Top

Published in:

2018 | OriginalPaper | Chapter

Auslander–Reiten Theory for Finite-Dimensional Algebras

Author : Piotr Malicki

Published in: Homological Methods, Representation Theory, and Cluster Algebras

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

This article is based on a course given at the CIMPA School “Homological Methods, Representation Theory and Cluster Algebras,” held in March 2016 in Mar del Plata. The aim of the course, consisting of four lectures, was to provide a brief introduction to the notion of an almost split sequence and its use in the representation theory of finite-dimensional algebras. The first two sections are reduced, and the next three sections are extended in comparison with the above-mentioned course.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

previous chapter Introduction to the Representation Theory of Finite-Dimensional Algebras: The Functorial Approach

next chapter Cluster Algebras from Surfaces

Assem, I., Coelho, F.U.: Two-sided gluings of tilted algebras. J. Algebra 269(2), 456–479 (2003). DOI https://doi.org/10.1016/S0021-8693(03)00436-8

Assem, I., Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras. Vol. 1. Techniques of Representation Theory, London Math. Soc. Stud. Texts, vol. 65. Cambridge Univ. Press, Cambridge (2006). DOI https://doi.org/10.1017/CBO9780511614309

Assem, I., Skowroński, A.: On some classes of simply connected algebras. Proc. London Math. Soc. (3) 56(3), 417–450 (1988). DOI https://doi.org/10.1112/plms/s3-56.3.417

Assem, I., Skowroński, A.: Minimal representation-infinite coil algebras. Manuscripta Math. 67(3), 305–331 (1990). DOI https://doi.org/10.1007/BF02568435MathSciNetCrossRef

Assem, I., Skowroński, A.: Indecomposable modules over multicoil algebras. Math. Scand. 71(1), 31–61 (1992). DOI https://doi.org/10.7146/math.scand.a-12409MathSciNetCrossRef

Assem, I., Skowroński, A.: Multicoil algebras. In: V. Dlab, H. Lenzing (eds.) Representations of Algebras (Ottawa, ON, 1992), CMS Conf. Proc., vol. 14, pp. 29–68. Amer. Math. Soc., Providence, RI (1993)

Auslander, M.: Representation theory of Artin algebras. II. Comm. Algebra 1(4), 269–310 (1974). DOI https://doi.org/10.1080/00927877409412807

Auslander, M., Bautista, R., Platzeck, M.I., Reiten, I., Smalø, S.O.: Almost split sequences whose middle term has at most two indecomposable summands. Canad. J. Math. 31(5), 942–960 (1979). DOI https://doi.org/10.4153/CJM-1979-089-5MathSciNetCrossRef

Auslander, M., Bridger, M.: Stable module theory. Mem. Amer. Math. Soc. (94) (1969)

10.

Auslander, M., Reiten, I.: Representation theory of Artin algebras. III. Almost split sequences. Comm. Algebra 3(3), 239–294 (1975). DOI https://doi.org/10.1080/00927877508822046MathSciNetCrossRef

11.

Auslander, M., Reiten, I.: Representation theory of Artin algebras. IV. Invariants given by almost split sequences. Comm. Algebra 5(5), 443–518 (1977). DOI https://doi.org/10.1080/00927877708822180MathSciNetCrossRef

12.

Auslander, M., Reiten, I.: Representation theory of Artin algebras. V. Methods for computing almost split sequences and irreducible morphisms. Comm. Algebra 5(5), 519–554 (1977). DOI https://doi.org/10.1080/00927877708822181MathSciNetCrossRef

13.

Auslander, M., Reiten, I.: Representation theory of Artin algebras. VI. A functorial approach to almost split sequences. Comm. Algebra 6(3), 257–300 (1978). DOI https://doi.org/10.1080/00927877808822246

14.

Auslander, M., Reiten, I.: Uniserial functors. In: Dlab and Gabriel [31], pp. 1–47. DOI https://doi.org/10.1007/BFb0088457

15.

Auslander, M., Reiten, I., Smalø, S.O.: Representation Theory of Artin Algebras, Cambridge Stud. Adv. Math., vol. 36. Cambridge Univ. Press, Cambridge (1995). DOI https://doi.org/10.1017/CBO9780511623608

16.

Barot, M.: Introduction to the Representation Theory of Algebras. Springer, Cham (2015). DOI https://doi.org/10.1007/978-3-319-11475-0

17.

Bautista, R.: Irreducible morphisms and the radical of a category. An. Inst. Mat. Univ. Nac. Autónoma México 22, 83–135 (1982)

18.

