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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
DOI: https://doi.org/10.1090/S0002-9947-1977-0444710-0
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.


References [Enhancements On Off] (What's this?)

  • [1] D. W. Barnes, On the cohomology of soluble Lie algebras, Math. Z. 101 (1967), 343-349. MR 36 #3836. MR 0220784 (36:3836)
  • [2] H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, N.J., 1956. MR 17, 1040. MR 0077480 (17:1040e)
  • [3] P. M. Curran, Fixed-point-free actions on a class of abelian groups, Proc. Amer. Math. Soc. 57 (1976), 189-193. MR 0414739 (54:2834)
  • [4] J. Dixmier, Cohomologie des algebres de Lie nilpotentes, Acta Sci. Math. (Szeged) 16 (1955), 246-250. MR 17, 645. MR 0074780 (17:645b)
  • [5] G. Hochschild, Cohomology and representations of associative algebras, Duke Math. J. 14 (1947), 921-948. MR 9, 267. MR 0022842 (9:267b)
  • [6] -, Cohomology of restricted Lie algebras, Amer. J. Math. 76 (1954), 555-580. MR 16, 109. MR 0063361 (16:109a)
  • [7] -, Lie algebra kernels and cohomology, Amer. J. Math. 76 (1954), 698-716. MR 16, 109. MR 0063362 (16:109b)
  • [8] M. Mori, On the three-dimensional cohomology group of Lie algebras, J. Math. Soc. Japan 5 (1953), 171-183. MR 15, 282. MR 0057854 (15:282e)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1977-0444710-0
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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