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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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