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The Derived Category of Surface Algebras: the Case of the Torus with One Boundary Component

Algebras and Representation Theory volume 19, pages 1059–1080 (2016)Cite this article

Abstract

In this paper we refine the main result of a previous paper of the author with Grimeland on derived invariants of surface algebras. We restrict to the case where the surface is a torus with one boundary component and give an easily computable derived invariant for such surface algebras. This result permits to give answers to open questions on gentle algebras: it provides examples of gentle algebras with the same AG-invariant (in the sense of Avella-Alaminos and Geiss) that are not derived equivalent and gives a partial positive answer to a conjecture due to Bobiński and Malicki on gentle 2-cycle algebras.

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References

  1. Amiot, C., Grimeland, Y.: Derived invariants for surface algebras, preprint, arXiv:1411.0383, to appear in Journal of Pure and Applied Algebra

  2. Amiot, C., Oppermann, S.: Cluster equivalence and graded derived equivalence. Documenta Math. 19, 1155–1206 (2014)

    MathSciNet  MATH  Google Scholar 

  3. Amiot, C., Oppermann, S.: Algebras of acyclic cluster type: tree type and type \(\widetilde {A}\). Nagoya Math. J. 211, 1–50 (2013)

    MathSciNet  MATH  Google Scholar 

  4. Angeleri-Hügel, L., Happel, D., Krause, H. (eds.): Handbook of Tilting Theory, London Mathematical Society, vol. 332. Cambridge University Press (2007)

  5. Assem, I., Brüstle, T., Charbonneau-Jodoin, G., Plamondon, P.-G.: Gentle algebras arising from surface triangulations. Algebra Number Theory 4(2), 201–229 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  6. Avella-Alaminos, D., Geiss, C.: Combinatorial derived invariants for gentle algebras. J. Pure Appl. Algebra 212(1), 228–243 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  7. Avella-Alaminos, D.: Derived classification of gentle algebras with two cycles. Bol. Soc. Mat. Mexicana (3) 14(2), 177–216 (2008)

    MathSciNet  MATH  Google Scholar 

  8. Bobiński, G., Malicki, P.: On derived equivalence classification of gentle two-cycle algebras. Colloq. Math. 112(1), 33–72 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  9. Bobiński, G.: The derived equivalence classification of gentle two-cycle algebras, preprint, arXiv:1509:08631

  10. Chow, W.: On the algebraical braid group. Ann. Math. 49, 654–658 (1948)

    MathSciNet  Article  MATH  Google Scholar 

  11. David-Roesler, L., Schiffler, R.: Algebras from surfaces without punctures. J. Algebra 350, 218–244 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  12. Farb, B., Margalit, D.: A Primer on Mapping class group, Princeton mathmatical series 49

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

    MathSciNet  Article  MATH  Google Scholar 

  14. Happel, D.: Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras, volume 119 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1988)

    Google Scholar 

  15. Paris, L., Rolfsen, D.: Geometric subgroups of mapping class groups. J. Reine Angew. Math. 521, 47–83 (2000)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institut Fourier, 100 rue des maths, 38402, Saint Martin d’Hères, France

    Claire Amiot

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  1. Claire Amiot
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Correspondence to Claire Amiot.

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Presented by Peter Littelmann.

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Amiot, C. The Derived Category of Surface Algebras: the Case of the Torus with One Boundary Component. Algebr Represent Theor 19, 1059–1080 (2016). https://doi.org/10.1007/s10468-016-9611-x

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  • DOI: https://doi.org/10.1007/s10468-016-9611-x

Keywords

  • Quiver representation
  • Derived categories
  • Triangulations of surfaces
  • Cluster combinatorics
  • Quiver mutation

Mathematics Subject Classification (2010)

  • 16E35
  • 16G20
  • 14F35