Top

Published in:

2018 | OriginalPaper | Chapter

Brauer Graph Algebras

A Survey on Brauer Graph Algebras, Associated Gentle Algebras and Their Connections to Cluster Theory

Author : Sibylle Schroll

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 survey starts with a motivation of the study of Brauer graph algebras by relating them to special biserial algebras. The definition of Brauer graph algebras is given in great detail with many examples to illustrate the concepts. An interpretation of Brauer graphs as decorated ribbon graphs is included. A section on gentle algebras and their associated ribbon graph, trivial extensions of gentle algebras, admissible cuts of Brauer graph algebras and a first connection of Brauer graph algebras with Jacobian algebras associated to triangulations of marked oriented surfaces follows. The interpretation of flips of diagonals in triangulations of marked oriented surfaces as derived equivalences of Brauer graph algebras and the comparison of derived equivalences of Brauer graph algebras with derived equivalences of frozen Jacobian algebras is the topic of the next section. In the last section, after defining Green’s walk around the Brauer graph, a complete description of the Auslander Reiten quiver of a Brauer graph algebra is given.

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 A Course on Cluster Tilted Algebras

Available only for authorised users

http://www.iaz.uni-stuttgart.de/LstAGeoAlg/Kuelshammer/topics/biserial.html

http://www.math.uni-bonn.de/people/schroer/fd-atlas.html

Adachi, T., Aihara, T., Chan, A.: Tilting Brauer graph algebras. I. Classification of two-term tilting complexes. arXiv:1504.04827

Aihara, T.: Derived equivalences between symmetric special biserial algebras. J. Pure Appl. Algebra 219(5), 1800–1825 (2015). DOI https://doi.org/10.1016/j.jpaa.2014.07.012MathSciNetCrossRef

Aihara, T., Iyama, O.: Silting mutation in triangulated categories. J. Lond. Math. Soc. (2) 85(3), 633–668 (2012). DOI https://doi.org/10.1112/jlms/jdr055MathSciNetCrossRef

Amiot, C.: On generalized cluster categories. In: A. Skowroński, K. Yamagata (eds.) Representations of Algebras and Related Topics, EMS Ser. Congr. Rep., pp. 1–53. Eur. Math. Soc., Zürich (2011). DOI https://doi.org/10.4171/101-1/1

Antipov, M.A.: Derived equivalence of symmetric special biserial algebras (Russian). Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 343(Vopr. Teor. Predts. Algebr. i Grupp. 15), 5–32, 272 (2007). DOI https://doi.org/10.1007/s10958-007-0524-4. English transl., J. Math. Sci. (N.Y.) 147(5), 6981–6994 (2007)MathSciNetCrossRef

Antipov, M.A., Zvonareva, A.: On stably biserial algebras. arXiv:1711.05021

Assem, I., Brüstle, T., Charbonneau-Jodoin, G., Plamondon, P.-G.: Gentle algebras arising from surface triangulations. Algebra Number Theory 4(2), 201–229 (2010). DOI https://doi.org/10.2140/ant.2010.4.201MathSciNetCrossRef

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

Auslander, M., Reiten, I., Smalø, S.O.: Representation Theory of Artin Algebras, Cambridge Stud. Adv. Math., vol. 36. Cambridge Univ. Press, Cambridge (1997)

10.

Avella-Alaminos, D., Geiss, C.: Combinatorial derived invariants for gentle algebras. J. Pure Appl. Algebra 212(1), 228–243 (2008). DOI https://doi.org/10.1016/j.jpaa.2007.05.014MathSciNetCrossRef

11.

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

12.

Benson, D.J.: Representations and Cohomology. I. Basic Representation Theory of Finite Groups and Associative Algebras, Cambridge Stud. Adv. Math., vol. 30, 2nd edn. Cambridge Univ. Press, Cambridge (1998)

13.

Benson, D.J.: Resolutions over symmetric algebras with radical cube zero. J. Algebra 320(1), 48–56 (2008). DOI https://doi.org/10.1016/j.jalgebra.2008.02.033

14.

