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

TL;DR

This paper investigates automorphisms and ideals of noncommutative deformations of quotient singularities, revealing transitive group actions, isomorphisms with Picard groups, and classifying Morita equivalence classes, extending previous results for Weyl algebras.

Contribution

It provides a detailed analysis of automorphism groups and ideal structures of quantized coordinate rings of $C^2/Z_2$, including explicit algebra classifications and generalizations to cyclic groups.

Findings

Automorphism group acts transitively on quiver varieties for $Z_2$.

The automorphism group is isomorphic to the Picard group of the algebra.

Countably many non-isomorphic Morita equivalent algebras are classified.

Abstract

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 $A_1$.