McKay correspondence, cohomological Hall algebras and categorification

Abstract

Let π : Y → X \pi \colon Y\to X denote the canonical resolution of the two dimensional Kleinian singularity X X of type ADE. In the present paper, we establish isomorphisms between the cohomological and K-theoretical Hall algebras of ω \omega -semistable properly supported sheaves on Y Y with fixed slope μ \mu and ζ \zeta -semistable finite-dimensional representations of the preprojective algebra of affine type ADE of slope zero respectively, under some conditions on ζ \zeta depending on the polarization ω \omega and μ \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 a finer description of the cohomological, K-theoretical and categorified Hall algebra of ω \omega -semistable properly supported sheaves on Y Y with fixed slope μ \mu : for example, in the cohomological case, the algebra can be given in terms of Yangians of finite type ADE Dynkin diagrams.