Bocian, R., Skowroński, A.: Symmetric special biserial algebras of Euclidean type. Colloq. Math. 96(1), 121–148 (2003). DOI https://doi.org/10.4064/cm96-1-11MathSciNetCrossRef

15.

Buan, A.B., Iyama, O., Reiten, I., Smith, D.: Mutation of cluster-tilting objects and potentials. Amer. J. Math. 133(4), 835–887 (2011). DOI https://doi.org/10.1353/ajm.2011.0031MathSciNetCrossRef

16.

Buan, A.B., Vatne, D.F.: Derived equivalence classification for cluster-tilted algebra of type \({{\rm A}_{n}}\). J. Algebra 319(7), 2723–2738 (2008). DOI https://doi.org/10.1016/j.jalgebra.2008.01.007

17.

Burman, Yu.M.: Triangulations of surfaces with boundary and the homotopy principle for functions without critical points. Ann. Global Anal. Geom. 17(3), 221–238 (1999). DOI https://doi.org/10.1023/A:1006556632099

18.

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/10.1080/00927878708823416

19.

Cline, E., Parshall, B., Scott, L.: Derived categories and Morita theory. J. Algebra 104(2), 397–409 (1986). DOI https://doi.org/10.1016/0021-8693(86)90224-3MathSciNetCrossRef

20.

Coelho Simões, R., Parsons, M.J.: Endomorphism algebras for a class of negative Calabi–Yau categories. J. Algebra 491, 32–57 (2017). DOI https://doi.org/10.1016/j.jalgebra.2017.07.016

21.

Craven, D.A.: The Brauer trees of non-crystallographic groups of Lie type. J. Algebra 398, 481–495 (2014). DOI https://doi.org/10.1016/j.jalgebra.2013.06.002MathSciNetCrossRef

22.

Crawley-Boevey, W.W.: Maps between representations of zero-relation algebras. J. Algebra 126(2), 259–263 (1989). DOI https://doi.org/10.1016/0021-8693(89)90304-9MathSciNetCrossRef

23.

Dade, E.C.: Blocks with cyclic defect groups. Ann. of Math. (2) 84, 20–48 (1966). DOI https://doi.org/10.2307/1970529MathSciNetCrossRef

24.

David-Roesler, L., Schiffler, R.: Algebras from surfaces without punctures. J. Algebra 350, 218–244 (2012). DOI https://doi.org/10.1016/j.jalgebra.2011.10.034MathSciNetCrossRef

25.

Demonet, L.: Algebras of partial triangulations. arXiv:1602.01592

26.

Demonet, L., Luo, X.: Ice quivers with potential associated with triangulations and Cohen–Macaulay modules over orders. Trans. Amer. Math. Soc. 368(6), 4257–4293 (2016). DOI https://doi.org/10.1090/tran/6463

27.

Dlab, V., Gabriel, P. (eds.): Representation Theory. I (Ottawa, ON, 1979), Lecture Notes in Math., vol. 831. Springer, Berlin (1980)

28.

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

29.

Donovan, P.W.: Dihedral defect groups. J. Algebra 56(1), 184–206 (1979). DOI https://doi.org/10.1016/0021-8693(79)90332-6

30.

Donovan, P.W., Freislich, M.R.: The indecomposable modular representations of certain groups with dihedral Sylow subgroup. Math. Ann. 238(3), 207–216 (1978). DOI https://doi.org/10.1007/BF01420248MathSciNetCrossRef

31.

Drozd, Ju.A.: Tame and wild matrix problems. In: Dlab and Gabriel [28], pp. 242–258. DOI https://doi.org/10.1007/BFb0088467CrossRef

32.

Dudas, O.: Coxeter orbits and Brauer trees. Adv. Math. 229(6), 3398–3435 (2012). DOI https://doi.org/10.1016/j.aim.2012.02.011MathSciNetCrossRef

33.

