Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-18T15:32:12.568Z Has data issue: false hasContentIssue false
  • English
  • Français

Selfinjective quivers with potential and 2-representation-finite algebras

Part of: Homological algebra Representation theory of rings and algebras

Published online by Cambridge University Press:  28 September 2011

Martin Herschend  and
Osamu Iyama
Martin Herschend
Affiliation:
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan (email: martin.herschend@gmail.com)
Osamu Iyama
Affiliation:
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan (email: iyama@math.nagoya-u.ac.jp)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study quivers with potential (QPs) whose Jacobian algebras are finite-dimensional selfinjective. They are an analogue of the ‘good QPs’ studied by Bocklandt whose Jacobian algebras are 3-Calabi–Yau. We show that 2-representation-finite algebras are truncated Jacobian algebras of selfinjective QPs, which are factor algebras of Jacobian algebras by certain sets of arrows called cuts. We show that selfinjectivity of QPs is preserved under iterated mutation with respect to orbits of the Nakayama permutation. We give a sufficient condition for all truncated Jacobian algebras of a fixed QP to be derived equivalent. We introduce planar QPs which provide us with a rich source of selfinjective QPs.

Keywords

2-representation-finite algebraquiver with potential3-preprojective algebraselfinjective algebracluster tiltingmutation

MSC classification

Secondary: 16G10: Representations of Artinian rings 18G20: Homological dimension
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 6 , November 2011 , pp. 1885 - 1920
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[Ami09a]Amiot, C., Cluster categories for algebras of global dimension 2 and quivers with potential, Ann. Inst. Fourier 59 (2009), 2525–2590.CrossRefGoogle Scholar
[Ami09b]Amiot, C., A derived equivalence between cluster equivalent algebras, arXiv:0911.5410.Google Scholar
[AO10]Amiot, C. and Oppermann, S., Cluster equivalence and graded derived equivalence, arXiv:1003.4916.Google Scholar
[BG09]Barot, M. and Geiss, C., Tubular cluster algebras I: categorification, Math. Z., to appear, arXiv:0905.0028.Google Scholar
[BFPPT10]Barot, M., Fernandez, E., Platzeck, M., Pratti, N. and Trepode, S., From iterated tilted algebras to cluster-tilted algebras, Adv. Math. 223 (2010), 1468–1494.CrossRefGoogle Scholar
[BD02]Berenstein, D. and Douglas, M., Seiberg duality for quiver gauge theories, arXiv:hep-th/0207027.Google Scholar
[BO11]Bertani-Okland, M. A. and Oppermann, S., Mutating loops and 2-cycles in 2-CY triangulated categories, J. Algebra 334 (2011), 195–218.CrossRefGoogle Scholar
[Boc08]Bocklandt, R., Graded Calabi Yau algebras of dimension 3, J. Pure Appl. Algebra 212 (2008), 14–32.CrossRefGoogle Scholar
[Bro11]Broomhead, N., Dimer models and Calabi–Yau algebras, Mem. Amer. Math. Soc., published online, doi:10.1090/S0065-9266-2011-00617-9.CrossRefGoogle Scholar
[BIRS11]Buan, A., Iyama, O., Reiten, I. and Smith, D., Mutation of cluster-tilting objects and potentials, Amer. J. Math. 133 (2011), 835–887.CrossRefGoogle Scholar
[BMR06]Buan, A., Marsh, R. and Reiten, I., Cluster-tilted algebras of finite representation type, J. Algebra 306 (2006), 412–431.CrossRefGoogle Scholar
[Dav08]Davison, B., Consistency conditions for brane tilings, arXiv:0812.4185.Google Scholar
[DWZ08]Derksen, H., Weyman, J. and Zelevinsky, A., Quivers with potentials and their representations. I. Mutations, Selecta Math. (N.S.) 14 (2008), 59–119.CrossRefGoogle Scholar
[DWZ10]Derksen, H., Weyman, J. and Zelevinsky, A., Quivers with potentials and their representations. II: applications to cluster algebras, J. Amer. Math. Soc. 23 (2010), 749–790.CrossRefGoogle Scholar
[Gin06]Ginzburg, V., Calabi–Yau algebras, arXiv:math/0612139.Google Scholar
[Hat02]Hatcher, A., Algebraic topology (Cambridge University Press, Cambridge, 2002).Google Scholar
[HI11]Herschend, M. and Iyama, O., n-representation-finite algebras and twisted fractionally Calabi–Yau algebras, Bull. Lond. Math. Soc. 43 (2011), 449–466.CrossRefGoogle Scholar
[HILO]Herschend, M., Iyama, O., Lenzing, H. and Oppermann, S., in preparation.Google Scholar
[IU09]Ishii, A. and Ueda, K., Dimer models and the special McKay correspondence, arXiv:0905.0059.Google Scholar
[Iya07a]Iyama, O., Higher-dimensional Auslander–Reiten theory on maximal orthogonal subcategories, Adv. Math. 210 (2007), 22–50.CrossRefGoogle Scholar
[Iya07b]Iyama, O., Auslander correspondence, Adv. Math. 210 (2007), 51–82.CrossRefGoogle Scholar
[Iya08]Iyama, O., Auslander–Reiten theory revisited, in Trends in representation theory of algebras and related topics (European Mathematical Society, Zürich, 2008), 349–398.CrossRefGoogle Scholar
[Iya11]Iyama, O., Cluster tilting for higher Auslander algebras, Adv. Math. 226 (2011), 1–61.CrossRefGoogle Scholar
[IO09]Iyama, O. and Oppermann, S., Stable categories of higher preprojective algebras, arXiv:0912.3412.Google Scholar
[IO11]Iyama, O. and Oppermann, S., n-representation-finite algebras and n-APR tilting, Trans. Amer. Math. Soc. 363 (2011), 6575–6614.CrossRefGoogle Scholar
[IY08]Iyama, O. and Yoshino, Y., Mutation in triangulated categories and rigid Cohen–Macaulay modules, Invent. Math. 172 (2008), 117–168.CrossRefGoogle Scholar
[Kel10]Keller, B., The periodicity conjecture for pairs of Dynkin diagrams, arXiv:1001.1531.Google Scholar
[KVdB11]Keller, B. and Van den Bergh, M., Deformed Calabi–Yau completions, J. Reine Angew. Math. 654 (2011), 125–180.Google Scholar
[KY11]Keller, B. and Yong, D., Derived equivalences from mutations of quivers with potential, Adv. Math. 226 (2011), 2118–2168.CrossRefGoogle Scholar
[Rie80]Riedtmann, C., Algebren, Darstellungsköcher, Überlagerungen und zurück, Comment. Math. Helv. 55 (1980), 199–224.CrossRefGoogle Scholar
[Rin08]Ringel, C. M., The self-injective cluster-tilted algebras, Arch. Math. (Basel) 91 (2008), 218–225.CrossRefGoogle Scholar
[Sei95]Seiberg, N., Electric–magnetic duality in supersymmetric non-abelian gauge theories, Nuclear Phys. B 435 (1995), 129–146.CrossRefGoogle Scholar