The invariant bilinear form and the generalized Casimir operator

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


In this chapter we introduce two important tools of our theory, the invariant bilinear form and the generalized Casimir operator Ω. The operator Ω is a “second order” operator which, in contrast to finite-dimensional theory, does not lie in the universal enveloping algebra of g(A) and is not defined for all representations. However, Ω is defined on the so-called restricted representations, and commutes with the action of g(A) in these representations. Remarkably, one can manage to prove a number of results (including classical ones) using only Ω.


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Bibliographical notes and comments

  1. Kac, V. G. [1968 B] Simple irreducible graded Lie algebras of finite growth, Izvestija AN USSR (ser. mat.) 32 (1968), 1923–1967.Google Scholar
  2. Moody, R. V. [ 1968 ] A new class of Lie algebras, J. Algebra 10 (1968), 211–230.MathSciNetCrossRefGoogle Scholar
  3. Kac, V. G. [ 1974 ] Infinite-dimensional Lie algebras and Dedekind’s n-function, English translation: Funct. Anal. Appl. 8 (1974), 68–70.zbMATHGoogle Scholar
  4. Kac, V. G., Peterson, D. H. [ 1983 C] Unitary structure in representations of infinite-dimensional groups and a convexity theorem, MIT, preprint.Google Scholar
  5. Kac, V. G. [ 1983 B] Laplace operators of infinite-dimensional Lie algebras and theta functions, Proc. Nat’l. Acad. Sci. USA (1983).Google Scholar
  6. Feigin, B. L., Fuchs, D. B. [1983 B] Verma modul over the Virasoro algebra, Funkt. analys i ego prilozh. 17 (1983), No. 3, 91–92 (in Russian).zbMATHGoogle 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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