| Summary: | Associated with each finite subgroup Γ of SL2(C) there is a family of noncommutative algebras Oτ (Γ) quantizing C2/Γ. Let GΓ be the group of Γ -equivariant automorphisms of Oτ. In [16], one of the authors defined and studied a natural action of GΓ on certain quiver varieties associated with Γ. He established a GΓ -equivariant bijective correspondence between quiver varieties and the space of isomorphism classes of Oτ -ideals. In this paper we prove that the action of GΓ on the quiver variety is transitive when Γ is a cyclic group. This generalizes an earlier result due to Berest and Wilson who showed the transitivity of the automorphism group of the 1st Weyl algebra on the Calogero-Moser spaces. Our result has two important implications. First, it confirms the Bocklandt-Le Bruyn conjecture for cyclic quiver varieties. Second, it will be used to give a complete classification of algebras Morita equivalent to Oτ (Γ), thus answering the question of Hodges. At the end of the introduction we explain why the result of this paper does not extend when Γ is not cyclic. © 2022 The Author(s).
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