The invariant bilinear form and the generalized Casimir operator
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
- Kac, V. G. [1968 B] Simple irreducible graded Lie algebras of finite growth, Izvestija AN USSR (ser. mat.) 32 (1968), 1923–1967.Google Scholar
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- Kac, V. G. [ 1983 B] Laplace operators of infinite-dimensional Lie algebras and theta functions, Proc. Nat’l. Acad. Sci. USA (1983).Google Scholar