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Associated with each finite subgroup $\Gamma$ of $\rm{SL}_2(\mathbb{C})$ there is a family of noncommutative algebras $O_\tau(\Gamma)$ quantizing $\mathbb{C}^2/\!\!/\Gamma$. Let $G_\Gamma$ be the group of $\Gamma$-equivariant automorphisms of $O_\tau$. One of the authors earlier defined and studied a natural action of $G_\Gamma$ on certain quiver varieties associated with $\Gamma$. He established a $G_\Gamma$-equivariant bijective correspondence between quiver varieties and the space of isomorphism classes of $O_\tau$-ideals. The main theorem in this paper states that when $\Gamma$ is a cyclic group, the action of $G_\Gamma$ on each quiver variety is transitive. This generalizes an earlier result due to Berest and Wilson who showed the transitivity of the automorphism group of the first Weyl algebra on the Calogero-Moser spaces. Our result has two important implications. First, it confirms the Bockland-Le Bruyn conjecture for cyclic quiver varieties. Second, it will be used to give a complete classification of algebras Morita equivalent to $O_\tau(\Gamma)$.

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 $A_1$., Comment: 39 pages

Let $A$ be a Koszul (or more generally, $N$-Koszul) Calabi-Yau algebra. Inspired by the works of Kontsevich, Ginzburg and Van den Bergh, we show that there is a derived non-commutative Poisson structure on $A$, which induces a graded Lie algebra structure on the cyclic homology of $A$; moreover, we show that the Hochschild homology of $A$ is a Lie module over the cyclic homology and the Connes long exact sequence is in fact a sequence of Lie modules. Finally, we show that the Leibniz-Loday bracket associated to the derived non-commutative Poisson structure on $A$ is naturally mapped to the Gerstenhaber bracket on the Hochschild cohomology of its Koszul dual algebra and hence on that of $A$ itself. Relations with some other brackets in literature are also discussed and several examples are given in detail., Comment: 40 pages. Some typos corrected

We describe the structure of the automorphism groups of algebras Morita equivalent to the first Weyl algebra $ A_1 $. In particular, we give a geometric presentation for these groups in terms of amalgamated products, using the Bass-Serre theory of groups acting on graphs. A key r\^ole in our approach is played by a transitive action of the automorphism group of the free algebra $ \c < x, y > $ on the Calogero-Moser varieties $ \CC_n $ defined in \cite{BW}. Our results generalize well-known theorems of Dixmier and Makar-Limanov on automorphisms of $ A_1 $, answering an old question of Stafford (see \cite{St}). Finally, we propose a natural extension of the Dixmier Conjecture for $ A_1 $ to the class of Morita equivalent algebras., Comment: 12 pages, 2 figures