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Noncommutative deformations of Kleinian singularities. (English) Zbl 0974.16007
A Kleinian singularity is the quotient $$K^2/\Gamma$$, where $$K$$ is an (algebraically closed) field (of characteristic zero) and $$\Gamma$$ is a nontrivial finite subgroup of $$\text{SL}_2(K)$$. More precisely, this is an object whose coordinate ring is $$K[x,y]^\Gamma$$, where the action of $$\Gamma$$ on $$K[x,y]$$ is extended from the given action of $$\Gamma$$ on the two-dimensional vector space spanned by $$x$$ and $$y$$. In the paper under review, the authors define and study a family $${\mathcal O}^\lambda$$ of deformations of $$K[x,y]^\Gamma$$, where $$\lambda\in Z(K\Gamma)$$. The definition of $${\mathcal O}^\lambda$$ is as follows. $$\Gamma$$ acts in an obvious way on the noncommuting polynomials $$K\langle x,y\rangle$$ and one forms the corresponding skew group ring $$K\langle x,y\rangle\Gamma$$. For $$\lambda\in Z(K\Gamma)$$, define $${\mathcal S}^\lambda$$ as the quotient $$K\langle x,y\rangle\Gamma/(xy-yx-\lambda)$$. Let $$e\in K\Gamma$$ be the average of the group elements. Then $${\mathcal O}^\lambda$$ is defined as $$e{\mathcal S}^\lambda e$$. These rings are Noetherian, finitely generated $$K$$-algebras, of Gelfand-Kirillov dimension 2. They are also Auslander-Gorenstein and Cohen-Macaulay. Other properties of $${\mathcal O}^\lambda$$ are studied by means of the so called deformed preprojective algebras.
In a subsequent paper by the second author [Comment. Math. Helv. 74, No. 4, 548-574 (1999; Zbl 0958.16014)], deformed preprojective algebras are embedded in a wider class of algebras, which provides a more conceptual approach to the study of deformations of Kleinian singularities. The reader is referred to that paper for more details.

##### MSC:
 16G10 Representations of associative Artinian rings 14B07 Deformations of singularities 16S80 Deformations of associative rings 14A22 Noncommutative algebraic geometry 16S32 Rings of differential operators (associative algebraic aspects)
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