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

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 $U(s{l}_2)$, 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 $A_1$.

39 pages

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• Flexible varieties and automorphism groups

Cited by in corpus (1)

• Morita equivalent unital locally matrix algebras