Around Hall algebras in 50 minutes

The talk will start with an introduction to Hall algebras, different versions and appearances thereof. Then we will turn our attention to McKay correspondence in dimensions two and three. This is a correspondence between nontrivial irreducible representations of a finite subgroup G⊂GLn(C) and irreducible subvarieties in the central fiber of resolution of the categorical quotient X=Cn//G. In examples relevant to our discussion X will have an isolated singularity at the origin. Another (more modern) version of McKay correspondence is the derived equivalence Ψ : Db(Coh_G(Cn)) → Db(Coh(Y)).
Sometimes Hall algebras 'arising' on the right hand side admit embeddings of subalgebras in universal enveloping algebras U(g). We will see examples of the latter result for Lie algebras g with simply laced Dynkin diagrams of finite or affine type. The aim of the talk will be to give an accessible introduction to the aforementioned topics.