Some properties of generalized Cartan matrices

  • Victor G. Kac
Part of the Progress in Mathematics book series (PM, volume 44)


In order to develop the theory of root systems of Kac-Moody algebras we need to know some properties of generalized Cartan matrices. It is convenient to work in a slightly more general situation. Unless otherwise stated, we will deal with a real n × n matrix A = (a ij ) which satisfies the following three properties:
  1. (m1)

    A is indecomposable;

  2. (m2)

    a ij ≤ 0 for ij;

  3. (m3)

    a ij = 0 implies a ji = 0.



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Bibliographical notes and comments

  1. Vinberg, E. B. [ 1971 ] Discrete linear groups generated by reflections, Izvestija AN USSR (ser. mat.) 35 (1971), 1072–1112.MathSciNetzbMATHGoogle Scholar
  2. Kac, V. G. [1968 B] Simple irreducible graded Lie algebras of finite growth, Izvestija AN USSR (ser. mat.) 32 (1968), 1923–1967.Google Scholar
  3. Moody, R. V. [ 1967 ] Lie algebras associated with generalized Cartan matrices, Bull. Amer. Math. Soc., 73 (1967), 217–221.MathSciNetzbMATHCrossRefGoogle Scholar
  4. Kac, V. G. [ 1969 A] Automorphisms of finite order of semi-simple Lie algebras, English translation: Funct. Anal. Appl. 3 (1969), 252–254.zbMATHGoogle Scholar
  5. Vinberg, E. B., Kac, V. G. [ 1967 ] Quasi-homogeneous cones, Math. Zametki 1 (1967), 347–354MathSciNetGoogle Scholar
  6. Bourbaki, N. [ 1968 ] Groupes et algèbres de Lie, Ch. 4–6, Hermann, Paris, 1968.Google Scholar

Copyright information

© Springer Science+Business Media New York 1983

Authors and Affiliations

  • Victor G. Kac
    • 1
  1. 1.Mathematics DepartmentMassachusetts Institute of TechnologyCambridgeUSA

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