Basic definitions

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


The central object of our study is a certain class of infinite-dimensional Lie algebras alternatively known as contragredient Lie algebras, generalized Cartan matrix Lie algebras or Kac-Moody algebras. Their definition is a rather straightforward “infinite-dimensional” generalization of the definition of semisimple Lie algebras via the Cartan matrix and Chevalley generators. The slight technical difficulty that occurs in the case det A = 0 is handled by introducing the “realization” in the “Cartan subalgebra” h. The Lie algebra o(A) is then a quotient of the Lie algebra õ(A) with generators e i , f i and h, and defining relations (1.2.1), by the maximal ideal intersecting h trivially. Some of the advantages of this definition as compared to the one given in the introduction, as we will see, are as follows: the definition of roots and weights is natural; the Weyl group acts on a nice convex cone; the characters have a nice region of convergence.


Cartan Matrix Formal Topology Chevalley Generator Generalize Cartan Matrix Root Space Decomposition 
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  1. Kac, V. G. [1967] Simple graded Lie algebras of finite growth, Funkt. analys i ego prilozh. 1 (1967), No. 4, 82–83.Google Scholar
  2. Kac, V. G. [ 1968 A] Graded Lie algebras and symmetric spaces, Funkt. analys. i ego prilozh. 2 (1968), No. 2, 93–94Google Scholar
  3. Kac, V. G. [ 1968 B] Simple irreducible graded Lie algebras of finite growth, Izvestija AN USSR (ser. mat.) 32 (1968), 1923–1967.Google Scholar
  4. Moody, R. V. [ 1967 ] Lie algebras associated with generalized Cartan matrices, Bull. Amer. Math. Soc., 73 (1967), 217–221.MathSciNetzbMATHCrossRefGoogle Scholar
  5. Moody, R. V. [ 1968 ] A new class of Lie algebras, J. Algebra 10 (1968), 211–230.MathSciNetCrossRefGoogle Scholar
  6. Moody, R. V. [ 1969 ] Euclidean Lie algebras, Canad. J. Math. 21 (1969), 1432–1454.MathSciNetzbMATHGoogle Scholar
  7. Vinberg, E. B. [ 1971 ] Discrete linear groups generated by reflections, Izvestija AN USSR (ser. mat.) 35 (1971), 1072–1112.MathSciNetzbMATHGoogle Scholar
  8. Chevalley, C. [ 1948 ] Sur la classification des algèbres de Lie simples et de leur representations, C. R., 227 (1948), 1136–1138.MathSciNetzbMATHGoogle Scholar
  9. Harish-Chandra [ 1951 ] On some applications of the universal enveloping algebra of a semi-simple Lie algebra, Trans. Amer. Math. Soc., 70 (1951), 28–96.MathSciNetzbMATHCrossRefGoogle Scholar
  10. Jacobson, N. [ 1962 ] Lie algebras, Interscience, New York, 1962.zbMATHGoogle Scholar
  11. Kantor, I. L. [ 1968 ] Infinite dimensional simple graded Lie algebras, Doklady AN SSR 179 (1968), 534–537.MathSciNetGoogle Scholar
  12. Kantor, I. L. [ 1970 ] Graded Lie algebras, Trudy sem. Vect. Tens. Anal. 15 (1970), 227–266 (in Russian).MathSciNetGoogle Scholar
  13. Weisfeiler, B. Ju., Kac, V. G. [ 1971 ] Exponentials in Lie algebras of characteristic p. English translation: Math. USSR-Izvestija 5 (1971), 777–803.CrossRefGoogle Scholar
  14. Berman, S. [ 1976 ] On derivations of Lie algebras, Canad. J. Math. 27 (1976), 174–180.Google Scholar
  15. Kac, V. G., Peterson, D. H. [ 1983 C] Unitary structure in representations of infinite-dimensional groups and a convexity theorem, MIT, preprint.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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