Dudas, O.: Coxeter orbits and Brauer trees. II. Int. Math. Res. Not. IMRN 2014(15), 4100–4123 (2014). DOI https://doi.org/10.1093/imrn/rnt070

34.

Dudas, O., Rouquier, R.: Coxeter orbits and Brauer trees. III. J. Amer. Math. Soc. 27(4), 1117–1145 (2014). DOI https://doi.org/10.1090/S0894-0347-2014-00791-8MathSciNetCrossRef

35.

Duffield, D.: Auslander–Reiten components of symmetric special biserial algebras. arXiv:1509.02478

36.

Dugas, A.: Tilting mutation of weakly symmetric algebras and stable equivalence. Algebr. Represent. Theory 17(3), 863–884 (2014). DOI https://doi.org/10.1007/s10468-013-9422-2MathSciNetCrossRef

37.

Erdmann, K.: Algebras with non-periodic bounded modules. J. Algebra 475, 308–326 (2017). DOI https://doi.org/10.1016/j.jalgebra.2016.05.004

38.

Erdmann, K., Skowroński, A.: On Auslander-Reiten components of blocks and self-injective biserial algebras. Trans. Amer. Math. Soc. 330(1), 165–189 (1992). DOI https://doi.org/10.2307/2154159MathSciNetCrossRef

39.

Fernández, E.: Extensiones triviales y álgebras inclinadas iteradas. Ph.D. Thesis, Universidad Nacional del Sur (1999)

40.

Fomin, S., Shapiro, M., Thurston, D.: Cluster algebras and triangulated surfaces. I. Cluster complexes. Acta Math. 201(1), 83–146 (2008). DOI https://doi.org/10.1007/s11511-008-0030-7

41.

Gabriel, P.: Auslander–Reiten sequences and representation-finite algebras. In: Dlab and Gabriel [27], pp. 1–71. DOI https://doi.org/10.1007/BFb0089778

42.

Gel\(^\prime \)fand, I.M., Ponomarev, V.A.: Indecomposable representations of the Lorentz group (Russian). Uspehi Mat. Nauk 23(2(140)), 3–60 (1968)

43.

Green, E.L., Schroll, S.: Almost gentle algebras and their trivial extensions. arXiv: 1603.03587

44.

Green, E.L., Schroll, S.: Multiserial and special multiserial algebras and their representations. Adv. Math. 302, 1111–1136 (2016). DOI https://doi.org/10.1016/j.aim.2016.07.006MathSciNetCrossRef

45.

Green, E.L., Schroll, S.: Brauer configuration algebras: A generalization of Brauer graph algebras. Bull. Sci. Math. 141(6), 539–572 (2017). DOI https://doi.org/10.1016/j.bulsci.2017.06.001MathSciNetCrossRef

46.

Green, E.L., Schroll, S.: Special multiserial algebras are quotients of symmetric special multiserial algebras. J. Algebra 473, 397–405 (2017). DOI https://doi.org/10.1016/j.jalgebra.2016.10.033MathSciNetCrossRef

47.

Green, E.L., Schroll, S., Snashall, N.: Group actions and coverings of Brauer graph algebras. Glasg. Math. J. 56(2), 439–464 (2014). DOI https://doi.org/10.1017/S0017089513000372

48.

Green, E.L., Schroll, S., Snashall, N., Taillefer, R.: The Ext algebra of a Brauer graph algebra. J. Noncommut. Geom. 11(2), 537–579 (2017). DOI https://doi.org/10.4171/JNCG/11-2-4MathSciNetCrossRef

49.

Green, J.A.: Walking around the Brauer Tree. J. Austral. Math. Soc. 17, 197–213 (1974)MathSciNetCrossRef

50.

Happel, D.: Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras, London Math. Soc. Lecture Note Ser., vol. 119. Cambridge Univ. Press, Cambridge (1988). DOI https://doi.org/10.1017/CBO9780511629228

51.

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 [28], pp. 280–294. DOI https://doi.org/10.1007/BFb0088469

52.

von Höhne, H.-J., Waschbüsch, J.: Die struktur \(n\)-reihiger Algebren. Comm. Algebra 12(9-10), 1187–1206 (1984). DOI https://doi.org/10.1080/00927878408823049

53.

