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Selfinjective algebras and tilting theory

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References

  1. I. Assem and D. Happel, Generalized tilted algebras of type An, Comm. Algebra 9 (1981), 2101–2125.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. I. Assem, D. Happel and O. Roldan, Representation-finite trivial extension algebras, preprint.

    Google Scholar 

  3. I. Assem and Y. Iwanaga, Stable equivalence of representation-finite trivial extension, preprint.

    Google Scholar 

  4. M. Auslander and I. Reiten, Representation theory of artin algebras III, V, Comm. Algebra 3 (1975), 239–294, 5 (1977), 519–554.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. M. Auslander and I. Reiten, On a generalized version of the Nakayama conjecture, Proc. Amer. Math. Soc. 52 (1975), 69–74.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. M. Auslander, M. I. Platzeck and I. Reiten, Coxeter functors without diagrams, Trans. Amer. Math. Soc. 250 (1979), 1–46.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. R. Bautista, P. Gabriel, A. V. Roiter and L. Salmerón, Representation-finite algebras and multiplicative bases, Publicacions Preliminares del Instituto de Mathematicas No 69.

    Google Scholar 

  8. I. N. Bernstein, I. M. Gelfand and V. A. Ponomareve, Coxeter functors and Gabriel's theorem, Uspechi Mat. Nauk. 28 (1973), translated in Russian Math. Surveys 28 (1973), 17–32.

    Google Scholar 

  9. K. Bongartz, Tilted algebras, LNM 903 (1981), 16–32.

    MathSciNet  MATH  Google Scholar 

  10. K. Bongartz and P. Gabriel, Covering spaces in representation theory, Invent. Math. 65 (1982), 331–378.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. S. Brenner and M. C. R. Butler, Generalization of Bernstein-Gelfand-Ponomarev reflection functors, LNM 832 (1980), 103–169.

    MathSciNet  MATH  Google Scholar 

  12. O. Bretscher, C. Läser and C. Riedtmann, Selfinjective and simply connected algebras, Manuscripta Math. 38 (1981), 253–308.

    CrossRef  MATH  Google Scholar 

  13. V. P. Camillo and K. R. Fuller, Balanced and QF-1 algebras, Proc. Amer. Math. Soc. 34 (1972), 373–378.

    MathSciNet  MATH  Google Scholar 

  14. H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Princeton, 1981.

    Google Scholar 

  15. C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Math. XI (1962), Wiley and Sons Inc.

    Google Scholar 

  16. E. Dade, Blocks with cyclic defect groups, Ann. Math. 84 (1966), 20–48.

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. S. C. Dickson, A torsion theory for abelian categories, Trans. Amer. Math. Soc. 121 (1966), 223–235.

    CrossRef  MathSciNet  MATH  Google Scholar 

  18. F. Dischinger and W. Müller, Einreihig zerlegbare artinsche Ringe sind selbstdual, Arch. Math. 43 (1984).

    Google Scholar 

  19. V. Dlab and C. M. Ringel, Exceptional rings, Colloquia Math. Soc. Janos Bolyai 6 (1971), 167–171.

    MathSciNet  MATH  Google Scholar 

  20. V. Dlab and C. M. Ringel, Indecomposable representations of graphes and algebras, Memoirs Amer. Math. Soc. 173 (1976).

    Google Scholar 

  21. P. Donovan and M. R. Freislich, The indecomposable modular representations of certain groups with dihedral Sylow subgroup. Math. Ann. 238 (1978), 207–216.

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. R. M. Fossum, P. A. Griffith and I. Reiten, Trivial extensions of abelian categories, LNM 456 (1975).

    Google Scholar 

  23. K. R. Fuller and I. Reiten, Note on rings of finite representation type and decompositions of modules, Proc. Amer. Math. Soc. 50 (1975), 92–94.

    CrossRef  MathSciNet  MATH  Google Scholar 

  24. P. Gabriel and C. Riedtmann, Group representations without groups, Comment. Math. Helve. 54 (1979), 240–287.

    CrossRef  MathSciNet  MATH  Google Scholar 

  25. P. Gabriel, Indecomposable representations II, Symposia Mathematica XI, Instituto Nag. di Alta Math., (1973), 81–104.

    Google Scholar 

  26. P. Gabriel, Chr. Riedtmann and the selfinjective algebras of finite representation-type, Proc. Conf. on Ring Theory, Antwerp 1978, Marcel Dekker (1979), 453–458.

    Google Scholar 

  27. P. Gabriel, Auslander-Reiten sequences and representation-finite algebras, LNM 831 (1980), 1–71.

    MathSciNet  MATH  Google Scholar 

  28. P. Gabriel, The universal cover of a representation-finite algebra, Representations of Algebras, LNM 903 (1981), 68–105.

