Quiver varieties and a non-commutative P^2

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dc.creatorBaranovsky, Vladimir
dc.creatorGinzburg, Victor
dc.creatorKuznetsov, Alexander
dc.date2001-03-12
dc.date2001-07-06
dc.date.accessioned2026-07-07T04:40:35Z
dc.date.available2026-07-07T04:40:35Z
dc.descriptionTo 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 $τ$-functions.
dc.description36pp, Crawley-Boevey & Holland conjecture generalized and proved
dc.identifierhttps://arxiv.org/abs/math/0103068
dc.identifierhttp://arxiv.org/abs/math/0103068
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/61074
dc.subjectAlgebraic Geometry
dc.subjectQuantum Algebra
dc.titleQuiver varieties and a non-commutative P^2
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