname{Y}\nolimits^{\flat}} is pure and we compute its Poincaré polynomials in terms of (nilpotent) Kac polynomials. We also provide a family of algebra generators. We conjecture that Y ♭ {\operatorname{Y}\nolimits^{\flat}} is equal, after a suitable extension of scalars, to the Yangian 𝕐 {\mathbb{Y}} 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.