Recent Advances in the Representation Theory of Finite Dimensional Algebras

  • Claus Michael Ringel
Part of the Progress in Mathematics book series (PM, volume 95)


This is a report on advances in the representation theory of finite dimensional algebras in the years 1984 – 1990. During these years, the German research council (DFG) has sponsered a Forschungsschwerpunkt devoted to the representation theory of finite groups and finite dimensional algebras; it started in 1984 and will be finished by 1991.


Endomorphism Ring Indecomposable Module Finite Dimensional Algebra Hall Algebra Hereditary Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [Ag]
    Agoston, I.: Quotients of quasi-hereditary algebras. C. R. Math. Rep. Acad. Sci. Canada 11 (1989), 99–103.MathSciNetzbMATHGoogle Scholar
  2. [AS1]
    Assem, I.; Skowronski, A.: Iterated tilted algebras of type à n, Math. Z. 195 (1987), 269–290MathSciNetzbMATHCrossRefGoogle Scholar
  3. [AS2]
    Assem, I.; Skowronski, A.: On some classes of simply connected algebras. Proc. London Math. Soc. 56 (1988), 417–450.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [AS3]
    Assem, I.; Skowronski, A.: Algebras with cycle finite derived categories. Math. Ann. 280 (1988), 441–463.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [AS4]
    Assem, I.; Skowronski, A.: Algèbres pré-inclinées et catégories dérivées. In: Sém. d’Algèbre Dubreil-Malliavin. Springer LNM. 1404 (1989), 1–34.CrossRefGoogle Scholar
  6. [AS5]
    Assem, L; Skowronski, A.: Minimal representation-infinite coil algebras. Manuscr. Math. 67 (1990), 305–331.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [AB]
    Auslander, M.; Buchweitz, R.-O.: The homology theory of maximal Cohen-Macaulay approximations. Mém. Soc. Math. France N.S. (1989).Google Scholar
  8. [APT]
    Auslander, M.; Platzeck, M. L; Todorov, G.: Homological theory of idempotent ideals. (To appear).Google Scholar
  9. [AR1]
    Auslander, M.; Reiten, I.: Applications of contravariantly finite subcategories. Preprint 8/1989 Univ. Trondheim.Google Scholar
  10. [AR2]
    Auslander, M.; Reiten, L: Cohen-Macaulay and Gorenstein Artin algebras. This volume.Google Scholar
  11. [BS]
    Bakke, Ø.; Smalø, S.O.: Modules with the same socles and tops as a directing module are isomorphic. Comm. Algebra 15 (1987), 1–10.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [Bau]
    Bautista, R.: On algebras of strongly unbounded representation type. Comment. Math. Helv. 60 (1985), 392–399.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [BK]
    Bautista, R.; Kleiner, M.: Almost split sequences for relatively projective modules. J. Algebra 135 (1990), 19–56MathSciNetzbMATHCrossRefGoogle Scholar
  14. [B1]
    Baer, D.: Homological properties of wild hereditary Artin algebras. In: Representation Theory I. Springer LNM 1177 (1986), 1–12Google Scholar
  15. [B2]
    Baer, D.: Wild hereditary Artin algebras and linear methods. Manuscr. Math. 55 (1986), 69–82.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [B3]
    Baer, D.: Tilting sheaves in representation theory of algebras. Manuscr. Math. 60 (1988), 323–347.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [B4]
    Baer, D.: A note on wild quiver algebras and tilting modules. Comm. Algebra 17 (1989), 751–757MathSciNetzbMATHCrossRefGoogle Scholar
  18. [BGL]
    Baer, D.; Geigle, W.; Lenzing, H.: The preprojective algebra of a tame hereditary Artin algebra. Comm. Algebra 15 (1987), 425–457.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [BG]
    Beilinson, A.; Ginsburg, V.: Mixed categories, Ext-duality abd representations (Results and conjectures). (To appear).Google Scholar
  20. [Be]
    Benson, D.: Modular Representation Theory: New Trends and Methods. Springer LNM 1081 (1984).Google Scholar
  21. [Bs]
    Bessenrodt-Timmerscheidt, Ch.: The Auslander-Reiten quiver of a modular group algebra revisited. Preprint SM-DU-159 Duisburg (1989).Google Scholar
  22. [Bo]
    Bongartz, K.: Indecomposables are standard. Comment. Math. Helv. 60 (1985), 400–410.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [BS]
