Indecomposable representations of the Kronecker quivers
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- by Claus Michael Ringel PDF
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Abstract:
Let $k$ be a field and $\Lambda$ the $n$-Kronecker algebra. This is the path algebra of the quiver with $2$ vertices, a source and a sink, and $n$ arrows from the source to the sink. It is well known that the dimension vectors of the indecomposable $\Lambda$-modules are the positive roots of the corresponding Kac-Moody algebra. Thorsten Weist has shown that for every positive root there are tree modules with this dimension vector and that for every positive imaginary root there are at least $n$ tree modules. Here, we present a short proof of this result. The considerations used also provide a calculation-free proof that all exceptional modules over the path algebra of a finite quiver are tree modules.References
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Additional Information
- Claus Michael Ringel
- Affiliation: Fakultät für Mathematik, Universität Bielefeld, P. O. Box 100 131, D-33 501 Bielefeld, Germany – and – King Abdulaziz University, P. O. Box 80200, Jeddah, Saudi Arabia
- MR Author ID: 148450
- Email: ringel@math.uni-bielefeld.de
- Received by editor(s): September 28, 2010
- Received by editor(s) in revised form: October 4, 2010, March 29, 2011, and June 10, 2011
- Published electronically: May 11, 2012
- Communicated by: Birge Huisgen-Zimmermann
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 115-121
- MSC (2010): Primary 16G20; Secondary 05C05, 11B39, 15A22, 16G60, 17B67, 65F50
- DOI: https://doi.org/10.1090/S0002-9939-2012-11296-1
- MathSciNet review: 2988715