McKay correspondence, cohomological Hall algebras and categorification

Let $\pi\colon Y\to \mathbb C^2/\mathbb Z_N$ denote the canonical resolution of the two dimensional Kleinian singularity of type $A_{N-1}$ for $N\geq 2$. In the present paper, we establish isomorphisms between the cohomological and K-theoretical Hall algebras of $\omega$-semistable compactly supported sheaves on $Y$ with fixed slope $\mu$ and $\zeta$-semistable finite-dimensional representations of the preprojective algebra $\Pi_{A_{N-1}^{(1)}}$ of slope zero respectively, under some conditions on $\zeta$ depending on the polarization $\omega$ and $\mu$. These isomorphisms are induced by the derived McKay correspondence. In addition, they are interpreted as decategorified versions of a monoidal equivalence between the corresponding categorified Hall algebras. Finally, we provide explicit descriptions of the cohomological, K-theoretical and categorified Hall algebra of $\omega$-semistable compactly supported sheaves on $Y$ with fixed slope $\mu$: for example, in the cohomological case, the algebra is given in terms of Yangians of finite type A Dynkin diagrams.