Quiver varieties and a non-commutative P^2

Abstract

To any finite group G in SL_2(C), and each `t' in the center of the group algebra of G, we associate a category, Coh_t. 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 Coh_t should be thought of as the category of coherent sheaves on a `noncommutative projective 2-space', equipped with a framing at the line at infinity. Our first result establishes an isomorphism between the moduli space of torsion free objects of Coh_t and the Nakajima quiver variety arising from G via the McKay correspodence. We apply the above isomorphism to deduce generalized Crawley-Boevey & Holland conjecture, saying that the moduli space of `rank 1' projective modules over the deformed preprojective algebra is isomorphic to a particular quiver variety. This reduces, for G=1, to the recently obtained parametrisation of the isomorphism classes of right ideals in the first Weyl algebra, A_1, by points of the Calogero-Moser space, due to Cannings-Holland and Berest-Wilson. Our approach is algebraic and is based on a monadic description of torsion free sheaves on the noncommutative projective 2-space. It is totally different from the one used by Berest-Wilson, involving $\tau$-functions.

36pp, Crawley-Boevey & Holland conjecture generalized and proved

References in corpus (2)

• Noncommutative Instantons and Twistor Transform
• La correspondance de McKay

Cited by in corpus (16)

• Lectures on Noncommutative Geometry
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• Quiver varieties and Hilbert schemes
• D-Bundles and Integrable Hierarchies
• From solitons to many-body systems
• Perverse Bundles and Calogero-Moser Spaces
• Koszul Algebras and Sheaves over Projective Space
• Homological realization of Nakajima varieties and Weyl group actions
• Frobenius bimodules between noncommutative spaces
• Ideal classes of three dimensional Sklyanin algebras
• Flows of Calogero-Moser Systems
• Intersection cohomology of the Uhlenbeck compactification of the Calogero-Moser space
• Cusps and $\D$-Modules