On cohomological Hall algebras of quivers: Generators

We study the cohomological Hall algebra {\operatorname{Y}\nolimits^{\flat}} of a Lagrangian substack {\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 {\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 {\operatorname{Y}\nolimits^{\flat}} is equal, after a suitable extension of scalars, to the Yangian {\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.

Borcherds–Bozec algebras, root multiplicities and the Schofield construction

Using the twisted denominator identity, we derive a closed form root multiplicity formula for all symmetrizable Borcherds–Bozec algebras and discuss its applications including the case of Monster Borcherds–Bozec algebra. In the second half of the paper, we provide the Schofield construction of symmetric Borcherds–Bozec algebras.