Timeline for Isometric embedding of a compact Lie Group in $M(n,\mathbb{C})$

5 events

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Dec 2, 2012 at 13:34	comment	added	Robert Bryant		The answer to your exact question is 'no'. It's not even possible for most invariant metrics on $G = S^1$, as is easy to see since you know all of its homomorphisms into $M(n,\mathbb{C})$ up to conjugacy. As Peter points out below, in the case $G$ is simple or $S^1$, you can easily get a homomorphism into $M(n,\mathbb{C})$ that is isometric up to a constant multiple; in fact, any homomorphism will do. However, as soon as the group is a nontrivial product, its bi-invariant metrics are not unique up to an overall constant multiple, and so most of them will not admit a homothetic homomorphism.
Dec 2, 2012 at 9:46	answer	added	Peter Michor		timeline score: 6
Dec 2, 2012 at 9:36	comment	added	Dmitri		I mean if this embedding is also a group homomorphism. By Nash it is clear that such an embedding exists. But is it also a grouphomomorphism ?
Dec 2, 2012 at 9:25	comment	added	Amin		Maybe I missed a point, but why does that not trivially follow from Nash ? At what point do you use seriously the group structure here (except for picking a particular metric) ?
Dec 2, 2012 at 8:55	history	asked	Dmitri	CC BY-SA 3.0