Note on faithful representations and a local property of Lie groups
HTML articles powered by AMS MathViewer
- by Nazih Nahlus PDF
- Proc. Amer. Math. Soc. 125 (1997), 2767-2769 Request permission
Abstract:
Let $G$ be any analytic group, let $T$ be a maximal toroid of the radical of $G$, and let $S$ be a maximal semisimple analytic subgroup of $G$. If $L=\mathcal {L}(G)$ is the Lie algebra of $G$, $\mathrm {rad}[L,L]$ is the radical of $[L,L]$, and $\mathcal {Z}(L)$ is the center of $L$, we show that $G$ has a faithful representation if and only if (i) $\mathrm {rad}[L,L]\cap \mathcal {Z}(L)\cap \mathcal {L}(T)=(0)$, and (ii) $S$ has a faithful representation.References
- Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 1–3, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French; Reprint of the 1975 edition. MR 979493
- G. Hochschild, The structure of Lie groups, Holden-Day, Inc., San Francisco-London-Amsterdam, 1965. MR 0207883
- G. Hochschild and G. D. Mostow, On the algebra of representative functions of an analytic group, Amer. J. Math. 83 (1961), 111–136. MR 141732, DOI 10.2307/2372724
- Martin Moskowitz, Faithful representations and a local property of Lie groups, Math. Z. 143 (1975), no. 2, 193–198. MR 374342, DOI 10.1007/BF01187063
Additional Information
- Nazih Nahlus
- Affiliation: Department of Mathematics, American University of Beirut, c/o New York Office, 850 Third Ave., 18th floor, New York, New York 10022
- Email: nahlus@layla.aub.edu.lb
- Received by editor(s): October 26, 1995
- Received by editor(s) in revised form: March 29, 1996
- Communicated by: Roe Goodman
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 2767-2769
- MSC (1991): Primary 22E15, 22E60
- DOI: https://doi.org/10.1090/S0002-9939-97-03893-8
- MathSciNet review: 1396990