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

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