Bautista, R., Brenner, S.: On the number of terms in the middle of an almost split sequence. In: M. Auslander, E. Lluis (eds.) Representations of Algebras (Puebla, 1980), Lecture Notes in Math., vol. 903, pp. 1–8. Springer, Berlin (1981). DOI https://doi.org/10.1007/BFb0092980

19.

Bautista, R., Gabriel, P., Roĭter, A.V., Salmerón, L.: Representation-finite algebras and multiplicative bases. Invent. Math. 81(2), 217–285 (1985). DOI https://doi.org/10.1007/BF01389052

20.

Bernsten, I.N., Gel’fand, I.M., Ponomarev, V.A.: Coxeter functors, and Gabriel’s theorem (Russian). Uspehi Mat. Nauk 28(2(170)), 19–33 (1973). English transl., Russian Math. Surveys 28(2), 17–32 (1973)

21.

Bongartz, K.: A criterion for finite representation type. Math. Ann. 269(1), 1–12 (1984). DOI https://doi.org/10.1007/BF01455993

22.

Bongartz, K.: Indecomposables are standard. Comment. Math. Helv. 60(3), 400–410 (1985). DOI https://doi.org/10.1007/BF02567423

23.

Bongartz, K., Gabriel, P.: Covering spaces in representation-theory. Invent. Math. 65(3), 331–378 (1981/82). DOI https://doi.org/10.1007/BF01396624

24.

Brenner, S., Butler, M.C.R.: Wild subquivers of the Auslander–Reiten quiver of a tame algebra. In: E.L. Green, B. Huisgen-Zimmermann (eds.) Trends in the Representation Theory of Finite-Dimensional Algebras (Seattle, WA, 1997), Contemp. Math., vol. 229, pp. 29–48. Amer. Math. Soc., Providence, RI (1998). DOI https://doi.org/10.1090/conm/229/03309

25.

Bretscher, O., Gabriel, P.: The standard form of a representation-finite algebra. Bull. Soc. Math. France 111(1), 21–40 (1983)MathSciNetCrossRef

26.

Butler, M.C.R., Ringel, C.M.: Auslander–Reiten sequences with few middle terms and applications to string algebras. Comm. Algebra 15(1-2), 145–179 (1987). DOI https://doi.org/10.1080/00927878708823416MathSciNetCrossRef

27.

Coelho, F.U., Lanzilotta, M.A.: Algebras with small homological dimensions. Manuscripta Math. 100(1), 1–11 (1999). DOI https://doi.org/10.1007/s002290050191

28.

Coelho, F.U., Marcos, E.d.N., Merklen, H.A., Skowroński, A.: Module categories with infinite radical square zero are of finite type. Comm. Algebra 22(11), 4511–4517 (1994). DOI https://doi.org/10.1080/00927879408825084

29.

Crawley-Boevey, W.: Regular modules for tame hereditary algebras. Proc. London Math. Soc. (3) 62(3), 490–508 (1991). DOI https://doi.org/10.1112/plms/s3-62.3.490MathSciNetCrossRef

30.

Crawley-Boevey, W.: Modules of finite length over their endomorphism rings. In: H. Tachikawa, S. Brenner (eds.) Representations of Algebras and Related Topics (Kyoto, 1990), London Math. Soc. Lecture Note Ser., vol. 168, pp. 127–184. Cambridge Univ. Press, Cambridge (1992)

31.

Dlab, V., Gabriel, P. (eds.): Representation Theory. II (Ottawa, ON, 1979), Lecture Notes in Math., vol. 832. Springer, Berlin (1980)

32.

Dlab, V., Ringel, C.M.: On algebras of finite representation type. J. Algebra 33, 306–394 (1975). DOI https://doi.org/10.1016/0021-8693(75)90125-8MathSciNetCrossRef

33.

Dlab, V., Ringel, C.M.: Indecomposable representations of graphs and algebras. Mem. Amer. Math. Soc. 6(173), v+57 (1976). DOI https://doi.org/10.1090/memo/0173MathSciNetCrossRef

34.

Dlab, V., Ringel, C.M.: The representations of tame hereditary algebras. In: Gordon [38], pp. 329–353

35.

Donovan, P., Freislich, M.R.: The Representation Theory of Finite Graphs and Associated Algebras, Carleton Math. Lecture Notes, vol. 5. Carleton Univ., Ottawa, ON (1973)

36.

