Slice modules over minimal 2-fundamental algebras

Zygmunt Pogorzały; Karolina Szmyt

Open Mathematics (2007)

  • Volume: 5, Issue: 1, page 164-180
  • ISSN: 2391-5455

Abstract

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We consider a class of algebras whose Auslander-Reiten quivers have starting components that are not generalized standard. For these components we introduce a generalization of a slice and show that only in finitely many cases (up to isomorphism) a slice module is a tilting module.

How to cite

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Zygmunt Pogorzały, and Karolina Szmyt. "Slice modules over minimal 2-fundamental algebras." Open Mathematics 5.1 (2007): 164-180. <http://eudml.org/doc/269709>.

@article{ZygmuntPogorzały2007,
abstract = {We consider a class of algebras whose Auslander-Reiten quivers have starting components that are not generalized standard. For these components we introduce a generalization of a slice and show that only in finitely many cases (up to isomorphism) a slice module is a tilting module.},
author = {Zygmunt Pogorzały, Karolina Szmyt},
journal = {Open Mathematics},
keywords = {Minimal 2-fundamental algebra; Auslander-Reiten quiver; slice module; tilting module; minimal 2-fundamental algebras; Auslander-Reiten quivers; tilting modules; Auslander-Reiten components; postprojective slice modules; cotilting modules; hereditary algebras; preinjective slice modules},
language = {eng},
number = {1},
pages = {164-180},
title = {Slice modules over minimal 2-fundamental algebras},
url = {http://eudml.org/doc/269709},
volume = {5},
year = {2007},
}

TY - JOUR
AU - Zygmunt Pogorzały
AU - Karolina Szmyt
TI - Slice modules over minimal 2-fundamental algebras
JO - Open Mathematics
PY - 2007
VL - 5
IS - 1
SP - 164
EP - 180
AB - We consider a class of algebras whose Auslander-Reiten quivers have starting components that are not generalized standard. For these components we introduce a generalization of a slice and show that only in finitely many cases (up to isomorphism) a slice module is a tilting module.
LA - eng
KW - Minimal 2-fundamental algebra; Auslander-Reiten quiver; slice module; tilting module; minimal 2-fundamental algebras; Auslander-Reiten quivers; tilting modules; Auslander-Reiten components; postprojective slice modules; cotilting modules; hereditary algebras; preinjective slice modules
UR - http://eudml.org/doc/269709
ER -

References

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  1. [1] I. Assem: “Tilting theory - an introduction”, In: Topics in Algebras, Banach Center Publications, Vol. 26, Part I, PWN, Warszawa, 1990, pp. 127–180. 
  2. [2] M. Auslander and I. Reiten: “Representation theory of artin algebras III”, Comm. Alg., Vol. 3, (1975), pp. 239–294. 
  3. [3] M. Auslander and I. Reiten: “Representation theory of artin algebras IV”, Comm. Alg., Vol. 5, (1977), pp. 443–518. 
  4. [4] M. Auslander, I. Reiten and S.O. Smalø: Representation Theory of Artin Algebras, Cambridge Stud. Adv. Math., Vol. 36, Cambridge Univ. Press, Cambridge, 1995. 
  5. [5] K. Bongartz: Tilted algebras, LNM 903, Springer, Berlin, 1981, pp. 26–38. 
  6. [6] M.C.R. Butler and C.M. Ringel: “Auslander-Reiten sequences with few middle terms and applications to string algebras”, Comm. Alg., Vol. 15, (1987), pp. 145–179. Zbl0612.16013
  7. [7] P. Dowbor and A. Skowroński: “Galois coverings of representation-infinite algebras”, Comment. Math. Helv., Vol. 62, (1987), pp. 311–337. Zbl0628.16019
  8. [8] P. Gabriel: Auslander-Reiten sequences and representation-finite algebras, INM 831, Springer, Berlin, 1980, pp. 1–71. Zbl0445.16023
  9. [9] D. Happel and C.M. Ringel: “Tilted algebras”, Trans. Amer. Math. Soc., Vol. 274, (1982), pp. 399–443. http://dx.doi.org/10.2307/1999116 
  10. [10] F. Huard: “Tilted gentle algebras”, Comm. Alg., Vol. 26(1), (1998), pp. 63–72. Zbl0898.16010
  11. [11] F. Huard and Sh. Liu: “Tilted special biserial algebras”, J. Algebra, Vol. 217, (1999), pp. 679–700. http://dx.doi.org/10.1006/jabr.1998.7828 
  12. [12] F. Huard and Sh. Liu: “Tilted string algebras”, J. Pure Appl. Algebra, Vol. 153, (2000), pp. 151–164. http://dx.doi.org/10.1016/S0022-4049(99)00101-2 Zbl0962.16009
  13. [13] Z. Pogorzały and M. Sufranek: “Starting and ending components of the Auslander-Reiten quivers of a class of special biserial algebras”, Colloq. Math., Vol. 99(1), (2004), pp. 111–144. Zbl1107.16022
  14. [14] C.M. Ringel: Tame algebras and integral quadratic forms, LNM 1099, Springer, Berlin, 1984. 
  15. [15] J. Schröer: “Modules without self-extensions over gentle algebras”, J. Algebra, Vol. 216, (1999), pp. 178–189. http://dx.doi.org/10.1006/jabr.1998.7696 
  16. [16] A. Skowroński: “Generalized standard Auslander-Reiten components”, J. Math. Soc. Japan, Vol. 46, (1994), pp. 517–543. http://dx.doi.org/10.2969/jmsj/04630517 Zbl0828.16011
  17. [17] A. Skowroński and J. Waschbüsch: “Representation-finite biserial algebras”, J. Reine Angew. Math., Vol. 345, (1983), pp. 172–181. Zbl0511.16021
  18. [18] B. Wald and J. Waschbüsch: “Tame biserial algebras”, J. Algebra, Vol. 95, (1985), pp. 480–500. http://dx.doi.org/10.1016/0021-8693(85)90119-X Zbl0567.16017

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