Cluster-tilted algebras
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- by Aslak Bakke Buan, Bethany R. Marsh and Idun Reiten PDF
- Trans. Amer. Math. Soc. 359 (2007), 323-332
Abstract:
We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory is very close to the representation theory of hereditary algebras. As an application of this, we prove a generalised version of so-called APR-tilting.References
- Maurice Auslander and Idun Reiten, Representation theory of Artin algebras. V. Methods for computing almost split sequences and irreducible morphisms, Comm. Algebra 5 (1977), no. 5, 519–554. MR 439882, DOI 10.1080/00927877708822181
- Maurice Auslander, María Inés Platzeck, and Idun Reiten, Coxeter functors without diagrams, Trans. Amer. Math. Soc. 250 (1979), 1–46. MR 530043, DOI 10.1090/S0002-9947-1979-0530043-2
- M. Auslander and Sverre O. Smalø, Preprojective modules over Artin algebras, J. Algebra 66 (1980), no. 1, 61–122. MR 591246, DOI 10.1016/0021-8693(80)90113-1
- Klaus Bongartz, Tilted algebras, Representations of algebras (Puebla, 1980) Lecture Notes in Math., vol. 903, Springer, Berlin-New York, 1981, pp. 26–38. MR 654701
- Sheila Brenner and M. C. R. Butler, Generalizations of the Bernstein-Gel′fand-Ponomarev reflection functors, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979) Lecture Notes in Math., vol. 832, Springer, Berlin-New York, 1980, pp. 103–169. MR 607151
- Buan, A., Marsh, B., Reineke, M., Reiten, I., Todorov, G., Tilting theory and cluster combinatorics, preprint math.RT/0402054, (2004), to appear in Adv. Math.
- P. Caldero, F. Chapoton, and R. Schiffler, Quivers with relations arising from clusters ($A_n$ case), Trans. Amer. Math. Soc. 358 (2006), no. 3, 1347–1364. MR 2187656, DOI 10.1090/S0002-9947-05-03753-0
- Sergey Fomin and Andrei Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497–529. MR 1887642, DOI 10.1090/S0894-0347-01-00385-X
- Sergey Fomin and Andrei Zelevinsky, Cluster algebras: notes for the CDM-03 conference, Current developments in mathematics, 2003, Int. Press, Somerville, MA, 2003, pp. 1–34. MR 2132323
- Dieter Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, vol. 119, Cambridge University Press, Cambridge, 1988. MR 935124, DOI 10.1017/CBO9780511629228
- Dieter Happel and Claus Michael Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), no. 2, 399–443. MR 675063, DOI 10.1090/S0002-9947-1982-0675063-2
- Bernhard Keller, On triangulated orbit categories, Doc. Math. 10 (2005), 551–581. MR 2184464
- Robert Marsh, Markus Reineke, and Andrei Zelevinsky, Generalized associahedra via quiver representations, Trans. Amer. Math. Soc. 355 (2003), no. 10, 4171–4186. MR 1990581, DOI 10.1090/S0002-9947-03-03320-8
- C. Riedtmann, Algebren, Darstellungsköcher, Überlagerungen und zurück, Comment. Math. Helv. 55 (1980), no. 2, 199–224 (German). MR 576602, DOI 10.1007/BF02566682
Additional Information
- Aslak Bakke Buan
- Affiliation: Institutt for Matematiske Fag, Norges Teknisk-Naturvitenskapelige Universitet, N-7491 Trondheim, Norway
- Email: aslakb@math.ntnu.no
- Bethany R. Marsh
- Affiliation: Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, England
- MR Author ID: 614298
- ORCID: 0000-0002-4268-8937
- Idun Reiten
- Affiliation: Institutt for Matematiske Fag, Norges Teknisk-Naturvitenskapelige Universitet, N-7491 Trondheim, Norway
- Email: idunr@math.ntnu.no
- Received by editor(s): February 26, 2004
- Received by editor(s) in revised form: November 12, 2004
- Published electronically: July 21, 2006
- Additional Notes: The first author was supported by a grant from the Norwegian Research Council. The second author was supported by a Leverhulme Fellowship and by the Norwegian Research Council
- © Copyright 2006 A. B. Buan, B. R. Marsh and I. Reiten
- Journal: Trans. Amer. Math. Soc. 359 (2007), 323-332
- MSC (2000): Primary 16G20, 16G70; Secondary 16E99, 16S99
- DOI: https://doi.org/10.1090/S0002-9947-06-03879-7
- MathSciNet review: 2247893
Dedicated: Dedicated to Claus Michael Ringel on the occasion of his sixtieth birthday