Quiver varieties and a noncommutative \(\mathbb P_2\). (English) Zbl 1048.14001

Baranovsky, V.; Ginzburg, V.; Kuznetsov, A.

Compos. Math. 134, No. 3, 283-318 (2002).

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

Keywords:

noncommutative algebraic geometry; deformed preprojective algebra; McKay correspondence; Nakajima quiver variety