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1993 | 142 | 1 | 59-84
Tytuł artykułu

On tame repetitive algebras

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let A be a finite dimensional algebra over an algebraically closed field, and denote by T(A) (respectively, Â) the trivial extension of A by its minimal injective cogenerator bimodule (respectively, the repetitive algebra of A). We characterise the algebras A such that  is tame and exhaustive, that is, the push-down functor mod  → mod T(A) associated with the covering functor  → T(A)\nsimto Â/(ν_A)$ is dense. We show that, if  is tame and exhaustive, then A is simply connected if and only if A is not an iterated tilted algebra of type $Â_m$. Then we prove that  is tame and exhaustive if and only if A is tilting-cotilting equivalent to an algebra which is either hereditary of Dynkin or Euclidean type or is tubular canonical.
Słowa kluczowe
Rocznik
Tom
142
Numer
1
Strony
59-84
Opis fizyczny
Daty
wydano
1993
otrzymano
1991-09-30
poprawiono
1992-06-03
Twórcy
  • Département de Mathématiques et d'Informatique, Université de Sherbrooke, Sherbrooke, Québec, Canada, J1K 2R1
  • Institute of Mathematics, Nicholas Copernicus University, Chopina 12/18, 87-100 Toruń, Poland
Bibliografia
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  • [2] I. Assem and D. Happel, Generalized tilted algebras of type $\sym A_n$, ibid. 9 (1981), 2101-2125.
  • [3] I. Assem, D. Happel and O. Roldán, Representation-finite trivial extension algebras, J. Pure Appl. Algebra 33 (1984), 235-242.
  • [4] I. Assem, J. Nehring and A. Skowroński, Domestic trivial extensions of simply connected algebras, Tsukuba J. Math. 13 (1989), 31-72.
  • [5] I. Assem and A. Skowroński, Iterated tilted algebras of type $Â_n$, Math. Z. 195 (1987), 269-290.
  • [6] I. Assem and A. Skowroński, On some classes of simply connected algebras, Proc. London Math. Soc. (3) 56 (1988), 417-450.
  • [7] I. Assem and A. Skowroński, Algebras with cycle-finite derived categories, Math. Ann. 280 (1988), 441-463.
  • [8] I. Assem and A. Skowroński, Quadratic forms and iterated tilted algebras, J. Algebra 128 (1990), 55-85.
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  • [15] P. Dowbor and A. Skowroński, On Galois coverings of tame algebras, Arch. Math. (Basel) 44 (1985), 522-529.
  • [16] P. Dowbor and A. Skowroński, Galois coverings of representation-infinite algebras, Comment. Math. Helv. 62 (1987), 311-337.
  • [17] Yu. A. Drozd, Tame and wild matrix problems, in: Proc. ICRA II (Ottawa 1979), Lecture Notes in Math. 832, Springer, Berlin 1980, 242-258.
  • [18] P. Gabriel, Auslander-Reiten sequences and representation-finite algebras, in: Proc. ICRA II (Ottawa 1979), Lecture Notes in Math. 831, Springer, Berlin 1980, 1-71.
  • [19] P. Gabriel, The universal cover of a representation-finite algebra, in: Proc. ICRA III (Puebla 1980), Lecture Notes in Math. 903, Springer, Berlin 1981, 68-105.
  • [20] D. Happel, Tilting sets on cylinders, Proc. London Math. Soc. (3) 51 (1985), 21-55.
  • [21] D. Happel, On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62 (1987), 339-389.
  • [22] D. Happel, J. Rickard and A. Schofield, Piecewise hereditary algebras, Bull. London Math. Soc. 20 (1988), 23-28.
  • [23] D. Happel and C. M. Ringel, Tilted algebras, Trans. Amer. ath. Soc. 274 (1982), 399-443.
  • [24] D. Happel and C. M. Ringel, Construction of tilted algebras, in: Proc. ICRA III (Puebla 1980), Lecture Notes in Math. 903, Springer, Berlin 1981, 125-144.
  • [25] D. Happel and C. M. Ringel, The derived category of a tubular algebra, in: Proc. ICRA IV (Ottawa 1984), Lecture Notes in ath. 1177, Springer, Berlin 1986, 156-180.
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  • [30] J. Nehring and A. Skowroński, Polynomial growth trivial extensions of simply connected algebras, Fund. Math. 132 (1989), 117-134.
  • [31] C. M. Ringel, Tame algebras, in: Proc. ICRA II (Ottawa 1979), Lecture Notes in Math. 831, Springer, Berlin 1980, 137-287.
  • [32] C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math. 1099, Springer, Berlin 1984.
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  • [34] O. Roldán, Tilted algebras of type $\widetilde\sym A_n$, $\widetilde\sym B_n$, $\widetilde\sym C_n$ and $\widetilde\sym B\sym C_n$, Ph.D. thesis, Carleton University, 1983.
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  • [36] A. Skowroński, On algebras of finite strong global dimension, Bull. Polish Acad. Sci. 35 (1987), 539-547.
  • [37] A. Skowroński, Group algebras of polynomial growth, anuscripta Math. 59 (1987), 499-516.
  • [38] A. Skowroński, Selfinjective algebras of polynomial growth, ath. Ann. 285 (1989), 177-199.
  • [39] A. Skowroński, Algebras of polynomial growth, in: Topics in Algebra, Banach Center Publ. 26, Part 1, PWN, Warszawa 1990, 535-568.
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  • [41] H. Tachikawa and T. Wakamatsu, Tilting functors and stable equivalences for selfinjective algebras, J. Algebra 109 (1987), 138-165.
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  • [43] J. L. Verdier, Catégories dérivées, état 0, in: SGA 4 1/2, Lecture Notes in Math. 569, Springer, Berlin 1977, 262-311.
  • [44] T. Wakamatsu, Stable equivalence between universal covers of trivial extension self-injective algebras, Tsukuba J. Math. 9 (1985), 299-316.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-fmv142i1p59bwm
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