Mathematics – Algebraic Geometry
Scientific paper
2008-11-17
Mathematics
Algebraic Geometry
50 pages. Major revision
Scientific paper
A superpotential algebra is square if its quiver admits an embedding into a two-torus such that the image of its underlying graph is a square grid, possibly with diagonal edges in the unit squares (examples are provided by brane tilings in physics). Such an embedding is special since it reveals much of the algebras representation theory through a device we introduce called an impression. Using impressions, we classify all simple representations of maximal k-dimension of all homogeneous square superpotential algebras and show that the localization of each algebra is a noncommutative crepant resolution with a 3 dimensional normal Gorenstein center, and hence a local Calabi-Yau algebra. Another special property of these algebras, equipped with an impression, is that crystal melting (a type of stability change) and quiver mutation may be regarded as a single operation. A particular class of square superpotential algebras, the Y^{p,q} algebras, is considered in detail. We show that the Azumaya and smooth loci of the centers coincide, and we make the proposal that the ``stack-like'' maximal ideals sitting over the singular locus are exceptional divisors of a blowup of the center shrunk to zero size.
Beil Charlie
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