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.
Similar content being viewed by others
References
Amiot, C., Grimeland, Y.: Derived invariants for surface algebras, preprint, arXiv:1411.0383, to appear in Journal of Pure and Applied Algebra
Amiot, C., Oppermann, S.: Cluster equivalence and graded derived equivalence. Documenta Math. 19, 1155–1206 (2014)
Amiot, C., Oppermann, S.: Algebras of acyclic cluster type: tree type and type \(\widetilde {A}\). Nagoya Math. J. 211, 1–50 (2013)
Angeleri-Hügel, L., Happel, D., Krause, H. (eds.): Handbook of Tilting Theory, London Mathematical Society, vol. 332. Cambridge University Press (2007)
Assem, I., Brüstle, T., Charbonneau-Jodoin, G., Plamondon, P.-G.: Gentle algebras arising from surface triangulations. Algebra Number Theory 4(2), 201–229 (2010)
Avella-Alaminos, D., Geiss, C.: Combinatorial derived invariants for gentle algebras. J. Pure Appl. Algebra 212(1), 228–243 (2008)
Avella-Alaminos, D.: Derived classification of gentle algebras with two cycles. Bol. Soc. Mat. Mexicana (3) 14(2), 177–216 (2008)
Bobiński, G., Malicki, P.: On derived equivalence classification of gentle two-cycle algebras. Colloq. Math. 112(1), 33–72 (2008)
Bobiński, G.: The derived equivalence classification of gentle two-cycle algebras, preprint, arXiv:1509:08631
Chow, W.: On the algebraical braid group. Ann. Math. 49, 654–658 (1948)
David-Roesler, L., Schiffler, R.: Algebras from surfaces without punctures. J. Algebra 350, 218–244 (2012)
Farb, B., Margalit, D.: A Primer on Mapping class group, Princeton mathmatical series 49
Fomin, S., Shapiro, M., Thurston, D.: Cluster algebras and triangulated surfaces. I. Cluster complexes. Acta Math. 201(1), 83–146 (2008)
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)
Paris, L., Rolfsen, D.: Geometric subgroups of mapping class groups. J. Reine Angew. Math. 521, 47–83 (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by Peter Littelmann.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-016-9611-x
Keywords
- Quiver representation
- Derived categories
- Triangulations of surfaces
- Cluster combinatorics
- Quiver mutation