Greetings,

Let $G$ be a compact Lie group with a bi-invariant inner product $h$ on it. Can one embedd $G$ in $M(n,\mathbb{C})$ isometrically for some $n \in \mathbb{N}$. By isometrically I mean that the restriction of the standard metric in $M(n,\mathbb{C}) = \mathbb{C}^{n^{2}}$ restricts to $h$ on $G$. Is this possible? If no, why it isnt possible? If yes, how could one prove this?

Dmitri

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. – Robert Bryant Dec 2 '12 at 13:34