Janusz, G.J.: Indecomposable modules for finite groups. Ann. of Math. (2) 89, 209–241 (1969). DOI https://doi.org/10.2307/1970666MathSciNetCrossRef

54.

Kalck, M.: Derived categories of quasi-hereditary algebras and their derived composition series. In: Krause et al. [59], pp. 269–308

55.

Kauer, M.: Derived equivalence of graph algebras. 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. 201–213. Amer. Math. Soc., Providence, RI (1998). DOI https://doi.org/10.1090/conm/229/03319

56.

Keller, B., Yang, D.: Derived equivalences from mutations of quivers with potential. Adv. Math. 226(3), 2118–2168 (2011). DOI https://doi.org/10.1016/j.aim.2010.09.019

57.

König, S., Zimmermann, A.: Derived Equivalences for Group Rings, Lecture Notes in Math., vol. 1685. Springer, Berlin (1998)CrossRef

58.

Krause, H.: Maps between tree and band modules. J. Algebra 137(1), 186–194 (1991). DOI https://doi.org/10.1016/0021-8693(91)90088-PMathSciNetCrossRef

59.

Krause, H., Littelmann, P., Malle, G., Neeb, K.-H., Schweigert, C. (eds.): Representation Theory—Current Trends and Perspectives. EMS Ser. Congr. Rep. Eur. Math. Soc., Zürich (2017)

60.

Labardini-Fragoso, D.: Quivers with potentials associated to triangulated surfaces. Proc. Lond. Math. Soc. (3) 98(3), 797–839 (2009). DOI https://doi.org/10.1112/plms/pdn051MathSciNetCrossRef

61.

Labourie, F.: Lectures on Representations of Surface Groups. Zur. Lect. Adv. Math. Eur. Math. Soc., Zürich (2013). DOI https://doi.org/10.4171/127

62.

Ladkani, S.: Derived equivalence classification of the gentle algebras arising from surface triangulations. Preprint

63.

Ladkani, S.: Mutation classes of certain quivers with potentials as derived equivalence classes. arXiv:1102.4108

64.

Ladkani, S.: From groups to clusters. In: Krause et al. [59], pp. 427–500

65.

Linckelmann, M.: On stable equivalences of Morita type. In: Derived equivalences for group rings [57], pp. 221–232. DOI https://doi.org/10.1007/BFb0096377

66.

Marsh, R.J., Schroll, S.: The geometry of Brauer graph algebras and cluster mutations. J. Algebra 419, 141–166 (2014). DOI https://doi.org/10.1016/j.jalgebra.2014.08.002MathSciNetCrossRef

67.

Nakayama, T.: Note on uni-serial and generalized uni-serial rings. Proc. Imp. Acad. Tokyo 16, 285–289 (1940)MathSciNetCrossRef

68.

Okuyama, T.: Some examples of derived equivalent blocks of finite groups. Unpublished

69.

Palu, Y.: Grothendieck group and generalized mutation rule for \(2\)-Calabi–Yau triangulated categories. J. Pure Appl. Algebra 213(7), 1438–1449 (2009). DOI https://doi.org/10.1016/j.jpaa.2008.12.012

70.

Pogorzały, Z., Skowroński, A.: Self-injective biserial standard algebras. J. Algebra 138(2), 491–504 (1991). DOI https://doi.org/10.1016/0021-8693(91)90183-9MathSciNetCrossRef

71.

Rickard, J.: A derived equivalence for the principal 2-blocks of \({A}_4\) and \({A}_5\). Manuscript

72.

Rickard, J.: Derived categories and stable equivalence. J. Pure Appl. Algebra 61(3), 303–317 (1989). DOI https://doi.org/10.1016/0022-4049(89)90081-9MathSciNetCrossRef

73.

