Is there a classification of non-compact Lie Groups? I am interested specifically in subgroups of GL(n,R).

Question

I know there is a classification of compact Lie groups, and that the subgroup structure of the compact Lie groups is well understood. What is the state of knowledge about non-compact Lie groups? Are they completely classified? Are all of the subgroups of GL(n,R) classified and understood?

Answer 1

No, non-compact Lie groups are not classified at all. For example, (connected and simply connected) nilpotent Lie groups have only been classified, in terms of nilpotent Lie algebras, up to dimension $7$. For references see here. It is also clear that a classification is impossible in the general case (some people call this "wild", i.e., algebras of wild type, see here). By Ado's theorem, nilpotent Lie algebras can be realised as subalgebras of the Lie algebra of strictly upper-triangular Lie algebras, corresponding to the Lie subgroup of upper unitriangular matrices in $GL(n,\mathbb{R})$.