Quiver Varieties and a Noncommutative P2
Tóm tắt
To any finite group
$$\Gamma \subset SL_2 (\mathbb{C})$$
and each element τ in the center of the group algebra of Γ, we associate a category,
$$\mathcal{C}oh(\mathbb{P}_{\Gamma ,\tau }^2 ,\mathbb{P}^1 )$$
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
$$\mathcal{C}oh(\mathbb{P}_{\Gamma ,\tau }^2 ,\mathbb{P}^1 )$$
should be thought of as the category of coherent sheaves on a `noncommutative projective space',
$$\mathbb{P}_{\Gamma ,\tau }^2 $$
equipped with a framing at
$$\mathbb{P}^1 $$
, the line at infinity. Our first result establishes an isomorphism between the moduli space of torsion free objects of
$$\mathcal{C}oh(\mathbb{P}_{\Gamma ,\tau }^2 ,\mathbb{P}^1 )$$
and the Nakajima quiver variety arising from Γ via the McKay correspondence. We apply the above isomorphism to deduce a generalization 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 quiver variety. This reduces, for
$$\Gamma \;{\text{ = }}\;{\text{\{ 1\} }}$$
, to the recently obtained parametrisation of the isomorphism 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
$$\mathbb{P}_{\Gamma ,\tau }^2 $$
. It is totally different from the one used by Berest and Wilson, involving τ-functions.
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