    MathSciNet  Google Scholar 

  29. R. Goldon and E. L. Green, Graded artin algebras, J. of Algebra 76 (198), 111–137.

    Google Scholar 

  30. W. H. Gustafson, The history of algebras and their representations, LNM 944 (1982), 1–28.

    MathSciNet  MATH  Google Scholar 

  31. D. Happel, U. Preiser and C. M. Ringel, Vinberg's characterization of Dynkin diagrams using subadditive functions with application to DTr-periodic modules, LNM 832 (1980), 280–294.

    MathSciNet  MATH  Google Scholar 

  32. D. Happel and C. M. Ringel, Construction of tilted algebras, LNM 903 (1981), 123–144.

    MathSciNet  MATH  Google Scholar 

  33. D. Happel and C. M. Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), 39–443.

    CrossRef  MathSciNet  MATH  Google Scholar 

  34. D. G. Higman, Indecomposable representations at characteristic P, Duke Math. J. 21 (1954), 377–381.

    CrossRef  MathSciNet  MATH  Google Scholar 

  35. M. Hoshino, Trivial extensions of titled algebras, Comm. Algebra 10 (1982), 1965–199

    CrossRef  MathSciNet  MATH  Google Scholar 

  36. M. Hoshino, On splitting torsion theories induced by tilting modules, Comm. Algebra 11 (1983), 427–440.

    CrossRef  MathSciNet  MATH  Google Scholar 

  37. M. Hoshino, Modules without self-extensions and Nakayama's conjecture, Arch. Math. 43 (1984), 493–500.

    CrossRef  MathSciNet  MATH  Google Scholar 

  38. D. Hughes and J. Waschbüsch, Trivial extensions of tilted algebras, Proc. London Math. Soc. 46 (1983), 347–364.

    CrossRef  MathSciNet  MATH  Google Scholar 

  39. Y. Iwanaga and T. Wakamatsu, Trivial extensions of artin algebras, LNM 832 (1980), 295–301.

    MathSciNet  MATH  Google Scholar 

  40. G. J. Janusz, Indecomposable modules for finite groups, Ann. Math. 89 (1969), 209–241.

    CrossRef  MathSciNet  MATH  Google Scholar 

  41. H. Kupisch, Symmetrische Algebren mit endlich vielen unzerlegbaren Darstellungen I, J. Reine Angew. Math. 219 (1965), 1–25.

    MathSciNet  MATH  Google Scholar 

  42. H. Kupisch, Projective Moduln endlicher Gruppen mit zyklischer p-Sylow Gruppe, J. of Algebra 10 (1968), 1–7.

    CrossRef  MathSciNet  MATH  Google Scholar 

  43. H. Kupisch, Unzerlegbare Moduln endlicher Gruppen mit zyklischer p-Sylow Gruppe, Math. Z. 108 (1968), 77–104.

    CrossRef  MathSciNet  MATH  Google Scholar 

  44. H. Kupisch and E. Scherzer, Symmetric algebras of finite representation type, LNM 832 (1980), 324–368.

    MathSciNet  Google Scholar 

  45. R. Makino, QF-1 algebras of local-colocal type, to appear in Math. Z.

    Google Scholar 

  46. R. Makino, QF-1 algebras with infinitely many minimal faithful modules, to appear in Comm. Algebra.

    Google Scholar 

  47. R. Martinez-Villa and J. A. dela Pena, Automorphisms of representation-finite algebras, Invent. Math. 72 (1983), 359–362.

