Talk:Cartan decomposition

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Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Please add useful comments here. --Cronholm144 02:34, 22 May 2007 (UTC) One question: for the map given by to be a diffeomorphism, we need to assume that G is connected, don't we? If not, K has to be replaced by the set of those such that Ad(g) commutes with the given Cartan involution Θ; this coincides with the analytic subgroup defined by if G is connected. HannoBecker (talk) 22:42, 24 February 2009 (UTC)

There is a mistake in first given example. The Killing form, evaluated at (X,Y) is being defined as the trace of [ad(X),ad(Y)]. It contains in its kernel the center of gl(n,R). In particular, each non zero element in the center (assume here n is at least 1) will also be in the kernel of Tr[ad(X),ad(t(X))]. So this last form is not definite for n>0. It is only semidefinite.

Helgason (p. 185) defines Cartan involution (as is done here) only for semisimple algebras, which means not for gl(n,R), if n>0.

I'll replace gl(n,R) with sl(n,R) in this example. —Preceding unsigned comment added by 77.206.130.58 (talk) 17:10, 14 April 2009 (UTC)