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The spectral radius of the Coxeter transformations for a generalized Cartan matrix. (English) Zbl 0819.15008
This work concerns the eigenvalues (more precisely, the spectral radius) of the Coxeter transformations \(C = C(A, \pi)\) for a generalized Cartan matrix \(A\) [for such matrices see V. Kac “Infinite dimensional Lie algebras”, 2nd ed. Cambridge Univ. Press (1985; Zbl 0574.17010); 3rd ed. (1990; Zbl 0716.17022)]. The following theorem is proved:
Let \(A\) be a generalized Cartan matrix which is connected and neither of finite nor of affine type. Let \(C\) be a Coxeter transformation for \(A\). Then the spectral radius \(\rho (C) > 1\), and \(\rho (C)\) is an eigenvalue of multiplicity one, whereas any other eigenvalue \(\lambda\) of \(C\) satisfies \(| \lambda | < \rho (C)\).

15A18 Eigenvalues, singular values, and eigenvectors
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
16G20 Representations of quivers and partially ordered sets
Full Text: DOI EuDML
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