On Transitive Action on Quiver Varieties

Abstract Associated with each finite subgroup $\Gamma $ of ${\textrm{SL}}_2({\mathbb{C}})$ there is a family noncommutative algebras $O_\tau (\Gamma )$ quantizing ${\mathbb{C}}^2/\!\!/\Gamma $. Let $G_\Gamma be the group $-equivariant automorphisms In [ 16], one authors defined and studied natural action on certain quiver varieties associated He established bijective correspondence between space isomorphism classes $-ideals. this paper we prove that variety transitive when cyclic group. This generalizes an earlier result due to Berest Wilson who showed transitivity automorphism 1st Weyl algebra Calogero–Moser spaces. Our has two important implications. First, it confirms Bocklandt–Le Bruyn conjecture for varieties. Second, will used give complete classification Morita equivalent ),$ thus answering question Hodges. At end introduction explain why does not extend cyclic.