I am reading Jacques Tits' paper "Homomorphismes `abstraits' de groupes de Lie" and he seems to be making a claim that if you have a simply connected Lie group then the derived subgroup is always closed. I was just wondering if this statement was true or not. When I consulted with my supervisor he seemed to think that that was not true and this made me wonder if I was misreading the text in some way.
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$\begingroup$ See here for a statement which is not true in this context. $\endgroup$– Dietrich BurdeCommented Jan 13, 2015 at 15:53
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This is true. The point is that the derived subgroup is the integral subgroup of the derived Lie algebra, therefore is a Lie subgroup (hence closed) if the group is simply connected. See Bourbaki, Lie Groups and Lie Algebras, ch. III, § 9, no. 2, Corollary of Prop. 4.
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1$\begingroup$ A Lie subgroup in a simply connected Lie group need not be closed, for instance consider a dense line contained in a 2-torus inside the simply connected Lie group $SL_3(\mathbf{C})$. $\endgroup$– YCorCommented Jan 13, 2015 at 16:38
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$\begingroup$ For me a Lie subgroup is a subvariety, hence it is locally closed -- hence closed because of the group structure. $\endgroup$– abxCommented Jan 13, 2015 at 17:48
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$\begingroup$ Ah OK, in Bourbaki they refer to III §6.6 Prop 14, which indeed says that an integral normal subgroup in a simply connected Lie group is closed. (In my previous comment I used "Lie subgroup" in the sense "integral subgroup", namely, in a connected Lie group, the subgroup generated by the exponential of some Lie subalgebra). And the corollary you quote indeed says the derived subgroup in a simply connected Lie group is closed. $\endgroup$– YCorCommented Jan 13, 2015 at 20:52