McKay correspondence, cohomological Hall algebras and categorification

Abstract

Let $\pi\colon Y\to X$ denote the canonical resolution of the two dimensional Kleinian singularity $X$ of type ADE. In the present paper, we establish isomorphisms between the cohomological and K-theoretical Hall algebras of $\omega$-semistable properly supported sheaves on $Y$ with fixed slope $\mu$ and $\zeta$-semistable finite-dimensional representations of the preprojective algebra of affine type ADE 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. In the type A case, we provide finer descriptions of the cohomological, K-theoretical and categorified Hall algebra of $\omega$-semistable properly supported sheaves on $Y$ with fixed slope $\mu$: for example, in the cohomological case, the algebra can be given in terms of Yangians of finite type ADE Dynkin diagrams.