Quiver GIT for Reconstruction Algebras.

Joe Karmazyn (University of Edinburgh)

Wednesday 5th November, 2014 16:00-17:00 Maths 516


A finite subgroup of \SL_2(\mathbb{C}) defines an invariant ring \mathbb{C}[x,y]^G and a surface quotient singularity \Spec \mathbb{C}[x,y]^G. It is also possible to associate a certain noncommutative algebra, the preprojective algebra, such that the minimal resolution of the quotient singularity can be realised as a moduli space of representations of this algebra (which can be constructed by quiver GIT). Moreover, the McKay correspondence shows that this moduli construction induces a derived equivalence between the minimal resolution and the preprojective algebra.

For G< GL_2(\mathbb{C}) quotient surface singularities (or more generally rational surface singularities) there is also an associated noncommutative algebra, the reconstruction algebra, that is derived equivalent to a geometric minimal resolution. However, it is currently only known that quiver GIT constructs the minimal resolution and induces a derived equivalence in certain cases.

I will recall the preprojective algebras and how they fit into the McKay correspondence via quiver GIT, and then I aim to discuss analogous results for the reconstruction algebras.

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