Eigenvalues of Coxeter transformations and the Gel′fand-Kirillov dimension of the preprojective algebras

Dlab, Vlastimil; Ringel, Claus Michael

doi:10.1090/S0002-9939-1981-0624903-6

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

Algebras of wild representation type

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