Eigenvalues of Coxeter transformations and the Gel′fand-Kirillov dimension of the preprojective algebras
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- by Vlastimil Dlab and Claus Michael Ringel PDF
- Proc. Amer. Math. Soc. 83 (1981), 228-232 Request permission
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.References
- Norbert A’Campo, Sur les valeurs propres de la transformation de Coxeter, Invent. Math. 33 (1976), no. 1, 61–67 (French). MR 424967, DOI 10.1007/BF01425505
- Walter Borho and Hanspeter Kraft, Über die Gelfand-Kirillov-Dimension, Math. Ann. 220 (1976), no. 1, 1–24. MR 412240, DOI 10.1007/BF01354525
- Vlastimil Dlab and Claus Michael Ringel, Indecomposable representations of graphs and algebras, Mem. Amer. Math. Soc. 6 (1976), no. 173, v+57. MR 447344, DOI 10.1090/memo/0173
- Vlastimil Dlab and Claus Michael Ringel, The preprojective algebra of a modulated graph, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979) Lecture Notes in Math., vol. 832, Springer, Berlin-New York, 1980, pp. 216–231. MR 607155
- 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 (French). MR 207918 C. M. Ringel, Algebras of wild representation type, Abstracts Conf. on Representations of Finite Dimensional Algebras, Oberwolfach 1977, pp. 95-102.
- V. F. Subbotin and R. B. Stekol′ščik, The Jordan form of the Coxeter transformation, and applications to representations of finite graphs, Funkcional. Anal. i Priložen. 12 (1978), no. 1, 84–85 (Russian). MR 0498732
- James S. Vandergraft, Spectral properties of matrices which have invariant cones, SIAM J. Appl. Math. 16 (1968), 1208–1222. MR 244284, DOI 10.1137/0116101
- V. G. Kac, Simple irreducible graded Lie algebras of finite growth, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 1323–1367 (Russian). MR 0259961
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 83 (1981), 228-232
- MSC: Primary 15A18; Secondary 15A48, 16A46, 16A64
- DOI: https://doi.org/10.1090/S0002-9939-1981-0624903-6
- MathSciNet review: 624903