$n$-representation-finite algebras and $n$-APR tilting
HTML articles powered by AMS MathViewer
- by Osamu Iyama and Steffen Oppermann PDF
- Trans. Amer. Math. Soc. 363 (2011), 6575-6614 Request permission
Abstract:
We introduce the notion of $n$-representation-finiteness, generalizing representation-finite hereditary algebras. We establish the procedure of $n$-APR tilting and show that it preserves $n$-representation-finiteness. We give some combinatorial description of this procedure and use this to completely describe a class of $n$-representation-finite algebras called “type A”.References
- Maurice Auslander and Mark Bridger, Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969. MR 0269685
- Claire Amiot, Cluster categories for algebras of global dimension 2 and quivers with potential, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 6, 2525–2590 (English, with English and French summaries). MR 2640929
- Claire Amiot. Sur les petites catégories triangulées. PhD thesis, Université Paris 7, 2008.
- Maurice Auslander, MarĂa InĂ©s 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
- Maurice Auslander, Idun Reiten, and Sverre O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1997. Corrected reprint of the 1995 original. MR 1476671
- Michael Barot, Elsa Fernández, MarĂa InĂ©s Platzeck, Nilda Isabel Pratti, and Sonia Trepode, From iterated tilted algebras to cluster-tilted algebras, Adv. Math. 223 (2010), no. 4, 1468–1494. MR 2581376, DOI 10.1016/j.aim.2009.10.004
- Aslak Bakke Buan, Robert J. Marsh, and Idun Reiten, Cluster-tilted algebras, Trans. Amer. Math. Soc. 359 (2007), no. 1, 323–332. MR 2247893, DOI 10.1090/S0002-9947-06-03879-7
- Aslak Bakke Buan, Idun Reiten, and Ahmet I. Seven, Tame concealed algebras and cluster quivers of minimal infinite type, J. Pure Appl. Algebra 211 (2007), no. 1, 71–82. MR 2333764, DOI 10.1016/j.jpaa.2006.12.007
- Christof Geiß, Bernard Leclerc, and Jan Schröer, Rigid modules over preprojective algebras, Invent. Math. 165 (2006), no. 3, 589–632. MR 2242628, DOI 10.1007/s00222-006-0507-y
- Christof Geiss, Bernard Leclerc, and Jan Schröer, Auslander algebras and initial seeds for cluster algebras, J. Lond. Math. Soc. (2) 75 (2007), no. 3, 718–740. MR 2352732, DOI 10.1112/jlms/jdm017
- 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
- Martin Herschend and Osamu Iyama. $n$-representation-finite algebras and twisted fractionally Calabi-Yau algebras. Bull. Lond. Math. Soc., 43(3):449–466, 2011.
- Wei Hu and Changchang Xi. $\mathcal {D}$-split sequences and derived equivalences. Adv. Math. 227:292–318, 2011.
- Zhaoyong Huang and Xiaojin Zhang. Higher Auslander Algebras Admitting Trivial Maximal Orthogonal Subcategories. J. Algebra 330(1):375–387, 2011.
- Zhaoyong Huang and Xiaojin Zhang. Trivial Maximal 1-Orthogonal Subcategories For Auslander’s 1-Gorenstein Algebras. preprint, arXiv:0903.0762.
- Zhaoyong Huang and Xiaojin Zhang, The existence of maximal $n$-orthogonal subcategories, J. Algebra 321 (2009), no. 10, 2829–2842. MR 2512629, DOI 10.1016/j.jalgebra.2009.01.036
- Osamu Iyama and Steffen Oppermann. Stable categories of higher preprojective algebras, 2009. preprint, arXiv:0912.3412.
- Osamu Iyama, Cluster tilting for higher Auslander algebras, Adv. Math. 226 (2011), no. 1, 1–61. MR 2735750, DOI 10.1016/j.aim.2010.03.004
- Osamu Iyama, Auslander correspondence, Adv. Math. 210 (2007), no. 1, 51–82. MR 2298820, DOI 10.1016/j.aim.2006.06.003
- Osamu Iyama, Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories, Adv. Math. 210 (2007), no. 1, 22–50. MR 2298819, DOI 10.1016/j.aim.2006.06.002
- Osamu Iyama, Auslander-Reiten theory revisited, Trends in representation theory of algebras and related topics, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2008, pp. 349–397. MR 2484730, DOI 10.4171/062-1/8
- Bernhard Keller. Deformed Calabi-Yau completions. to appear in J. Reine Angew. Math., arXiv:0908.3499.
- Bernhard Keller, Derived categories and tilting, Handbook of tilting theory, London Math. Soc. Lecture Note Ser., vol. 332, Cambridge Univ. Press, Cambridge, 2007, pp. 49–104. MR 2384608, DOI 10.1017/CBO9780511735134.005
- Bernhard Keller, Calabi-Yau triangulated categories, Trends in representation theory of algebras and related topics, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2008, pp. 467–489. MR 2484733, DOI 10.4171/062-1/11
- Jeremy Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), no. 3, 436–456. MR 1002456, DOI 10.1112/jlms/s2-39.3.436
- Christine Riedtmann and Aidan Schofield, On a simplicial complex associated with tilting modules, Comment. Math. Helv. 66 (1991), no. 1, 70–78. MR 1090165, DOI 10.1007/BF02566636
Additional Information
- Osamu Iyama
- Affiliation: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602 Japan
- MR Author ID: 634748
- Email: iyama@math.nagoya-u.ac.jp
- Steffen Oppermann
- Affiliation: Institutt for Matematiske fag, Norwegian University of Science and Technology, 7491 Trondheim, Norway
- MR Author ID: 810235
- Email: steffen.oppermann@math.ntnu.no
- Received by editor(s): September 3, 2009
- Received by editor(s) in revised form: January 28, 2010
- Published electronically: July 11, 2011
- Additional Notes: The first author was supported by JSPS Grant-in-Aid for Scientific Research 21740010
The second author was supported by NFR Storforsk grant no. 167130. - © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 6575-6614
- MSC (2010): Primary 16G10, 16E35
- DOI: https://doi.org/10.1090/S0002-9947-2011-05312-2
- MathSciNet review: 2833569