Skip to main content
Log in

Singularity Categories of some 2-CY-tilted Algebras

  • Published:
Algebras and Representation Theory Aims and scope Submit manuscript

Abstract

We define a class of finite-dimensional Jacobian algebras, which are called (simple) polygon-tree algebras, as a generalization of cluster-tilted algebras of type \(\mathbb {D}\). They are 2-CY-tilted algebras. Using a suitable process of mutations of quivers with potential (which are also BB-mutations) inducing derived equivalences, and one-pointed (co)extensions which preserve singularity equivalences, we find a connected selfinjective Nakayama algebra whose stable category is equivalent to the singularity category of a simple polygon-tree algebra. Furthermore, we also give a classification of algebras of this kind up to representation type.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Amiot, C.: Cluster categories for algebras of global dimension 2 and quivers with potential. Ann. Ins. Fourier (Grenoble) 59(6), 2525–2590 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Auslander, M., Reiten, I.: Applications of contravariantly finite subcategories. Adv. Math. 86(1), 111–152 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  3. Auslander, M., Reiten, I.: Cohen-Macaulay and Gorenstein Artin algebras. In: Progress in Math. 95, pp 221–245. Birkhäuser Verlag, Basel (1991)

  4. Barot, M.: C. Geißand A. Zelevinsky, C.uster algebras of finite type and positive symmetrizable matrices. J. London Math. Soc 73(3), 545–564 (2006)

    MathSciNet  Google Scholar 

  5. Barot, M., Kussin, D., Lenzing, H.: The cluster category of a canonical algebra. Trans. Amer. Math. Soc. 362(8), 4313–4330 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Barot, M., Trepode, S.: Cluster tilted algebras with a cyclically oriented quiver. Comm. Algebra 41(10), 3613–3628 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bastian, J., Holm, T., Ladkani, S.: Derived equivalence classification of cluster-tilted algebras of Dynkin type E. Algebr. Represent. Theor. 5(5), 567–594 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bastian, J., Holm, T., Ladkani, S.: Towards derived equivalence classification of the cluster-tilted algebras of Dynkin type D. J. Algebra 410, 277–332 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bernstein, I.N., Gelfand, I.M., Ponomarev, V.A.: Coxeter functors and Gabriel’s theorem. Russ. Math. Surv. 28(2), 17–32 (1973). (English)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brenner, S., Butler, M.C.R.: Generalizations of the Bernstein-Gelfand-Ponomarev reflection functors. In: Representation theory II (Proceedings Second International Conference, Carleton University, Ottawa, Ontario, 1979). Lecture Notes in Mathematics 832, pp 103–169. Springer-Verlag, Berlin (1980)

  11. Buan, A.B., Iyama, O., Reiten, I., Scott, J.: Cluster structures for 2-Calabi-Yau categories and unipotent groups. Compos. Math. 145(4), 1035–1079 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Buan, A.B., Iyama, O., Reiten, I., Smith, D.: Mutation of cluster-tilting objects and potentials. Am. J. Math. 133(4), 835–887 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Buan, A., Marsh, R., Reiten, I.: Cluster-tilted algebras. Trans. Amer. Math. Soc. 359(1), 323–332 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Buan, A., Marsh, R., Reiten, I.: Cluster mutation via quiver representation. Comment. Math. Helv. 83(1), 143–177 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Buan, A., Marsh, R., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. Adv. Math. 204, 572–618 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Buan, A., Reiten, I.: Acyclic quivers of finite mutation type. International Mathematics Research Notices(2006). 10pp (Art. ID 12804)

  17. Buchweitz, R.: Maximal Cohen-Macaulay modules and Tate cohomology over Gorenstein Rings. Unpublished Manuscript, 1987. Availble at: http://hdl.handle.net/1807/16682