    CrossRef  MathSciNet  MATH  Google Scholar 

  48. B. J. Müller, The classification of algebras by dominant dimensions, Can. J. Math. 20 (1968), 398–409.

    CrossRef  Google Scholar 

  49. W. Müller, Unzerlegbare Moduln über artinschen Ringen, Math. Z. 137 (1974), 197–226.

    CrossRef  MathSciNet  MATH  Google Scholar 

  50. T. Nakayama, On Frobeniusean algebras I, Ann. Math. 40 (1939), 611–633.

    CrossRef  MathSciNet  MATH  Google Scholar 

  51. T. Nakayama, On Frobeniusean algebras II, Ann. Math. 42 (1941), 1–21.

    CrossRef  MathSciNet  MATH  Google Scholar 

  52. T. Nakayama, On algebras with complete homology, Abh. Math. Sem. Univ. Hamburg 22 (1958), 300–307.

    CrossRef  MathSciNet  MATH  Google Scholar 

  53. I. Reiten, Stable equivalence of self-injective algebras, J. Algebra 14 (1976), 63–74.

    CrossRef  MathSciNet  MATH  Google Scholar 

  54. C. Riedtmann, Algebren, Darstellungsköcher, Ueberlagerungen und zurück, Comment. Math. Helv. 55 (1980), 199–224.

    CrossRef  MathSciNet  MATH  Google Scholar 

  55. C. Riedtmann, Representation-finite selfinjective algebras of class An, LNM 832 (1980), 449–520.

    MathSciNet  MATH  Google Scholar 

  56. C. Riedtmann, Many algebras with the some Auslander-Reiten quiver, Bull. London Math. Soc. 15 (1983), 43–47.

    CrossRef  MathSciNet  MATH  Google Scholar 

  57. C. Riedtmann, Representation-finite selfinjective algebras of class Dn, Compositio Math. 49 (1983), 231–282.

    MathSciNet  MATH  Google Scholar 

  58. C. M. Ringel, Socle conditions for QF-1 rings, Pacific J. Math. 41 (1973), 309–336.

    CrossRef  MathSciNet  MATH  Google Scholar 

  59. C. M. Ringel, Representations of K-species and bimodules, J. Algebra 41 (1976), 269–302.

    CrossRef  MathSciNet  MATH  Google Scholar 

  60. C. M. Ringel, Finite dimensional hereditary algebras of wild representation type, Math. Z. 169 (1978), 235–255.

    CrossRef  MathSciNet  MATH  Google Scholar 

  61. R. Schulz, Modules without self-extensions are projective, preprint.

    Google Scholar 

  62. H. Tachikawa, On dominant dimension of QF-3 algebras, Trans. Amer. Math. Soc. 112 (1964), 249–266.

    MathSciNet  MATH  Google Scholar 

  63. H. Tachikawa, Quasi-Frobenius rings and generalizations, LNM 351 (1973).

    Google Scholar 

  64. H. Tachikawa, Balancedness and left serial algebras of finite type LNM 488 (1975), 351–378.

    MathSciNet  MATH  Google Scholar 

  65. H. Tachikawa, Representations of trivial extensions of hereditary algebras, LNM 832 (1980), 579–599.

    MathSciNet  MATH  Google Scholar 

  66. H. Tachikawa, Reflection functors and Auslander-Reiten translations for trivial extensions of hereditary algebras, J. Algebra 90 (1984).

    Google Scholar 

  67. H. Tachikawa, Selfinjective algebras and tilting theory, Carleton-Ottawa Mathematical Notes Series No.1 (1985).

    Google Scholar 

  68. H. Tachikawa and T. Wakamatsu, Extensions of tilting functors and QF-3 algebras, to apear in J. Algebra.

    Google Scholar 

  69. H. Tachikawa and T. Wakamatsu, Stable equivalences for selfinjective algebras which are extensions of reflection functors, preprint.

    Google Scholar 

  70. H. Tachikawa and T. Wakamatsu, Applications of reflection functors for self-injective algebras, Carleton-Ottawa Mathematical Notes Series No.2 (1984).

    Google Scholar 

  71. T. Tanisaki, Foldings of root systems and Gabriel's theorem, Tsukuba J. Math. 4 (1980), 89–97.

    MathSciNet  MATH  Google Scholar 

  72. R. M. Thrall, Some generalizations of quasi-Frobenius algebras, Trans. Amer. Math. Soc. 64 (1948), 173–183.

    MathSciNet  MATH  Google Scholar 

  73. T. Wakamatsu, Partial Coxeter functors of selfinjective algebras, preprint.

    Google Scholar 

  74. T. Wakamatsu, Stable equivalence between universal covers of trivial extensions, preprint.

    Google Scholar 

  75. J. Waschbüsch, Symmetrische Algebren von endlichen modultyp, J. Reine Angew. Math. 321 (1981), 78–98.

    MathSciNet  MATH  Google Scholar 

  76. J. Waschbüsch, Universal coverings of selfinjective algebras, LNM 903 (1981), 331–349.

    Google Scholar 

  77. J. Waschbüsch, Ueber die Stammalgebra und Standardform einer darstellungsendlichen Algebra, Arch. Math. 41 (1983), 25–29.

    CrossRef  Google Scholar 

  78. C. V. Wilson, The Cartan map on categories of graded modules, J. Algebra 85 (1983), 390–398.

    CrossRef  MathSciNet  MATH  Google Scholar 

  79. K. Yamagata, Extensions over hereditary artinian rings with self-dualities I, J. Algebra 73 (1981), 386–433.

    CrossRef  MathSciNet  MATH  Google Scholar 

  80. K. Yamagata, On algebras whose trivial extensions are of finite representation type, LNM 93 (1981), 364–371.

    MathSciNet  MATH  Google Scholar 

  81. K. Yamagata, On algebras whose trivial extensions are of finite representation type II, preprint.

    Google Scholar 

  82. D. Zacharia, The finitistic global dimension for graded algebras, preprint.

    Google Scholar 

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Tachikawa, H. (1986). Selfinjective algebras and tilting theory. In: Dlab, V., Gabriel, P., Michler, G. (eds) Representation Theory I Finite Dimensional Algebras. Lecture Notes in Mathematics, vol 1177. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0075270

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