Note on faithful representations and a local property of Lie groups |
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| Authors: | Nazih Nahlus |
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| Institution: | Department of Mathematics, American University of Beirut, c/o New York Office, 850 Third Ave., 18th floor, New York, New York 10022 |
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| Abstract: | Let  be any analytic group, let  be a maximal toroid of the radical of  , and let  be a maximal semisimple analytic subgroup of  . If  is the Lie algebra of  , ![$\mathrm {rad}L,L]$](http://www.ams.org/proc/1997-125-09/S0002-9939-97-03893-8/gif-abstract/img8.gif) is the radical of ![$L,L]$](http://www.ams.org/proc/1997-125-09/S0002-9939-97-03893-8/gif-abstract/img9.gif) , and  is the center of  , we show that  has a faithful representation if and only if (i) ![$\mathrm {rad}L,L]\cap \mathcal{Z}(L)\cap \mathcal{L}(T)=(0)$](http://www.ams.org/proc/1997-125-09/S0002-9939-97-03893-8/gif-abstract/img13.gif) , and (ii)  has a faithful representation. |
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