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Transactions of the American Mathematical Society

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



Invariant-free representations of augmented rings

Author: Peter M. Curran
Journal: Trans. Amer. Math. Soc. 230 (1977), 313-319
MSC: Primary 16A62; Secondary 18H15
MathSciNet review: 0444710
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Abstract: Let $ \Gamma $ be an augmented ring in the sense of Cartan-Eilenberg, and let there be given a representation of $ \Gamma $ in $ {\text{End}_k}\;A$, where A is a finite dimensional vector space over the field k. We show that all cohomology of $ \Gamma $ in A is trivial if there are no invariants in A under the action of a suitable commutative subring of $ \Gamma $. This generalizes a previous result of the author for group cohomology, and is applied to obtain sufficient conditions for the vanishing of the cohomology of Lie algebras and associative algebras.

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Keywords: Augmented ring, cohomology of Lie algebras, cohomology of associative algebras, extensions of Lie algebras, extensions of associative algebras
Article copyright: © Copyright 1977 American Mathematical Society

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