    Bongartz, K.; Smalø, S.O.: Modules determined by their tops and socles. Proc. Amer. Math. Soc. 96 (1986), 34–38.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [Br]
    Brenner. Sh.: A combinatorial characterization of finite Auslander-Reiten quivers. In: Representation Theory I. Springer LNM 1177 (1986), 13–49.Google Scholar
  25. [BT]
    Bretscher, O; Todorov, G.: On a theorem of Nazarova and Roiter. In: Representation Theory I. Springer LNM 1177 (1986), 50–54.Google Scholar
  26. [Bé]
    Broué, M.: News about perfect isometries. (Unpublished).Google Scholar
  27. [BB]
    Burt, W.L.; Butler, M.C.R.: Almost split sequences for bocses. (To appear).Google Scholar
  28. [BR]
    Butler, M.C.R.; Ringel, C.M.: Auslander-Reiten sequences with few middle terms, with applications to string algebras. Comm. Alg. 15 (1987), 145–179.MathSciNetzbMATHCrossRefGoogle Scholar
  29. [CPS1]
    Cline, E.; Parshall, B.; Scott, L.: Algebraic statification in representation categories. J. Algebra. 117 (1988), 504–521MathSciNetzbMATHCrossRefGoogle Scholar
  30. [CPS2]
    Cline, E.; Parshall, B.; Scott, L.: Finite dimensional algebras and highest weight categories. J. reine angew. Math. 391 (1988), 85–99MathSciNetzbMATHGoogle Scholar
  31. [CPS3]
    Cline, E.; Parshall, B.; Scott, L.: Duality in highest weight categories. In: Classical Groups and Related Topics. Contemp. Math. 82 (1989), 7–22.MathSciNetCrossRefGoogle Scholar
  32. [CI]
    Collingwood, D.H.; Irving, R.S.: A decomposition theorem for certain self-dual modules in the categorie O. Duke Math. J. 58 (1989), 89–102.MathSciNetzbMATHCrossRefGoogle Scholar
  33. [C1]
    Crawley-Boevey, W.: On tame algebras and bocses. Proc. London Math. Soc. (3), 56 (1988), 451–483.MathSciNetzbMATHCrossRefGoogle Scholar
  34. [C2]
    Crawley-Boevey, W.: Functorial filtrations and the problem of an idempotent and a square-zero matrix. J. London Math. Soc. (2) 38 (1988), 385–402.MathSciNetzbMATHGoogle Scholar
  35. [C3]
    Crawley-Boevey, W.: Functorial filtrations II: Clans and the Gelfand problem. J. London Math. Soc. (2) 40 (1989), 9–30.MathSciNetzbMATHCrossRefGoogle Scholar
  36. [C4]
    Crawley-Boevey, W.: Functorial filtrations III: Semi-dihedral algebras. J. London Math. Soc. (2) 40 (1989), 31–39.MathSciNetzbMATHCrossRefGoogle Scholar
  37. [C5]
    Crawley-Boevey, W.: Maps between representations of zero-relation algebras. J. Algebra 126 (1989), 259–263MathSciNetzbMATHCrossRefGoogle Scholar
  38. [C6]
    Crawley-Boevey, W.: Tame hereditary algebras, hereditary orders and their curves. (To appear).Google Scholar
  39. [C7]
    Crawley-Boevey, W.: Matrix reduction for artinian rings, and an application to rings of finite representation type. Preprint 90-020 SFB 343 Bielefeld.Google Scholar
  40. [C8]
    Crawley-Boevey, W.: Tame algebras and generic modules. Preprint 89-028 SFB 343 Bielefeld.Google Scholar
  41. [C9]
    Crawley-Boevey, W.: Lectures at the Tsukuba Workshop 1990. (To appear).Google Scholar
  42. [CR]
    Crawley-Boevey, W.; Ringel, C.M.: Algebras whose Auslander-Reiten components have large regular components. Preprint 90-052 SFB 343 Bielefeld.Google Scholar
  43. [CU]
    Crawley-Boevey, W.; Unger, L.: Dimensions of Auslander-Reiten translates for representation-finite algebras. Comm. Algebra 17 (1989), 837–842.MathSciNetzbMATHCrossRefGoogle Scholar
  44. [DH]
    D’Este, G.; Happel, D.: Representable equivalences are represented by tilting modules. Rend. Sem. Mat. Univ. Padova 82 (1990), 77–80.MathSciNetGoogle Scholar
  45. [Di]
    Dieterich, E.: Solution of a non-domestic tame classification problem from intergal representation theory of finite groups (Λ = RC 3, v(3)= A.) Memoirs Amer. Math. Soc. (To appear).Google Scholar
  46. [DR1]
    Dlab, V.; Ringel, C.M.: Quasi-hereditary algebras. Illinois J. Math. 33 (1989), 280–291MathSciNetzbMATHGoogle Scholar
  47. [DR2]
    Dlab, V.; Ringel, C.M.: Auslander algebras as quasi-hereditary algebras. J. London Math. Soc. (2) 39 (1989), 457–466.MathSciNetzbMATHCrossRefGoogle Scholar
  48. [DR3]
    Dlab, V.; Ringel, C.M.: Every semiprimary ring is the endomorphism ring of a projective module over a quasi-hereditary ring. Proc. Amer. Math. Soc. 107 (1989), 1–5.MathSciNetzbMATHGoogle Scholar
