Abstract:

    Let $O_\tau(\Gamma)$ be a family of algebras quantizing the coordinate ring of $\mathbb C^2/\Gamma$, where $\Gamma$ is a cyclic subgroup of $SL_2(\mathbb C)$. 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 this space breaks into a countable number of finite-dimensional algebraic (quiver) varieties.

    In the case when $\Gamma \cong \mathbb Z_2$, we show each of these quiver varieties are $G_\Gamma$-orbits. This will be used to show that there are discrete number non-isomorpic algebras Morita equivalent to $O_\tau(\mathbb Z_2)$, these algebras are exactly endomorphism rings of right ideals. We will also prove the natural embedding of $G_\Gamma$ into $\Pic(O_\tau(\mathbb Z_2))$, the Picard group of $O_\tau(\mathbb Z_2)$, is an isomorphism. The last two results are especially interesting, since $O_\tau(\mathbb Z_2)$'s are isomorphic to primitive factor rings  of $U(sl_2)$.  A structure of the group $G_{\Gamma}$, where $\Gamma$ is an arbitrary cyclic group,  is also investigated.