Automorphisms and Ideals of Noncommutative Deformations of $\mathbb{C}^2/\mathbb{Z}_2$

Let $O_τ(Γ)$ be a family of algebras \textit{quantizing} the coordinate ring of $\mathbb{C}^2 / Γ$, where $Γ$ is a finite subgroup of $\mathrm{SL}_2(\mathbb{C})$, and let $G_Γ$ be the automorphism group of $O_τ$. We study the natural action of $G_Γ$ on the space of right ideals of $O_τ$ (equivalently, finitely generated rank $1$ projective $O_τ$-modules). It is known that the later can be identified with disjoint union of algebraic (quiver) varieties, and this identification is $G_Γ$-equivariant. In the present paper, when $Γ\cong \mathbb{Z}_2$, we show that the $G_Γ$-action on each quiver variety is transitive. We also show that the natural embedding of $G_Γ$ into $\mathrm{Pic}(O_τ)$, the Picard group of $O_τ$, is an isomorphism. These results are used to prove that there are countably many non-isomorphic algebras Morita equivalent to $O_τ$, and explicit presentation of these algebras are given. Since algebras $O_τ(\mathbb{Z}_2)$ are isomorphic to primitive factors of $U(sl_2)$, we obtain a complete description of algebras Morita equivalent to primitive factors. A structure of the group $G_Γ$, where $Γ$ is an arbitrary cyclic group, is also investigated. Our results generalize earlier results obtained for the (first) Weyl algebra $A_1$.

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