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tried to bring the entry Lie group a bit into shape: added plenty of sections and cross links to other nLab material. But there is still much that deserves to be done.
There is the recent preprint
Linus Kramer, The topology of a simple Lie group is essentially unique, arXiv:1009.5457
Abstract: We study locally compact group topologies on simple Lie groups. We show that the Lie group topology on such a group $S$ is very rigid: every ’abstract’ isomorphism between $S$ and a locally compact and $\sigma$-compact group $\Gamma$ is automatically a homeomorphism, provided that $S$ is absolutely simple. If $S$ is complex, then non-continuous field automorphisms of the complex numbers have to be considered, but that is all.
Not to put pressure on you, Urs. This is as much just a marker for me to put it in later.
Notice that since $\mathbb{R}^n$ as Lie groups are not simple, this doesn’t apply to the example at Lie group regarding number if Lie group structures.
Thanks!
I have now pasted that into the entry.
added this pointer (will also add it at Lie algebra):
A. L. Onishchik (ed.) Lie Groups and Lie Algebras
I. A. L. Onishchik, E. B. Vinberg, Foundations of Lie Theory,
II. V. V. Gorbatsevich, A. L. Onishchik, Lie Transformation Groups
Encyclopaedia of Mathematical Sciences, Volume 20, Springer 1993
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