Drozd, Ju.A.: Tame and wild matrix problems. In: Dlab and Gabriel [31], pp. 242–258. DOI https://doi.org/10.1007/BFb0088467CrossRef

37.

Gabriel, P.: Unzerlegbare Darstellungen. I. Manuscripta Math. 6, 71–103 (1972). DOI https://doi.org/10.1007/BF01298413

38.

Gordon, R. (ed.): Representation Theory of Algebras (Philadelphia, PA, 1976), Lecture Notes in Pure Appl. Math., vol. 37. Dekker, New York (1978)

39.

Happel, D., Preiser, U., Ringel, C.M.: Vinberg’s characterization of Dynkin diagrams using subadditive functions with application to \(D\) Tr-periodic modules. In: Dlab and Gabriel [31], pp. 280–294

40.

Happel, D., Ringel, C.M.: Tilted algebras. Trans. Amer. Math. Soc. 274(2), 399–443 (1982). DOI https://doi.org/10.2307/1999116MathSciNetCrossRefMATH

41.

Kerner, O.: Tilting wild algebras. J. London Math. Soc. (2) 39(1), 29–47 (1989). DOI https://doi.org/10.1112/jlms/s2-39.1.29MathSciNetCrossRef

42.

Lenzing, H., Skowroński, A.: Quasi-tilted algebras of canonical type. Colloq. Math. 71(2), 161–181 (1996). DOI https://doi.org/10.4064/cm-71-2-161-181MathSciNetCrossRefMATH

43.

Liu, S.P.: Degrees of irreducible maps and the shapes of Auslander–Reiten quivers. J. London Math. Soc. (2) 45(1), 32–54 (1992). DOI https://doi.org/10.1112/jlms/s2-45.1.32MathSciNetCrossRef

44.

Liu, S.P.: Almost split sequences for nonregular modules. Fund. Math. 143(2), 183–190 (1993). DOI https://doi.org/10.4064/fm-143-2-183-190MathSciNetCrossRefMATH

45.

Liu, S.P.: Semi-stable components of an Auslander–Reiten quiver. J. London Math. Soc. (2) 47(3), 405–416 (1993). DOI https://doi.org/10.1112/jlms/s2-47.3.405MathSciNetCrossRef

46.

Malicki, P., de la Peña, J.A., Skowroński, A.: On the number of terms in the middle of almost split sequences over cycle-finite Artin algebras. Cent. Eur. J. Math. 12(1), 39–45 (2014). DOI https://doi.org/10.2478/s11533-013-0328-3MathSciNetCrossRefMATH

47.

Malicki, P., Skowroński, A.: Algebras with separating almost cyclic coherent Auslander–Reiten components. J. Algebra 291(1), 208–237 (2005). DOI https://doi.org/10.1016/j.jalgebra.2005.03.021MathSciNetCrossRefMATH

48.

Nazarova, L.A.: Representations of quivers of infinite type (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 37, 752–791 (1973). English transl.,Math. USSR-Izv. 7(4) 749–792 (1973)CrossRef

49.

de la Peña, J.A., Skowroński, A.: Algebras with cycle-finite Galois coverings. Trans. Amer. Math. Soc. 363(8), 4309–4336 (2011). DOI https://doi.org/10.1090/S0002-9947-2011-05256-6MathSciNetCrossRefMATH

50.

de la Peña, J.A., Takane, M.: On the number of terms in the middle of almost split sequences over tame algebras. Trans. Amer. Math. Soc. 351(9), 3857–3868 (1999). DOI https://doi.org/10.1090/S0002-9947-99-02137-6MathSciNetCrossRefMATH

51.

de la Peña, J.A., Tomé, B.: Iterated tubular algebras. J. Pure Appl. Algebra 64(3), 303–314 (1990). DOI https://doi.org/10.1016/0022-4049(90)90064-OMathSciNetCrossRefMATH

52.

Platzeck, M.I., Auslander, M.: Representation theory of hereditary Artin algebras. In: Gordon [38], pp. 389–424

53.

Pogorzały, Z., Skowroński, A.: On algebras whose indecomposable modules are multiplicity-free. Proc. London Math. Soc. (3) 47(3), 463–479 (1983). DOI https://doi.org/10.1112/plms/s3-47.3.463MathSciNetCrossRef

54.

Reiten, I., Skowroński, A.: Characterizations of algebras with small homological dimensions. Adv. Math. 179(1), 122–154 (2003). DOI https://doi.org/10.1016/S0001-8708(02)00029-4MathSciNetCrossRefMATH

55.

