Can the projective unitary group be realized as a matrix Lie group

Question

Is the projective unitary group $U(n)/U(1)$ isomorphic to some closed subgroup of the $GL(n,\mathbb{C})$ (i.e., a matrix Lie group)?

By the way, as a consequence of the Peter-Weyl theorem, it follows that all compact Lie groups can be realized as matrix groups. — Jason DeVito - on hiatus, Aug 25 '20 at 02:15

score 2 · Answer 1 · answered Aug 25 '20 at 02:10

2

The standard way: Take the Lie algebra of that group. It is finite dimensional. The group acts on it (adjoint action). This gives a representation.

answered Aug 25 '20 at 02:10

markvs

1 Answers1