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- Title
McKay correspondence, cohomological Hall algebras and categorification.
- Authors
Diaconescu, Duiliu-Emanuel; Porta, Mauro; Sala, Francesco
- 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 a finer description 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.
- Subjects
ALGEBRA; DYNKIN diagrams; HECKE algebras; REPRESENTATIONS of algebras; ISOMORPHISM (Mathematics); HOMOLOGY theory; SHEAF theory
- Publication
Representation Theory, 2023, Vol 27, p933
- ISSN
1088-4165
- Publication type
Article
- DOI
10.1090/ert/649