Invariant-free representations of augmented rings

Curran, Peter M.

doi:10.1090/S0002-9947-1977-0444710-0

Invariant-free representations of augmented rings
HTML articles powered by AMS MathViewer

by Peter M. Curran PDF

Trans. Amer. Math. Soc. 230 (1977), 313-319 Request permission

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

Donald W. Barnes, On the cohomology of soluble Lie algebras, Math. Z. 101 (1967), 343–349. MR 220784, DOI 10.1007/BF01109799
Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
P. M. Curran, Fixed-point-free actions on a class of abelian groups, Proc. Amer. Math. Soc. 57 (1976), no. 2, 189–193. MR 414739, DOI 10.1090/S0002-9939-1976-0414739-1
J. Dixmier, Cohomologie des algèbres de Lie nilpotentes, Acta Sci. Math. (Szeged) 16 (1955), 246–250 (French). MR 74780
G. Hochschild, Cohomology and representations of associative algebras, Duke Math. J. 14 (1947), 921–948. MR 22842
G. Hochschild, Cohomology of restricted Lie algebras, Amer. J. Math. 76 (1954), 555–580. MR 63361, DOI 10.2307/2372701
G. Hochschild, Lie algebra kernels and cohomology, Amer. J. Math. 76 (1954), 698–716. MR 63362, DOI 10.2307/2372712
Mitsuya Mori, On the three-dimensional cohomology group of Lie algebras, J. Math. Soc. Japan 5 (1953), 171–183. MR 57854, DOI 10.2969/jmsj/00520171