  49. [DR4]
    Dlab, V.; Ringel, C.M.: A construction for quasi-hereditary algebras. Compositio Math. 70 (1989), 155–175.MathSciNetzbMATHGoogle Scholar
  50. [DR5]
    Dlab, V.; Ringel, C.M.: The dimension of a quasi-hereditary algebra. In: Topics in Algebra. Banach Center Publ. 26. (To appear).Google Scholar
  51. [DR6]
    Dlab, V.; Ringel, C.M. Filtrat ions of right ideals related to projectivity of left ideals. In: Sém. d’Algèbre Dubreil-Malliavin. Springer LNM. 1404 (1989), 95–107.CrossRefGoogle Scholar
  52. [DR7]
    Dlab, V.; Ringel, C.M.: The Hochschild cocycle corresponding to a long exact sequence. Tsukuba J. Math (To appear).Google Scholar
  53. [DR8]
    Dlab, V.; Ringel, C.M.: Towers of semi-simple algebras. J. Funct. Anal. (To appear)Google Scholar
  54. [DGL]
    Dowbor, P.; Geigle, W.; Lenzing, H.: Graded Sheaf Theory and Group Quotients, with Applications to Representations of Finite Dimensional Algebras. (To appear).Google Scholar
  55. [DM]
    Dowbor, P.; Meltzer, H.: On equivalences of Bernstein-Gelfand-Gelfand, Beilinson and Happel. Preprint 257 Humboldt-Universität Berlin (1990).Google Scholar
  56. [DS]
    Dowbor, P.; Skowroński, A.: Galois coverings of representation-infinite algebras. Comment. Math. Helv. 62 (1987), 311–337.MathSciNetzbMATHCrossRefGoogle Scholar
  57. [D1]
    Dräxler, P.: U-Fasersummen in darstellungsendlichen Algebren. J. Algebra 113 (1988), 430–437.MathSciNetzbMATHCrossRefGoogle Scholar
  58. [D2]
    Dräxler, P.: Aufrichtige gerichtete Ausnahmealgebren. Bayreuther Math. Schriften 29 (1989).Google Scholar
  59. [D3]
    Dräxler, P.: Fasersummen über dünnen s-Startmoduln. Arch. Math. 54 (1990), 252–257.CrossRefGoogle Scholar
  60. [D4]
    Dräxler, P.: On indecomposable modules over directed algebras. (To appear).Google Scholar
  61. [Df]
    Drinfeld, V.G.: Quantum groups. In: Proc. Intern. Congr. Math. 1986. Amer. Math. Soc. (1987), Vol.1, 798–820.MathSciNetGoogle Scholar
  62. [El]
    Erdmann, K.: Tame Blocks and Related Algebras. Springer LNM 1428 (1990).zbMATHGoogle Scholar
  63. [E2]
    Erdmann, K.: On Auslander-Reiten components for wild blocks. This volume.Google Scholar
  64. [ES]
    Erdmann, K.; Skowroñski, A.: On Auslander-Reiten components of blocks and self-injective biserial algebras. (To appear).Google Scholar
  65. [Fi]
    Fischbacher, U.: Une nouvelle preuve d’un théorème de Nazarova et Roiter. C.R. Acad. Sc. Paris I 300 (1985), 259–262.MathSciNetzbMATHGoogle Scholar
  66. [FP]
    Fischbacher, U.; de la Peña, J.: Algorithms in representation theory of algebras. In: Representation Theory I. Springer LNM 1177 (1986), 94–114.Google Scholar
  67. [GP]
    Gabriel, P.; de la Peña, J.: Quotients of representation-finite algebras. Comm. Algebra 15 (1987), 279–308.MathSciNetzbMATHCrossRefGoogle Scholar
  68. [GL1]
    Geigle, W.; Lenzing, H.: A class of weighted projective curves arising in the representation theory of finite dimensional algebras. In: Singularities, Representations of Algebras, and Vector Bundles. Springer LNM 1273 (1987), 265–297.CrossRefGoogle Scholar
  69. [GL2]
    Geigle, W.; Lenzing, H.: Perpendicular categories with applications to representations and sheaves. J. Algebra (To appear).Google Scholar
  70. [Gr]
    Green, J.A.: On certain subalgebras of the Schur algebra. (To appear).Google Scholar
  71. [GKK]
    Green, E.L.; Kirkman, E.; Kuzmanovich, J.: Finitistic dimension of finite dimensional monomial algebras. J. Algebra 136 (1991), 37–50.MathSciNetzbMATHCrossRefGoogle Scholar
  72. [GZ]
    Green, E.L.; Zimmermann-Huisgen, B.: Finitistic dimension of artinian rings with vanishing radical cube zero. Math. Z. (To appear).Google Scholar
  73. [GHJ]
    Goodman, F.M., de la Harpe, P.; Jones, V.F.R.: Coxeter Graphs of Algebras. Math. Sci. Res. Inst. Publ. 14 (1989).Google Scholar
  74. [G1]
    Guo, J.Y.: The isomorphism of Hall algebras. Preprint 90-060 SFB 343 Bielefeld.Google Scholar
  75. [G2]
    Guo, J.Y.: The center of a Hall algebra. Preprint 90-061 SFB 343 Bielefeld.Google Scholar
  76. [G3]
