Available in electronic format
Available in print format		 Mathematics of Computation
Journal of the American Mathematical Society
ISSN 1088-6842(e) ISSN 0025-5718(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

An algorithm for finding all preprojective components of the Auslander-Reiten quiver

Author(s): Peter Dräxler; Klara Kögerler.
Journal: Math. Comp. 71 (2002), 743-759.
MSC (2000): Primary 16G20, 16G70; Secondary 05C38, 05E99
Posted: December 21, 2001
Retrieve article in: PDF DVI PostScript
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: The Auslander-Reiten quiver of a finite-dimensional associative algebra $A$encodes information about the indecomposable finite-dimensional representations of $A$and their homomorphisms. A component of the Auslander-Reiten quiver is called preprojective if it does not admit oriented cycles and each of its modules can be shifted into a projective module using the Auslander-Reiten translation. Preprojective components play an important role in the present research on algebras of finite and tame representation type. We present an algorithm which detects all preprojective components of a given algebra.

References:

[AR]
M. Auslander, I. Reiten, Representation theory of Artin algebras III, Commun. Algebra 3 (1975), 239-294. MR 52:504

[ARS]
M. Auslander, I. Reiten, S.O. Smalø, Representation theory of Artin algebras, Cambridge, 1995. MR 96c:16015

[BL]
R. Bautista, F. Larrión, Auslander-Reiten quivers for certain algebras of finite representation type, J. London Math. Soc. 26 (1982), 43-52. MR 83k:16014

[BGRS]
R. Bautista, P. Gabriel, A.V. Roiter, L. Salmerón, Representation-finite algebras and multiplicative bases, Invent. Math. 81 (1985), 217-285. MR 87g:16031

[Bo]
K. Bongartz, A criterion for finite representation type, Math Ann. 269 (1984), 1-12. MR 86k:16023

[BG]
K. Bongartz, P. Gabriel, Covering spaces in representation theory, Invent. Math. 65 (1982), 331-378. MR 84i:16030

[DR]
V. Dlab, C.M. Ringel, Indecomposable representations of graphs and algebras, Mem. Amer. Math. Soc. 173 (1973). MR 56:5657

[DP]
P. Dräxler, J.A. de la Peña, One the existence of postprojective components in the Auslander-Reiten quiver of an algebra, Tsukuba J. Math. 20 (2) (1996), 457-469. MR 98a:16020

[Ga1]
P. Gabriel, Unzerlegbare Darstellungen I, Manuscr. Math. 6 (1972), 71-103. MR 48:11212

[Ga2]
P. Gabriel, Auslander-Reiten sequences and representation-finite algebras, Lecture Notes in Math. 831 (1980), 1-71. MR 82i:16030

[GR]
P. Gabriel, A.V. Roiter, Representations of finite-dimensional algebras, Encyclopedia of the Mathematical Sciences, Vol. 73, Algebra VIII, A.I. Kostrikin and I.V. Shafarevich (Eds.), Berlin, Heidelberg, New York, 1992, pp. 1-177. MR 94h:16001b

[Ha]
D. Happel, Composition factors for indecomposable modules, Proc. Amer. Math. Soc. 86 (1982), 29-31. MR 84i:16031

[HR]
D. Happel, C.M. Ringel, Directing projective modules, Arch. Math. 60 (1993), 237-246. MR 94b:16016

[KP]
S. Kasjan, J.A. de la Peña, Constructing the preprojective components of an algebra, J. Algebra 179 (1996), 793-807. MR 97c:16015

[Li]
S. Liu, Shapes of connected components of the Auslander-Reiten quivers of Artin algebras, Representation theory of algebras and related topics (Mexico City, 1994), 109-137, CMS Conf. Proc., 19, Amer. Math. Soc., Providence, RI, 1996. MR 97e:16037

[Ri]
C.M. Ringel, Tame algebras and integral quadratic forms, Springer LNM 1099 (1984). MR 87f:16027

[Sc]
T. Scheuer, The canonical decomposition of the poset of a hammock, J. London Math. Soc. (2) 49 (1994), 232-243. MR 95a:16017

Similar Articles:

Retrieve articles in Mathematics of Computation with MSC (2000): 16G20, 16G70, 05C38, 05E99

Retrieve articles in all Journals with MSC (2000): 16G20, 16G70, 05C38, 05E99

Additional Information:

Peter Dräxler
Affiliation: Fakultät für Mathematik, Universität Bielefeld, P.O. Box 100131, D-33501 Bielefeld, Germany

Klara Kögerler
Affiliation: Fakultät für Mathematik, Universität Bielefeld, P.O. Box 100131, D-33501 Bielefeld, Germany

DOI: 10.1090/S0025-5718-01-01404-1
PII: S 0025-5718(01)01404-1
Keywords: Representations of algebras, Auslander-Reiten quiver, preprojective components
Received by editor(s): April 6, 1999
Received by editor(s) in revised form: July 7, 2000
Posted: December 21, 2001
Copyright of article: Copyright 2001, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google