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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Eigenvalues of Coxeter transformations and the Gelfand-Kirillov dimension of the preprojective algebras


Authors: Vlastimil Dlab and Claus Michael Ringel
Journal: Proc. Amer. Math. Soc. 83 (1981), 228-232
MSC: Primary 15A18; Secondary 15A48, 16A46, 16A64
MathSciNet review: 624903
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Abstract: The spectral radius of a Coxeter transformation is shown to be an eigenvalue which can be expressed in terms of lengths of certain positive roots of the corresponding valued graph. This result is used to determine the Gelfand-Kirillov dimension of the preprojective algebras: This dimension is equal to 0, 1 or $ \infty $ according to whether the underlying graph is Dynkin, Euclidean or otherwise.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0624903-6
PII: S 0002-9939(1981)0624903-6
Keywords: Cartan matrix, valued graph, Dynkin diagram, Euclidean diagram, positive root, Coxeter transformation, eigenvalue, spectral radius, Gelfand-Kirillov dimension, preprojective algebra, finite, tame and wild representation types
Article copyright: © Copyright 1981 American Mathematical Society