Abstract
We show that every nondegenerate dimer algebra A on a torus admits a cyclic contraction to a cancellative dimer algebra. This implies, for example, that A is Calabi-Yau if and only if it is noetherian; and that the center of A has Krull dimension 3.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Baur, K., King, A., Marsh, R.: Dimer models and cluster categories of Grassmannians. Proc. London Math. Soc. 113(2), 213–260 (2016)
Beil, C.: Homotopy algebras, dimer algebras, and cyclic contractions, arXiv:1711.09771
Beil, C.: Morita equivalences and Azumaya loci from Higgsing dimer algebras,. J. Algebra 453, 429–455 (2016)
Beil, C.: Noetherian criteria for dimer algebras, arXiv:1805.08047
Beil, C.: Nonnoetherian homotopy dimer algebras and noncommutative crepant resolutions. Glasgow Math. J. 10, 1–33 (2017)
Beil, C.: On the central geometry of nonnoetherian dimer algebras, in preparation
Beil, C.: On the noncommutative geometry of square superpotential algebras. J. Algebra 371, 207–249 (2012)
Beil, C.: The central nilradical of nonnoetherian dimer algebras, in preparation
Beil, C., Ishii, A., Ueda, K.: Cancellativization of dimer models, arXiv:1301.5410
Bocklandt, R.: A dimer ABC. Bull. London Math. Soc. 48, 3, 387–451 (2016)
Bocklandt, R.: Graded Calabi Yau algebras of dimension 3. J. Pure Appl. Algebra 212(1), 14–32 (2008)
Bose, S., Gundry, J., He, Y.: Gauge theories and dessins d’enfants: Beyond the torus. J. High Energy Phys. 1, 135 (2015)
Broomhead, N.: Dimer models and Calabi-Yau algebras. Memoirs AMS, 1011 (2012)
Craw, A., Bocklandt, R., Quintero Vélez, A.: Geometric Reid’s recipe for dimer models. Math. Ann. 361, 689–723 (2015)
Feng, B., He, Y., Kennaway, K. D., Vafa, C.: Dimer models from mirror symmetry and quivering amoebae. Adv. Theor. Math. Phys., 12 (2008)
Franco, S., Hanany, A., Martelli, D., Sparks, J., Vegh, D., Wecht, B.: Gauge theories from toric geometry and brane tilings. J. High Energy Phys. 1, 128 (2006)
Franco, S., Hanany, A., Vegh, D., Wecht, B., Kennaway, K.: Brane dimers and quiver gauge theories. J. High Energy Phys. 1, 096 (2006)
Futaki, M., Ueda, K.: Exact Lefschetz fibrations associated with dimer models. Math. Res. Lett. 17(6), 1029–1040 (2010)
Goncharov, A., Kenyon, R.: Dimers and cluster integrable systems. Annales scientifiques de l’ENS 46(5), 747–813 (2013)
Hanany, A., Kennaway, K.D.: Dimer models and toric diagrams, arXiv:0503149
Ishii, A., Ueda, K.: Dimer models and the special McKay correspondence. Geom. Topol. 19(6), 3405–3466 (2015)
Iyama, O., Nakajima, Y.: On steady non-commutative crepant resolutions, J. Noncommut. Geom., to appear
Acknowledgments
Open access funding provided by Austrian Science Fund (FWF). The author would like to thank Akira Ishii, Kazushi Ueda, and Ana Garcia Elsener for useful discussions, as well as an anonymous referee for comments that have helped improve the article. The author was supported by the Austrian Science Fund (FWF) grant P 30549-N26.
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by: Henning Krause
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Beil, C. Cyclic Contractions of Dimer Algebras Always Exist. Algebr Represent Theor 22, 1083–1100 (2019). https://doi.org/10.1007/s10468-018-9812-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-018-9812-6