  18. Chen, X-W.: Singularity categories, Schur functors and triangular matrix rings. Algebr. Represent. Theor. 12, 181–191 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chen, X. -W.: Singular equivalences induced by homological epimorphisms. Proc. Amer. Math. Soc. 142(8), 2633–2640 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chen, X., Geng, S., Lu, M.: The singularity categories of the Cluster-tilted algebras of Dynkin type. Algebr. Represent. Theor. 18(2), 531–554 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chen, X., Lu, M.: Singularity categories of skewed-gentle algebras. Colloq. Math. 141(2), 183–198 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Derksen, H., Weyman, J., Zelevinsky, A.: Quivers with potentials and their representations I: Mutations. Sel. Math., New Ser. 14, 59–119 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Felikson, A., Shapiro, M., Tumarkin, P.: Skew-symmetric cluster algebras of finite mutation type. J. Eur. Math. Soc. 14(4), 1135–1180 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Fomin, S., Shapiro, M., Thurston, D.: Cluster algebras and triangulated surfaces. Part I: Cluster complexes. Acta Math. 201(1), 83–146 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Fomin, S., Zelevinsky, A.: Cluster algebras i: Foundations. J. Amer. Math. Soc. 15(2), 497–529 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  26. Geiß, C., Labardini-Fragoso, D., Schröer, J.: The representation type of Jacobian algebras. Adv. Math. 290, 394–452 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Geng, S., Peng, L.: A classification of 2-finite diagrams. Algebra Colloq. 17 (1), 131–152 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Happel, D.: On Gorenstein algebras. In: Progress in Mathematics 95, pp 389–404. Birkhäuser Verlag, Basel (1991)

  29. Iyama, O., Yoshino, Y.: Mutation in triangulated categories and rigid cohen-Macaulay modules. Invent. Math. 172(1), 117–168 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kalck, M.: Singularity categories of gentle algebras. Bull. London Math. Soc. 47 (1), 65–74 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Keller, B.: On triangulated orbit categories. Doc. Math. 10, 551–581 (2005)

    MathSciNet  MATH  Google Scholar 

  32. Keller, B.: Quiver mutation in Java. www.math.jussieu.fr/~keller/quivermutation

  33. Keller, B., Reiten, I.: Cluster-tilted algebras are Gorenstein and stably Calabi-Yau. Adv. Math. 211, 123–151 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. Keller, B., Yang, D.: Derived equivalences from mutations of quivers with potential. Adv. Math. 226(3), 2118–2168 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Koenig, S., Nagase, H.: Hochschild cohomology and stratifying ideals. J. Pure Appl. Algebra 213, 886–891 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. Koenig, S., Zhu, B.: From triangulated categories to abelian categories-cluster tilting in a general framework. Math. Zeit. 258, 143–160 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ladkani, S.: Perverse equivalences, BB-tilting, mutations and applications. Preprint, available at arXiv:1001.4765

  38. Marsh, R., Reineke, M., Zelevinsky, A.: Generalized associahedra via quiver representations. Trans. Amer. Math. Soc. 355(10), 4171–4186 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  39. Orlov, D.: Triangulated categories of singularities and D-branes in Landau-Ginzburg models. Proc. Steklv Inst. Math. 246(3), 227–248 (2004)

    MathSciNet  MATH  Google Scholar 

  40. Reiten, I.: Calabi-Yau categories. Talk at the meeting: Calabi-Yau algebras and N-Koszul algebras (CIRM, 2007)

  41. Riedtmann, Chr., Darstellungsköcher, A.: Überlagerungen und zurück. Comment. Math. Helv. 55, 199–224 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  42. Riedtmann, Chr.: Representation-finite selfinjective algebras of class A n . Lecture Notes in Math. 832, pp 449–520. Springer-Verlag, Berlin (1980)

    Google Scholar 

  43. Seven, A.: Quivers of finite mutation type and skew-symmetric matrices. Linear Algebra Appl. 433, 1154–1169 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  44. Trepode, S., Valdivieso-Díaz, Y.: On finite dimensional Jacobian algebras. Bol. Soc. Mat. Mex., 1–14 (2012)

  45. Vatne, D.F.: The mutation class of D n quivers. Comm. Algebra 38(3), 1137–1146 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  46. Vitória, J.: Mutations vs. Seiberg duality. J. Algebra 321(3), 816–828 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ming Lu.

Additional information

Presented by Henning Krause.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lu, M. Singularity Categories of some 2-CY-tilted Algebras. Algebr Represent Theor 19, 1257–1295 (2016). https://doi.org/10.1007/s10468-016-9618-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10468-016-9618-3

Keywords

Mathematics Subject Classification (2010)

Navigation