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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The cohomology of semisimple Lie algebras with coefficients in a Verma module
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by Floyd L. Williams PDF
Trans. Amer. Math. Soc. 240 (1978), 115-127 Request permission

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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Additional Information
  • © Copyright 1978 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 240 (1978), 115-127
  • MSC: Primary 17B10
  • DOI: https://doi.org/10.1090/S0002-9947-1978-0486012-3
  • MathSciNet review: 0486012