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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Note on faithful representations
and a local property of Lie groups

Author: Nazih Nahlus
Journal: Proc. Amer. Math. Soc. 125 (1997), 2767-2769
MSC (1991): Primary 22E15, 22E60
MathSciNet review: 1396990
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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.

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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

Received by editor(s): October 26, 1995
Received by editor(s) in revised form: March 29, 1996
Communicated by: Roe Goodman
Article copyright: © Copyright 1997 American Mathematical Society