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Quiver varieties and a noncommutative $$\mathbb P_2$$. (English) Zbl 1048.14001
The authors associate to a finite subgroup $$\Gamma$$ of $$SL(2,\mathbb C)$$ and an element $$\tau$$ in the center of the group algebra $$\mathbb C[\Gamma]$$ a category, which is interpreted as a noncommutative analog of the complex projective plane. This category arises as a quotient of the category of graded modules over a graded version of the deformed preprojective algebra introduced by W. Crawley-Boevey and M. P. Holland [Duke Math. J. 92, No. 3, 605–635 (1998; Zbl 0974.16007)].
On the other hand, the authors associate to $$\Gamma$$, $$\tau$$ and each pair of finite dimensional $$\Gamma$$-modules a quiver variety, defined as a geometric invariant theory quotient of a variety of certain triples of homomorphisms. This quiver variety is isomorphic to a Nakajima quiver variety via the McKay correspondence.
The first main result is that the quiver variety is naturally isomorphic to a space of isomorphism classes of certain torsion free objects. This is then used to prove a generalized version of a conjecture by Crawley-Boevey and Holland.
Reviewer: Andreas Cap (Wien)

##### MSC:
 14A22 Noncommutative algebraic geometry 16S38 Rings arising from noncommutative algebraic geometry
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