    Guo, J.Y.: The Hall algebra of a cyclic serial algebra. (In preparation)Google Scholar
  77. [H1]
    Happel, D.: On the derived category of a finite-dimensional algebra. Comment. Math. Helv. 62 (1987), 339–389.MathSciNetzbMATHCrossRefGoogle Scholar
  78. [H2]
    Happel, D.: Iterated tilted algebras of affine type. Comm. Alg. 15 (1987), 29–46MathSciNetzbMATHCrossRefGoogle Scholar
  79. [H3]
    Happel, D.: Repetitive categories. In: Singularities, Representations of Algebras, and Vector Bundles. Springer LNM 1273 (1987), 298–317CrossRefGoogle Scholar
  80. [H4]
    Happel, D.: Triangulated Categories in the Representation Theory of Finite-dimensional Algebras. London Math. Soc. LNS 119 (1988).zbMATHCrossRefGoogle Scholar
  81. [H5]
    Happel, D.: Hochschild cohomology of finite-dimensional algebras. In: Sérn. d’Algbre. Dubreil-Malliavin. Springer LNM 1404 (1989), 108–126.CrossRefGoogle Scholar
  82. [H6]
    Happel, D.: A family of algebras with two simple modules and Fibonacci numbers. Archiv Math. (To appear)Google Scholar
  83. [H7]
    Happel, D.: Partial tilting modules and recollement. Preprint 89-016 SFB 343 Bielefeld.Google Scholar
  84. [H8]
    Happel, D.: Auslander-Reiten triangles in derived categories of finite-dimensional algebras. Proc. Amer. Math. Soc. (To appear).Google Scholar
  85. [H9]
    Happel, D.: On Gorenstein algebras. This volume.Google Scholar
  86. [H10]
    Happel, D.: Hochschild cohomology of Auslander algebras. In: Topics in Algebra. Banach Center Publ. 26. (To appear).Google Scholar
  87. [HRS]
    Happel, D.; Rickard, J.; Schofield, A.: Piecewise hereditary algebras. Bull. London Math. Soc. 20 (1988), 23–28.MathSciNetzbMATHCrossRefGoogle Scholar
  88. [HR]
    Happel, D.; Ringel, C.M.: The derived category of a tubular algebra. In: Representation Theory I. Springer LNM 1177 (1986). 156–180.Google Scholar
  89. [HU1]
    Happel, D.; Unger, L.: Factors of concealed algebras. Math. Z. 201 (1989), 477–483.MathSciNetzbMATHCrossRefGoogle Scholar
  90. [HU2]
    Happel, D.; Unger, L.: A family of infinite-dimensional non-selfextending bricks for wild hereditary algebras. Proc. Tsukuba Conf. 1990 (To appear).Google Scholar
  91. [Hi]
    Hille, L.: Assoziative gestufte Algebren und Kippfolgen mit dim(X)+l Stufen auf projektiven glatten algebraischen Mannigfaltigkeiten. Diplomarbeit Humboldt Univ. Berlin (1990).Google Scholar
  92. [HM]
    Hoshino, M.; Miyachi, J.: Tame triangular matrix algebras over self-injective algebras. Tsukuba J. Math. 11 (1987), 383–391.MathSciNetzbMATHGoogle Scholar
  93. [Hö]
    v. Höhne; H. J.: On weakly positive unit forms. Comm. Math. Helv. 63 (1988), 312–336.zbMATHCrossRefGoogle Scholar
  94. [Hü]
    Hübner, T.: (In preparation).Google Scholar
  95. [Ig]
    Igusa, K.: Notes on the loop conjecture. J. Pure and Appl. Algebra. (To appear).Google Scholar
  96. [IST]
    Igusa, K.; Smalø, S.; Todorov, G.: Finite projectivity and contravariant finiteness. (To appear).Google Scholar
  97. [IT]
    Igusa, K.; Todorov, G.: A numerical characterization of finite Auslander-Reiten quivers. In: Representation Theory I. Springer LNM 1177 (1986), 181–198.Google Scholar
  98. [IZ]
    Igusa, K.; Zacharia, D.: Syzygy pairs in a monomial algebra. Proc. Amer. Math. Soc. 108 (1990) 601–604.MathSciNetzbMATHCrossRefGoogle Scholar
  99. [Ir]
    Irving, R.: BGG algebras and the BGG reciprocity principle. J. Algebra 135 (1990), 363–380.MathSciNetzbMATHCrossRefGoogle Scholar
  100. [Kc]
    Kac, V. G.: Root systems, representations of quivers and invariant theory. In: Invariant Theory. Springer LNM 996 (1983), 74–108.CrossRefGoogle Scholar
  101. [Kp]
    Kapranov, M.M.: On the derived categories of coherent sheaves on some homogeneous spaces. Inv. Math. 92 (1988), 479–508.MathSciNetzbMATHCrossRefGoogle Scholar
  102. [Kl]
    Keller, B.: Chain complexes and stable categories. Manuscr. Math. 67 (1990), 379–417.zbMATHCrossRefGoogle Scholar
  103. [KV]
    Keller, B.; Vossieck, D: Sous les catégories dérivées. C.R. Acad. Sci. Paris I 305 (1987) 225–228.MathSciNetzbMATHGoogle Scholar
  104. [K1]
