Affine Lie algebras: the normalized invariant bilinear form, the root system and the Weyl group

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


The results of Chapter 4 show that a Kac-Moody algebra g(A) is finite-dimensional if and only if all principal minors of A are positive. These Lie algebras are semisimple; moreover, by the classical structure theory, they exhaust all finite-dimensional semisimple Lie algebras. So, the classical Killing-Cartan theory of simple Lie algebras is, in our terminology, the theory of Kac-Moody algebras associated to a matrix of finite type.


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Bibliographical notes and comments

  1. Kac, V. G. [1968B] Simple irreducible graded Lie algebras of finite growth, Izvestija AN USSR (ser. mat.) 32 (1968), 1923-1967. English translation: Math. USSRIzvestija 2 (1968), 1271 - 1311.Google Scholar
  2. Moody, R. V. [ 1969 ] Euclidean Lie algebras, Canad. J. Math. 21 (1969), 1432 - 1454.MathSciNetzbMATHGoogle Scholar
  3. Macdonald, I. G. [1972] Affine root systems and Dedekind’s ri-function, Inventiones Math. 15 (1972), 91 - 143.MathSciNetzbMATHCrossRefGoogle Scholar
  4. Kac, V. G.:[1978A] Infinite-dimensional algebras, Dedekind’s rI-finction, classical Möbious function and the very strange formula, Advances in Math. 30 (1978), 85 - 136.zbMATHCrossRefGoogle Scholar
  5. Kac, V. G., Peterson, D. H. [1983 A] Infinite dimensional Lie algebras, theta functions and modular forms, Advances in Math., 50 (1983).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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