Reiten, I., Skowroński, A.: Generalized double tilted algebras. J. Math. Soc. Japan 56(1), 269–288 (2004). DOI https://doi.org/10.2969/jmsj/1191418706MathSciNetCrossRefMATH

56.

Ringel, C.M.: Representations of \(k\)-species and bimodules. J. Algebra 41(2), 269–302 (1976). DOI https://doi.org/10.1016/0021-8693(76)90184-8MathSciNetCrossRefMATH

57.

Ringel, C.M.: Finite dimensional hereditary algebras of wild representation type. Math. Z. 161(3), 235–255 (1978). DOI https://doi.org/10.1007/BF01214506MathSciNetCrossRefMATH

58.

Ringel, C.M.: The spectrum of a finite-dimensional algebra. In: F. van Oystaeyen (ed.) Ring Theory (Antewrp, 1978), Lecture Notes in Pure and Appl. Math., vol. 51, pp. 535–597. Dekker, New York (1979)

59.

Ringel, C.M.: Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math., vol. 1099. Springer, Berlin (1984). DOI https://doi.org/10.1007/BFb0072870

60.

Roiter, A.V.: Unboundedness of the dimensions of the indecomposable representations of an algebra which has infinitely many indecomposable representations (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 32, 1275–1282 (1968)MathSciNet

61.

Schiffler, R.: Quiver representations. CMS Books Math./Ouvrages Math. SMC. Springer, Cham (2014). DOI https://doi.org/10.1007/978-3-319-09204-1MATH

62.

Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras. Vol. 2. Tubes and Concealed Algebras of Euclidean Type, London Math. Soc. Stud. Texts, vol. 71. Cambridge Univ. Press, Cambridge (2007). DOI https://doi.org/10.1017/CBO9780511619212

63.

Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras. Vol. 3. Representation-Infinite Tilted Algebras, London Math. Soc. Stud. Texts, vol. 72. Cambridge Univ. Press, Cambridge (2007). DOI https://doi.org/10.1017/CBO9780511619403

64.

Skowroński, A.: Generalized standard Auslander–Reiten components. J. Math. Soc. Japan 46(3), 517–543 (1994). DOI https://doi.org/10.2969/jmsj/04630517MathSciNetCrossRefMATH

65.

Skowroński, A.: Cycle-finite algebras. J. Pure Appl. Algebra 103(1), 105–116 (1995). DOI https://doi.org/10.1016/0022-4049(94)00094-YMathSciNetCrossRefMATH

66.

Skowroński, A.: Simply connected algebras of polynomial growth. Compositio Math. 109(1), 99–133 (1997). DOI https://doi.org/10.1023/A:1000245728528MathSciNetCrossRefMATH

67.

Skowroński, A.: Tame quasi-tilted algebras. J. Algebra 203(2), 470–490 (1998). DOI https://doi.org/10.1006/jabr.1997.7328MathSciNetCrossRefMATH

68.

Skowroński, A., Smalø, S.O., Zacharia, D.: On the finiteness of the global dimension for Artinian rings. J. Algebra 251(1), 475–478 (2002). DOI https://doi.org/10.1006/jabr.2001.9130MathSciNetCrossRefMATH

69.

Skowroński, A., Waschbüsch, J.: Representation-finite biserial algebras. J. Reine Angew. Math. 345, 172–181 (1983). DOI https://doi.org/10.1515/crll.1983.345.172MathSciNetCrossRefMATH

70.

Skowroński, A., Yamagata, K.: Frobenius Algebras. I. Basic Representation Theory. EMS Textbk. Math. Eur. Math. Soc., Zürich (2011). DOI https://doi.org/10.4171/102

71.

Wald, B., Waschbüsch, J.: Tame biserial algebras. J. Algebra 95(2), 480–500 (1985). DOI https://doi.org/10.1016/0021-8693(85)90119-XMathSciNetCrossRefMATH

72.

Zhang, Y.B.: The structure of stable components. Canad. J. Math. 43(3), 652–672 (1991). DOI https://doi.org/10.4153/CJM-1991-038-1MathSciNetCrossRefMATH

Title: Auslander–Reiten Theory for Finite-Dimensional Algebras
Author: Piotr Malicki
Publisher: Springer International Publishing
Book: Homological Methods, Representation Theory, and Cluster Algebras
Print ISBN: 978-3-319-74584-8

Electronic ISBN: 978-3-319-74585-5

Copyright Year: 2018
DOI: https://doi.org/10.1007/978-3-319-74585-5_2

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Premium Partner