    Kerner, O.: Preprojective components of wild hereditary algebras. Manuscr. Math. 61 (1988), 429–445.MathSciNetzbMATHCrossRefGoogle Scholar
  105. [K2]
    Kerner, O.: Tilting wild algebras. J. London Math. Soc. (2) 39 (1989), 29–47.MathSciNetzbMATHCrossRefGoogle Scholar
  106. [K3]
    Kerner, O.: Universal exact sequences for torsion theories. In: Topics in Algebra. Banach Center Publ. 26. (To appear).Google Scholar
  107. [K4]
    Kerner, O.: Stable components of wild tilted algebras. J. Algebra (To appear).Google Scholar
  108. [KSk]
    Kerner, O.; Skowronski, A.: On module categories with nilpotent infinite radical. (To appear).Google Scholar
  109. [KK]
    Kirkman, E.; Kuzmanovich, J.: Algebras with large homological dimension. (To appear)Google Scholar
  110. [KSi]
    Klemp, B.; Simson, D.: Schurian sp-representation-finite right peak PI-rings and their indecomposable socle projective modules. J. Algebra 134 (1990), 390–468.MathSciNetzbMATHCrossRefGoogle Scholar
  111. [Kö]
    König, S.: Tilting complexes, perpendicular categories, and recollement of derived module categories of rings. (To appear)Google Scholar
  112. [Kr1]
    Krause, H.: Maps between tree and band modules. J. Algebra (To appear).Google Scholar
  113. [Kr2]
    Krause, H.: Endomorphismem von Worten in einem Köcher. Dissertation. Bielefeld 1991.Google Scholar
  114. [Le]
    Le Bruyn, L.: Counterexamples to the Kac-conjecture on Schur roots. Bull. Sci. Math. 110 (1986), 437–448.MathSciNetzbMATHGoogle Scholar
  115. [L1]
    Lenzing, H.: Nilpotente Elemente in Ringen von endlicher globaler Dimension. Math. Z. 108 (1969), 313–324.MathSciNetzbMATHCrossRefGoogle Scholar
  116. [L2]
    Lenzing, H.: Curve singularities arising from the representation theory of tame hereditary Artin algebras. In: Representation Theory I. Springer LNM 1177 (1986), 199–231.Google Scholar
  117. [L3]
    Lenzing, H.: Canonical algebras and rings of automorphic forms. (In preparation).Google Scholar
  118. [LP]
    Lenzing, H.; de la Peña, J. A.: The Auslander-Reiten components of a canonical algebra. (In preparation).Google Scholar
  119. [Lu]
    Lusztig, G.: Canonical bases arising from quantized enveloping algebras. J. Amer. Math. Soc. 3 (1990), 447–498MathSciNetzbMATHCrossRefGoogle Scholar
  120. [MP]
    Marmaridis, N.; de la Peña, J.A.: Quadratic forms and preinjective modules. J. Algebra 134 (1990), 326–343.zbMATHCrossRefGoogle Scholar
  121. [MR]
    Marmolejo, E.; Ringel, C.M.: Modules of bounded length in Auslander-Reiten components, Arch. Math. 50 (1988), 128–133.MathSciNetzbMATHCrossRefGoogle Scholar
  122. [Ma1]
    Martínez-Villa, R.: The stable equivalence for algebras of finite representation type. Comm. Algebra 13 (1985), 991–1018.MathSciNetzbMATHCrossRefGoogle Scholar
  123. [Ma2]
    Martínez-Villa, R.: Some remarks on stably equivalent algebras. Comm. Algebra (To appear).Google Scholar
  124. [Me]
    Meltzer, H.: Tilting bundles, repetitive algebras, and derived categories of coherent sheaves. Preprint 193 Humboldt-Universität Berlin (1988)Google Scholar
  125. [MO]
    Menini, C.; Orsatti, A.: Representable equivalences between categories of modules and applications. Rend. Sem. Mat. Univ. Padova 82 (1989), 203–231.MathSciNetzbMATHGoogle Scholar
  126. [MV]
    Mirollo, R.; Vilonen, K.: Bernstein-Gelfand-Gelfand reciprocity on perverse sheaves. Ann. Sci. E.N.S. 20 (1987), 311–324.MathSciNetzbMATHGoogle Scholar
  127. [My]
    Miyashita, Y.: Tilting modules of finite projective dimension. Math. Z. 193 (1986), 113–146.MathSciNetzbMATHCrossRefGoogle Scholar
  128. [NR1]
    Nazarova, L.A.; Rojter, A.V.: Representations of completed partially ordered sets. In: Proc. Fourth Intern. Conf. Representations of Algebras. Vol.1. Carleton-Ottawa LNS 1 (1985).Google Scholar
  129. [NR2]
    Nazarova, L.A.; Rojter, A.V.: Representations of bipartite completed posets. Comment. Math. Helv. 63 (1988), 498–526.MathSciNetzbMATHCrossRefGoogle Scholar
  130. [NS]
    Nehring, J.; Skowroński, A.: Polynomial growth trivial extensions of simply connected algebras. Fund. Math. 132 (1989), 117–134.MathSciNetzbMATHGoogle Scholar
  131. [Ok]
