A characterization of representation-finite algebras

Andrzej Skowroński; M. Wenderlich

Fundamenta Mathematicae (1991)

  • Volume: 140, Issue: 1, page 31-34
  • ISSN: 0016-2736

Abstract

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Let A be a finite-dimensional, basic, connected algebra over an algebraically closed field. Denote by Γ(A) the Auslander-Reiten quiver of A. We show that A is representation-finite if and only if Γ(A) has at most finitely many vertices lying on oriented cycles and finitely many orbits with respect to the action of the Auslander-Reiten translation.

How to cite

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Skowroński, Andrzej, and Wenderlich, M.. "A characterization of representation-finite algebras." Fundamenta Mathematicae 140.1 (1991): 31-34. <http://eudml.org/doc/211926>.

@article{Skowroński1991,
abstract = {Let A be a finite-dimensional, basic, connected algebra over an algebraically closed field. Denote by Γ(A) the Auslander-Reiten quiver of A. We show that A is representation-finite if and only if Γ(A) has at most finitely many vertices lying on oriented cycles and finitely many orbits with respect to the action of the Auslander-Reiten translation.},
author = {Skowroński, Andrzej, Wenderlich, M.},
journal = {Fundamenta Mathematicae},
keywords = {finite-dimensional, basic, connected algebra; Auslander-Reiten quiver; representation-finite; Auslander-Reiten translation},
language = {eng},
number = {1},
pages = {31-34},
title = {A characterization of representation-finite algebras},
url = {http://eudml.org/doc/211926},
volume = {140},
year = {1991},
}

TY - JOUR
AU - Skowroński, Andrzej
AU - Wenderlich, M.
TI - A characterization of representation-finite algebras
JO - Fundamenta Mathematicae
PY - 1991
VL - 140
IS - 1
SP - 31
EP - 34
AB - Let A be a finite-dimensional, basic, connected algebra over an algebraically closed field. Denote by Γ(A) the Auslander-Reiten quiver of A. We show that A is representation-finite if and only if Γ(A) has at most finitely many vertices lying on oriented cycles and finitely many orbits with respect to the action of the Auslander-Reiten translation.
LA - eng
KW - finite-dimensional, basic, connected algebra; Auslander-Reiten quiver; representation-finite; Auslander-Reiten translation
UR - http://eudml.org/doc/211926
ER -

References

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  1. [1] W. W. Crawley-Boevey, On tame algebras and bocses, Proc. London Math. Soc. 56 (1988), 451-483. Zbl0661.16026
  2. [2] U. Fischbacher, Une nouvelle preuve d'un théorème de Nazarova et Roiter, C. R. Acad. Sci. Paris Sér. I 300 (9) (1985), 259-262. Zbl0586.16012
  3. [3] M. Harada and Y. Sai, On categories of indecomposable modules I, Osaka J. Math. 7 (1970), 323-344. Zbl0248.18018
  4. [4] O. Kerner and A. Skowroński, On module categories with nilpotent infinite radical, Compositio Math. 77 (1991), 313-333. Zbl0717.16012
  5. [5] C. M. Ringel, Finite-dimensional hereditary algebras of wild representation type, Math. Z. 161 (1978), 235-255. Zbl0415.16023
  6. [6] C. M. Ringel, Report on the Brauer-Thrall conjectures, in: Proc. ICRA II (Ottawa 1979), Lecture Notes in Math. 831, Springer, 1980, 104-136. 
  7. [7] C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math. 1099, Springer, 1984. 
  8. [8] A. Skowroński, Algebras of polynomial growth, in: Topics in Algebra, Banach Center Publ. 26, Part 1, PWN, Warszawa 1990, 535-568. 
  9. [9] A. Skowroński and S. O. Smalø, Directing modules, J. Algebra, to appear. Zbl0746.16008

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