I have beard a bit about so-called matrix Lie groups. From what I understand (and I don't understand it well) a matrix Lie group is a closed subgroup of $GL_n(\mathbb{C})$.
There is also the notion of a Lie group. It is something about a smooth manifold of the manifold $M_n(\mathbb{C})$.
I have also hear something saying that all Lie groups are in fact isomorphic to a matrix Lie group. Is this correct? Could someone give me a bit more detail about this? What, for example, is the isomorphism? Is it of abstract groups, manifolds, or …?
Best Answer
As other answers mention, it is not true that any Lie group is a matrix group; counterexamples include the universal cover of $SL_2(\mathbb{R})$ and the metaplectic group.
However it is true that all compact Lie groups are matrix groups, as a consequence of the Peter-Weyl theorem.
It is also true that every finite-dimensional Lie group has a finite-dimensional Lie algebra $\mathfrak{g}$ which is a matrix algebra. (This is Ado's theorem.)
In some sense, the Lie algebra of a Lie group captures "most" of the information about the Lie group. Finite-dimensional Lie algebras are in bijective correspondence with finite-dimensional simply-connected Lie groups. So given an arbitrary Lie group $G$, passing to its Lie algebra amounts to passing to the universal cover of the connected component of the identity $\widetilde{G_1}$. Note though that simply-connected Lie groups are not in general matrix groups; $\widetilde{SL_2(\mathbb{R})}$ is a counterexample.