    Okuyama, T.: On the Auslander-Reiten quiver of a finite group. J. Algebra 110 (1987), 425–430.MathSciNetzbMATHCrossRefGoogle Scholar
  132. [OP]
    Ostermann, A.; Pott, A.: Schwach positive ganze quadratische Formen, die eine aufrichtige, positive Wurzel mit einem Koeffizienten 6 besitzen. J. Algebra 126 (1989), 80–118.MathSciNetzbMATHCrossRefGoogle Scholar
  133. [Pa]
    Parshall, B.: Finite dimensional algebras and algebraic groups. In: Classical Groups and Related Topics. Contemp. Math. 82 (1989).Google Scholar
  134. [PS]
    Parshall, B; Scott, L.L.: Derived Categories, Quasi-hereditary Algebras, and Algebraic Groups. Proc. Ottawa-Moosonee Workshop Algebra. Carleton-Ottawa Math. LNS 3 (1988), 1–105Google Scholar
  135. [Pt]
    Partharasarathy, R.: t-structures dans la catégorie dérivée associée aux représentations d’un carquois. C.R. Acad. Sci. Paris I 304 (1987), 355–357.Google Scholar
  136. [P1]
    de la Peña, J.: On omnipresent modules in simply connected algebras. J. London Math. Soc. (2) 36 (1987), 385–392.MathSciNetzbMATHCrossRefGoogle Scholar
  137. [P2]
    de la Peña, J.: Quadratic forms and the representation type of an algebra. Ergänzungsreihe 90-003 SFB 343 Bielefeld.Google Scholar
  138. [P3]
    de la Peña, J.: Tame algebras with sincere directing modules. (To appear).Google Scholar
  139. [PS]
    de la Peña, J., Simson, D.: Prinjective modules, reflection functors, quadratic forms and Auslander-Reiten sequences. (To appear).Google Scholar
  140. [PTa]
    de la Peña, J.; Takane, M.: Spectral properties of Coxeter transformations and applications. Arch. Math. 55 (1990), 120–134.zbMATHCrossRefGoogle Scholar
  141. [PTo]
    de la Peña, J.; Tomé, B.: Iterated tubular algebras. J. Pure Appl. Algbera (To appear).Google Scholar
  142. [Rc1]
    Rickard, J.: Morita theory for derived categories. J. London Math. Soc. 39 (1989), 436–456MathSciNetzbMATHCrossRefGoogle Scholar
  143. [Rc2]
    Rickard, J.: Derived categories and stable equivalence. J. Pure Appl. Algebra 61 (1989), 436–456.MathSciNetCrossRefGoogle Scholar
  144. [Rc3]
    Rickard, J.: Derived equivalences as derived functors. (To appear).Google Scholar
  145. [RS]
    Rickard, J.; Schofield, A.: Cocovers and tilting modules. Proc. Cambridge Phil. Soc. 106 (1989) 1–5.MathSciNetzbMATHCrossRefGoogle Scholar
  146. [RS]
    Riedtmann, Ch.; Schofield, A.: On a simplicial complex associated with tilting modules. Prépubl. l’Inst. Fourier. 137 Grenoble (1989)Google Scholar
  147. [R1]
    Ringel, C.M.: Tame Algebras and Integral Quadratic Forms. Springer LNM 1099 (1984).zbMATHGoogle Scholar
  148. [R2]
    Ringel, C.M.: The regular components of the Auslander-Reiten quiver of a tilted algebra. Chinese Ann. Math. B9 (1988), 1–18.MathSciNetGoogle Scholar
  149. [R3]
    Ringel, C.M.: The canonical algebras. (With an appendix by W. Crawley-Boevey). In: Topics in Algebra. Banach Center Publ. 26. (To appear).Google Scholar
  150. [R4]
    Ringel, C.M.: The category of modules with good filtrat ions over a quasi-hereditary algebra has almost split sequences. Math. Z. (To appear).Google Scholar
  151. [R5]
    Ringel, C.M.: On contravariantly finite subcategories. In: Proc. Tsukuba Conf. 1990 (To appear)Google Scholar
  152. [R6]
    Ringel, C.M.: The category of good modules over a quasi-hereditary algebra. In: Proc. Tsukuba Conf. 1990 (To appear)Google Scholar
  153. [R7]
    Ringel, C.M.: Hall algebras. In: Topics in Algebra. Banach Centre Publ. 26. Warszawa (To appear).Google Scholar
  154. [R8]
    Ringel, C.M.: Hall polynomials for the representation-finite hereditary algebras. Adv. Math. 84 (1990), 137–178.MathSciNetzbMATHCrossRefGoogle Scholar
  155. [R9]
    Ringel, C.M.: From representations of quivers via Hall and Loewy algebras to quantum groups. In: Proc. Novosibirsk Conf. Algebra 1989 (To appear)Google Scholar
  156. [R10]
    Ringel, C.M.: Hall algebras and quantum groups. Invent. Math. 101 (1990), 583–592MathSciNetzbMATHCrossRefGoogle Scholar
  157. [R11]
    Ringel, C.M.: The composition algebra of a cyclic quiver. (To appear)Google Scholar
  158. [RV]
    Ringel, C.M.; Vossieck, D.: Hammocks. Proc. London Math. Soc. (3) 54 (1987), 216–246.MathSciNetzbMATHCrossRefGoogle Scholar
