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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The cohomology of semisimple Lie algebras with coefficients in a Verma module

Author: Floyd L. Williams
Journal: Trans. Amer. Math. Soc. 240 (1978), 115-127
MSC: Primary 17B10
MathSciNet review: 0486012
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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.

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Keywords: Complex semisimple Lie algebra, Lie algebra cohomology, Verma module, highest weight, spectral sequence
Article copyright: © Copyright 1978 American Mathematical Society