Abstract
Given a continous representation of a Lie group in a Banach space we study its 1-cohomology. We prove that the computation of the 1-cocycles can be reduced to that of the 1-cocycles of the differential of the representation. When the group is semi-simple and the representation is K-finite, we prove that the cohomology is equivalent to the cohomology of the Lie algebra representation on K-finite vectors. We prove, using Casimir operators, that there exist only a finite number of irreducible representation of a semi-simple Lie group with a non-trivial cohomology. Exemples of such representations are given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
ArakiH., Publ. Res. Inst. Math. Sci. Kyoto Univ. 5, 361 (1970).
DixmierJ., Les C *-algèbres et leurs représentations, Gauthier-Villars, Paris, 1964.
GardingL., Bull. Soc. Math. France, 88, 73 (1960).
Harish-Chandra, Amer. J. Math. 77, 753 (1955).
HattoriA., J. Math. Soc. Japan 16, 226 (1964).
Kisynski, J., On the Integrability of Banach Space Representations of Lie Algebras, Universitet Warsawski, Institut Matematyki, 1973.
SimonJ., Comm. Math. Phys. 28, 39 (1972).
WarnerG., Harmonic Analysis on Semi-Simple Lie Groups I, Springer-Verlag, Berlin, 1972.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pinczon, G., Simon, J. On the 1-cohomology of Lie groups. Lett Math Phys 1, 83–91 (1975). https://doi.org/10.1007/BF00405591
Issue Date:
DOI: https://doi.org/10.1007/BF00405591