Quiver Varieties and a Noncommutative P2

Abstract

To any finite group GSL2ðCÞ and each element t in the center of the group alge- bra of G, we associate a category, CohðP 2 G;t ; P 1 Þ: It is defined as a suitable quotient of the category of graded modules over (a graded version of) the deformed preprojective algebra introduced by Crawley-Boevey and Holland. The category CohðP 2 G;t ; P 1 Þ should be thought of as the category of coherent sheaves on a 'noncommutative projective space', P 2;t ; equipped with a framing at P 1 , the line at infinity. Our first result establishes an isomorphism between the moduli space of torsion free objects of CohðP 2 G;t ; P 1 Þ and the Nakajima quiver variety arising from G via the McKay correspondence. We apply the above isomorphism to deduce a general- ization of the Crawley-Boevey and Holland conjecture, saying that the moduli space of 'rank 1' projective modules over the deformed preprojective algebra is isomorphic to a particular qui- ver variety. This reduces, for G¼f1g, to the recently obtained parametrisation of the iso- morphism classes of right ideals in the first Weyl algebra, A1, by points of the Calogero- Moser space, due to Cannings and Holland and Berest and Wilson. Our approach is algebraic and is based on a monadic description of torsion free sheaves on P 2 G;t . It is totally different