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Representations of the generalized Kronecker quiver with countably many arrows


Author: Nils Mahrt
Journal: Proc. Amer. Math. Soc. 137 (2009), 815-824
MSC (2000): Primary 16G20; Secondary 16G70
DOI: https://doi.org/10.1090/S0002-9939-08-09552-X
Published electronically: September 10, 2008
MathSciNet review: 2457419
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Abstract: Let $ Q$ be the generalized Kronecker quiver with countably many arrows and let $ k$ be a field. We prove that the category of representations of $ Q$ over $ k$ has no right almost split morphism whose domain is projective. More precisely, we show that any indecomposable non-projective representation is the image of an epimorphism whose domain has no non-zero projective direct summand. This result does not hold for any finite subquiver of $ Q$.


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  • [ASS06] Ibrahim Assem, Daniel Simson, and Andrzej Skowroński, Elements of the representation theory of associative algebras. Vol. 1, London Mathematical Society Student Texts, vol. 65, Techniques of representation theory, Cambridge University Press, Cambridge, 2006. MR 2197389 (2006j:16020)
  • [AS80] M. Auslander and Sverre O. Smalø, Preprojective modules over Artin algebras, J. Algebra 66 (1980), no. 1, 61-122. MR 591246 (83a:16039)
  • [Ben98] D. J. Benson, Representations and cohomology. I, second ed., Basic representation theory of finite groups and associative algebras, Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1998. MR 1644252 (99f:20001a)
  • [CDT97] Riccardo Colpi, Gabriella D'Este, and Alberto Tonolo, Quasi-tilting modules and counter equivalences, J. Algebra 191 (1997), no. 2, 461-494. MR 1448804 (98g:16003)
  • [D'E00] Gabriella D'Este, Free modules obtained by means of infinite direct products, Algebra and its applications (Athens, OH, 1999), Contemp. Math., vol. 259, Amer. Math. Soc., Providence, RI, 2000, pp. 161-173. MR 1778499 (2001g:16006)
  • [EE05] Edgar Enochs and Sergio Estrada, Projective representations of quivers, Comm. Algebra 33 (2005), no. 10, 3467-3478. MR 2175445 (2007b:16035)
  • [EOT04] Edgar Enochs, Luis Oyonarte, and Blas Torrecillas, Flat covers and flat representations of quivers, Comm. Algebra 32 (2004), no. 4, 1319-1338. MR 2100360 (2006d:16024)
  • [HU91] Dieter Happel and Luise Unger, A family of infinite-dimensional non-self-extending bricks for wild hereditary algebras, Representations of finite-dimensional algebras (Tsukuba, 1990), CMS Conf. Proc., vol. 11, Amer. Math. Soc., Providence, RI, 1991, pp. 181-189. MR 1143851 (93c:16014)
  • [Hov01] Mark Hovey, Classifying subcategories of modules, Trans. Amer. Math. Soc. 353 (2001), no. 8, 3181-3191 (electronic). MR 1828603 (2002i:13007)
  • [Mah06] Nils Mahrt, Darstellungen des verallgemeinerten Kronecker-Köchers mit abzählbar vielen Pfeilen, Diplomarbeit, Fakultät für Mathematik, Universität Bielefeld, July 2006.
  • [Rin76] Claus Michael Ringel, Representations of $ K$-species and bimodules, J. Algebra 41 (1976), no. 2, 269-302. MR 0422350 (54:10340)

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Additional Information

Nils Mahrt
Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany
Email: nmahrt@math.uni-bielefeld.de

DOI: https://doi.org/10.1090/S0002-9939-08-09552-X
Keywords: Infinite generalized Kronecker quiver, almost split morphism
Received by editor(s): November 9, 2006
Received by editor(s) in revised form: May 23, 2007, August 31, 2007, December 20, 2007, and February 18, 2008
Published electronically: September 10, 2008
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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