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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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  • [1] N. A'Campo. Sur les valeurs propres de la transformation de Coxeter, Invent. Math. 33 (1976), 61-67. MR 0424967 (54:12925)
  • [2] W. Borho und H. Kraft, Über die Gelfand-Kirillov-Dimension, Math. Ann. 220 (1976), 1-24. MR 0412240 (54:367)
  • [3] V. Dlab and C. M. Ringel, Indecomposable representations of graphs and algebras, Mem. Amer. Math. Soc. No. 173 (1976). MR 0447344 (56:5657)
  • [4] -, The preprojective algebra of a modulated graph, Proc. 2nd Internat. Conf. on Representations of Algebras (Ottawa 1979), Lecture Notes in Math., vol. 832, Springer, Berlin and New York, 1980, pp. 216-231. MR 607155 (83c:16022)
  • [5] I. M. Gelfand and A. A. Kirillov, Sur les corps liés aux algèbres enveloppantes des algèbres de Lie, Inst. Hautes Études Sci. Publ. Math. 31 (1966), 5-19. MR 0207918 (34:7731)
  • [6] C. M. Ringel, Algebras of wild representation type, Abstracts Conf. on Representations of Finite Dimensional Algebras, Oberwolfach 1977, pp. 95-102.
  • [7] V. F. Subbotin and R. B. Stekolščik, Jordan form of Coxeter transformations and applications to representations of finite graphs, Funkcional Anal. i Priložen 12 (1978), 84-85 = Functional Appl. 12 (1978), 67-68. MR 0498732 (58:16800)
  • [8] J. S. Vandergraft, Spectral properties of matrices which have invariant cones, SIAM J. Appl. Math. 16 (1968), 1208-1222. MR 0244284 (39:5599)
  • [9] V. G. Kac, Simple irreducible graded Lie algebras of finite growth, Math. USSR Izv. 2 (1968), 1271-1311. MR 0259961 (41:4590)

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Additional Information

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

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