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On page 13 of this PDF the existence of KAN decompositions of semisimple Lie groups is proven. The proof uses the fact that the matrix multiplication map is regular to conclude that the image of $(K,A,N)\mapsto KAN$ is an open set. I believe that this is using facts about the Zariski topology to conclude Zariski-openness. How does one prove that the image is Zariski-open?

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I'm guessing that the dimensionality is key to showing Zariski-openness. A Zariski closed subset $S$ of a variety $V$ has the same dimensionality as $V$ iff $S$ is... also Zariski open?

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Actually, no that's wrong. I think maybe the detour to proving Zariski openness is unnecessary and wrong. Actually, the image is Zariski closed, but also has the same dimensionality as the codomain, and therefore by facts about Zariski topology the image is the whole space. Talking about Zariski openness is totally unnecessary here.

wlad
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  • The whole source reads as if $G$ is a real Lie group, and the topology they use is the one coming from the Lie structure, which should be much finer than the Zariski topology. It talks about diffeomorphisms and compact subgroups and simply-connected and all that jazz. I don't think Zariski comes ever into play here. By the way, the lemma 6.44 which is used on the page you quote is absent from the file, not good. – Torsten Schoeneberg Nov 17 '22 at 18:09
  • I mean, yes it mentions regular maps, which would work even for the Zariski topology, but aren't regular maps open even for the manifold topology as long as the dimensions match? That seems to be what they are using there. – Torsten Schoeneberg Nov 17 '22 at 18:15
  • @TorstenSchoeneberg Are you aware of a reference to a complete proof of the existence of the KAN decomposition for semisimple Lie groups? – wlad Nov 20 '22 at 09:48
  • I would be surprised if the standard sources (https://math.stackexchange.com/q/194419/96384, https://math.stackexchange.com/q/832624/96384, https://math.stackexchange.com/q/461029/96384) did not contain proofs of what is often called the Iwasawa decomposition. Apparently, Knapp uses it, so might have proven it: https://math.stackexchange.com/a/3072648/96384 – Torsten Schoeneberg Nov 20 '22 at 17:20

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