  159. [Ru]
    Rudakov, A. N. et. al.: Helices and Vector Bundles. Seminaire Rudakov. London Math. Soc. LNS 148 (1990).zbMATHCrossRefGoogle Scholar
  160. [Sr1]
    Scheuer, T.: More hammocks. In: Topics in Algebra. Banach Centre Publ. 26. Warszawa (To appear).Google Scholar
  161. [Sr2]
    Scheuer, T.: The canonical decomposition of the poset of a hammock. Proc. London Math. Soc. (To appear).Google Scholar
  162. [Sw1]
    Schewe, W.: The set of Z-coverings for a finite translation quiver. Dissertation Bielefeld 1989.Google Scholar
  163. [Sw2]
    Schewe, W.: Fundamental domains for representation-finite algebras. Rapport 74 Univ. Sherbrooke (1990).Google Scholar
  164. [Sf1]
    Schofield, A.: Bounding the global dimension in terms of the dimension. Bull. London Math. Soc. 17 (1985), 393–394.MathSciNetzbMATHCrossRefGoogle Scholar
  165. [Sf2]
    Schofield, A.: The field of definition of a real representation of a quiver Q. (To appear)Google Scholar
  166. [Sf3]
    Schofield, A.: The internal structure of real Schur representations. (To appear)Google Scholar
  167. [Sf4]
    Schofield, A.: (To appear).Google Scholar
  168. [Sc]
    Scott, L.L.: Simulating algebraic geometry with algebra I: The algebraic theory of derived categories. Proc. Symp. Pure Math. 47 (1987), 271–281.Google Scholar
  169. [Si]
    Simson, D.: Module categories and adjusted modules over traced rings. Diss. Math. 269 (1990).Google Scholar
  170. [Sk1]
    Skowroński, A.: Group algebras of polynomial growth. Manuscr. Math. 59 (1987), 499–516.zbMATHCrossRefGoogle Scholar
  171. [Sk2]
    Skowroński, A.: Selfinjective algebras of polynomial growth. Math. Ann. 285 (1989), 177–199.MathSciNetzbMATHCrossRefGoogle Scholar
  172. [Sk3]
    Skowroński, A.: Algebras of polynomial growth. Topics in Algebra. Banach centre publ. 26. (To appear).Google Scholar
  173. [SS]
    Skowroński, A.; Smalø, S.O.: Directing modules. Preprint 8/1990 Univ. Trondheim.Google Scholar
  174. [Sm]
    Smalø, S.O.: Functorial finite subcategories over triangular matrix rings. Preprint 6/1989 Trondheim.Google Scholar
  175. [Soe1]
    Soergel, W.: Kategorie O, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe. J. Amer. Math. Soc. 3 (1990), 421–445.MathSciNetzbMATHGoogle Scholar
  176. [Soe2]
    Soergel, W.: Parabolisch-singuläre Dualität für Kategorie O. Preprint MPI Bonn 89–68.Google Scholar
  177. [Soe3]
    Soergel, W.: Construction of projectives and reciprocity in an abstract setting. (To appear).Google Scholar
  178. [St]
    Strauß, H.: Tilting modules over wild hereditary algebras. Thesis. Carleton University 1986. Abstract: C.R. Math., Acad. Sci. Canada 9 (1987), 161–166.zbMATHGoogle Scholar
  179. [TW]
    Tachikawa, H., Wakamatsu, T.: Cartan matrices and Grothendieck groups of stable categories. (To appear).Google Scholar
  180. [UY]
    Uematsu, M.; Yamagata, K.: On serial quasi-hereditary rings. Hokkaido Math. J. 19 (1990), 165–174.MathSciNetzbMATHGoogle Scholar
  181. [U1]
    Unger, L.: Lower bounds for indecomposable, faithful, preinjective modules. Manuscr. Math. 57 (1986), 1–31.MathSciNetzbMATHCrossRefGoogle Scholar
  182. [U2]
    Unger, L.: The concealed algebras of the minimal wild hereditary algebras. Bayreuther Math. Schriften 31 (1990), 145–154.MathSciNetzbMATHGoogle Scholar
  183. [U3]
    Unger, L.: On wild tilted algebras which are squids. Archiv Mathematik (To appear).Google Scholar
  184. [U4]
    Unger, L.: Schur modules over wild, finite dimensional path algebras with three non isomorphic simple modules. J. Pure Appl. Algebra. (To appear).Google Scholar
  185. [U5]
    Unger, L.: On the number of maximal sincere modules over sincere directed algebras. J. Algebra 133 (1990), 211–231.MathSciNetzbMATHCrossRefGoogle Scholar
  186. [U6]
    Unger, L.: One-dimensional links of the simplicial complex of partial tilting modules. (In preparation).Google Scholar
  187. [W1]
    Wakamatsu, T.: On modules with trivial selfextensions. J. Algebra 114 (1988), 106–114.MathSciNetzbMATHCrossRefGoogle Scholar
  188. [W2]