Rickard, J.: Morita theory for derived categories. J. London Math. Soc. (2) 39(3), 436–456 (1989). DOI https://doi.org/10.1112/jlms/s2-39.3.436MathSciNetCrossRef

74.

Rickard, J.: Derived equivalences as derived functors. J. London Math. Soc. (2) 43(1), 37–48 (1991). DOI https://doi.org/10.1112/jlms/s2-43.1.37MathSciNetCrossRef

75.

Rickard, J.: Triangulated categories in the modular representation theory of finite groups. In: Derived Equivalences for Group Rings [57], pp. 177–198. DOI https://doi.org/10.1007/BFb0096375

76.

Riedtmann, C.: Algebren, Darstellungsköcher, überlagerungen und zurück. Comment. Math. Helv. 55(2), 199–224 (1980). DOI https://doi.org/10.1007/BF02566682MathSciNetCrossRef

77.

Ringel, C.M.: The indecomposable representations of the dihedral \(2\)-groups. Math. Ann. 214, 19–34 (1975). DOI https://doi.org/10.1007/BF01428252MathSciNetCrossRef

78.

Ringel, C.M.: On algorithms for solving vector space problems. II. Tame algebras. In: Dlab and Gabriel [27], pp. 137–287

79.

Roggenkamp, K.W.: Biserial algebras and graphs. In: I. Reiten, S.O. Smalø, Ø. Solberg (eds.) Algebras and Modules. II (Geiranger, 1996), CMS Conf. Proc., vol. 24, pp. 481–496. Amer. Math. Soc., Providence, RI (1998)

80.

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

81.

Schröer, J., Zimmermann, A.: Stable endomorphism algebras of modules over special biserial algebras. Math. Z. 244(3), 515–530 (2003). DOI https://doi.org/10.1007/s00209-003-0492-4MathSciNetCrossRef

82.

Schroll, S.: Trivial extensions of gentle algebras and Brauer graph algebras. J. Algebra 444, 183–200 (2015). DOI https://doi.org/10.1016/j.jalgebra.2015.07.037MathSciNetCrossRef

83.

Skowroński, A.: Group algebras of polynomial growth. Manuscripta Math. 59(4), 499–516 (1987). DOI https://doi.org/10.1007/BF01170851MathSciNetCrossRef

84.

Skowroński, A.: Algebras of polynomial growth. In: S. Balcerzyk, T. Józefiak, J. Krempa, D. Simson, W. Vogel (eds.) Topics in Algebra. Part 1. Rings and Representations of Algebras (Warsaw, 1988), Banach Center Publ., vol. 26, pp. 535–568. PWN, Warsaw (1990)

85.

Skowroński, A.: Selfinjective algebras: finite and tame type. In: J.A. de la Peña, R. Bautista (eds.) Trends in Representation Theory of Algebras and Related Topics (Querétaro, 2004), Contemp. Math., vol. 406, pp. 169–238. Amer. Math. Soc., Providence, RI (2006). DOI https://doi.org/10.1090/conm/406/07658

86.

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

87.

Vossieck, D.: The algebras with discrete derived category. J. Algebra 243(1), 168–176 (2001). DOI https://doi.org/10.1006/jabr.2001.8783MathSciNetCrossRef

88.

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

89.

Weibel, C.A.: An Introduction to Homological Algebra, Cambridge Stud. Adv. Math., vol. 38. Cambridge Univ. Press, Cambridge (1994). DOI https://doi.org/10.1017/CBO9781139644136

90.

Zvonareva, A.: Two-term tilting complexes over algebras that correspond to Brauer trees (Russian). Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 423(Vopr. Teor. Predts. Algebr. i Grupp. 26), 132–165 (2014). English transl., J. Math. Sci. (N.Y.) 209(4), 568–587 (2015)

91.

Zvonareva, A.: Mutations and the derived Picard group of the Brauer star algebra. J. Algebra 443, 270–299 (2015). DOI https://doi.org/10.1016/j.jalgebra.2015.06.038MathSciNetCrossRef

Title: Brauer Graph Algebras
Author: Sibylle Schroll
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_6

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"