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- W2003959897 abstract "The structure of the cohomology of a complex semisimple Lie algebra with coefficients in an arbitrary Verma module is completely determined. Because the Verma modules are infinite-dimensional, the cohomology need not vanish (as it does for nontrivial finite-dimensional modules). The methods presented exploit the homological machinery of Cartan-Eilenberg (3). The results of (3), when applied to the universal enveloping algebra of a semisimple Lie algebra and when coupled with key results of Kostant (12), Hochschild-Serre (9), yield the basic structure theorem-Theorem 4.19. Our results show, incidently, that an assertion of H. Kimura, Theorem 2 of (13) is false. A counterexample is presented in ?6. 1. Introduction. Let g be a 'finite-dimensional complex semisimple Lie algebra, let b be a Borel subalgebra of g, and let Ug, Ub denote the universal algebras of g, b respectively. If W is a finite-dimensional irreducible left Ub module (W is one-dimensional by Lie's theorem) then U4 ?Ub W is a left Ug module. Here we regard Ug as a (Ug, Ub) bimodule; i.e. Ug is a left Ug, right Ub module with commuting action because of the associative law. The induced modules Ug ?Ub W, where W varies, are called Verma modules (1), (14), (15). They enjoy many pleasant properties and are beginning to play an increasingly important role in the unitary representation theory of semi- simple Lie groups. In this paper we give the structure of the cohomology Hk (g, Ug ? Ub W) of g with coefficients in Ug ? ub W for all k and for all W. The main result is Theorem 4.19. We compute the cohomology of the dual module" @default.
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- W2003959897 date "1978-01-01" @default.
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- W2003959897 title "The cohomology of semisimple Lie algebras with coefficients in a Verma module" @default.
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- W2003959897 doi "https://doi.org/10.1090/s0002-9947-1978-0486012-3" @default.
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