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Indecomposable representations of the Kronecker quivers

Author: Claus Michael Ringel
Journal: Proc. Amer. Math. Soc. 141 (2013), 115-121
MSC (2010): Primary 16G20; Secondary 05C05, 11B39, 15A22, 16G60, 17B67, 65F50
Published electronically: May 11, 2012
MathSciNet review: 2988715
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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.

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Additional Information

Claus Michael Ringel
Affiliation: Fakultät für Mathematik, Universität Bielefeld, P. O. Box 100131, D-33501 Bielefeld, Germany – and – King Abdulaziz University, P. O. Box 80200, Jeddah, Saudi Arabia

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
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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