    Wakamatsu, T.: Stable equivalences for selfinjective algebras and a generalization of tilting modules. J. Algebra 134 (1990), 298–325.MathSciNetzbMATHCrossRefGoogle Scholar
  189. [Wi]
    Wiedemann, A.: Quotients of quasi-hereditary algebras. (To appear).Google Scholar
  190. [X1]
    Xi, Ch.: Die Vektorraumkategorie zu einem unzerlegbaren projektiven Modul einer Algebra. J. Algebra (To appear).Google Scholar
  191. [X2]
    Xi, Ch.: Die Vektorraumkategorie zu einem Punkt einer zahmen verkleideten Algebra. J. Algebra (To appear).Google Scholar
  192. [X3]
    Xi, Ch.: Die Vektorraumkategorie zu einem unzerlegbaren projektiven Modul einer tubularen Algebra. Manuscr. Math. 69 (1990), 223–235.zbMATHCrossRefGoogle Scholar
  193. [X4]
    Xi, Ch.: On wild hereditary algebras with small growth numbers. Comm. Algebra 18 (1990), 3413–3422.MathSciNetzbMATHCrossRefGoogle Scholar
  194. [X5]
    Xi, Ch.: Minimal elements of the poset of a hammock. Preprint 90-016 SFB 343 Bielefeld.Google Scholar
  195. [X6]
    Xi, Ch.: Quasi-heredity of algebras and their factor algebras. Preprint 90-59 SFB 343 Bielefeld.Google Scholar
  196. [X7]
    Xi, Ch.: Symmetric algebras as endomorphism rings of large projective modules over quasi-hereditary algebras. Preprint 90-59 SFB 343 Bielefeld.Google Scholar
  197. [Za]
    Zavadskij, A. G.: Classification of representations of posets of finite growth. in: Proc. Fourth Intern. Conf. Representations of Algebras. Vol.2. Carleton-Ottawa LNS 2 (1984), 36.01–36.15Google Scholar
  198. [Z1]
    Zhang, Y.: The modules in any component of the AR-quivaer of a wild hereditary artin algebra are uniquely determined by their composition factors. Archiv Math. 53 (1989), 250–251.zbMATHCrossRefGoogle Scholar
  199. [Z2]
    Zhang, Y.: Eigenvalues of Coxeter transformations and the structure of the regular components of the Auslander-Reiten quiver. Comm. Algebra. 17 (1989), 2347–2362.MathSciNetzbMATHCrossRefGoogle Scholar
  200. [Z3]
    Zhang, Y.: The structure of stable components. Can. J. Math. (To appear).Google Scholar
  201. [Zi1]
    Zimmermann-Huisgen, B.: Bounds on finitistic and global dimension for artinian rings with vanishing radical cube. (To appear).Google Scholar
  202. [Zi2]
    Zimmermann-Huisgen, B.: Predicting syzygies over monomial algebras. (To appear).Google Scholar

Older surveys

  1. [1]
    Auslander, M.: The what, where and why of almost split sequences. Proc. Intern. Congr. Math. 1986. Amer. Math. Soc. (1987), Vol.1, 338–345.MathSciNetGoogle Scholar
  2. [2]
    Gabriel, P.: Darstellungen endlichdimensionaler Algebren Proc. Intern. Congr. Math. 1986. Amer. Math. Soc. (1987), Vol.1, 378–388.MathSciNetGoogle Scholar
  3. [3]
    Reiten, I.: An introduction to the representation theory of Artin algebras. Bull. London Math. Soc. 17 (1985), 209–233.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Reiten, I.: Finite dimensional algebras and singularities. In: Singularities, Representations of Algebras, and Vector Bundles. Springer LNM 1273 (1987), 35–57.CrossRefGoogle Scholar
  5. [5]
    Riedtmann, Ch.: Algèbre de type de représentation fini. Séminaire Bour-baki. 37e année 1984–1985, n. 650. Astérisque 133–134 (1986), 335–350.MathSciNetGoogle Scholar
  6. [6]
    Ringel, C.M.: The representation type of local algebras. In: Representations of Algebras. Springer LNM 488 (1975), 282–305.CrossRefGoogle Scholar
  7. [7]
    Ringel, C.M.: Report on the Brauer-Thrall conjectures. In: Representation Theory I. Springer LNM 831 (1980), 104–136CrossRefGoogle Scholar
  8. [8]
    Ringel, C.M.: Tame algebras. In: Representation Theory I. Springer LNM 831 (1980), 137–287CrossRefGoogle Scholar
  9. [9]
    Ringel, C.M.: Indecomposable representations of finite dimensional algebras. Proceedings Intern. Conf. Math. Warszawa 1983, (1984), 425–436.Google Scholar
  10. [10]
    Ringel, C.M.: Representation theory of finite dimensional algebras. Durham Lectures 1985. London Math. Soc. Lecture Note Series 116 (1986), 7–79.MathSciNetGoogle Scholar

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Authors and Affiliations

  • Claus Michael Ringel

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