Classification Lie subgroups of $\textrm{Gl}(n;\mathbb R)$

Question

I just wonder are there any Lie subgroups of $\textrm{Gl}(n;\mathbb R)$ besides $\textrm{Sl}(n;\mathbb R)$, $\textrm{O}(n;\mathbb R)$ and $\textrm{SO}(n;\mathbb R)$, and is there any classification of them?

Answer 1

Sure. Let uss see:

• the diagonal matrices with non-zero determinant;
• the upper (or lower) trianguar matrices with non-zero determinant;
• the Heisenberg group;
• $\mathrm{SU}(n,\mathbb{C})$ can be seen as a subgroup of $\mathrm{GL}(2n,\mathbb{R})$;
• the symplectic group can be seen as a subgroup of $\mathrm{GL}(4n,\mathbb{R})$

And so on... Actually, it is hard (although not impossible) to find a Lie group wich is not a subgroup of some $\mathrm{GL}(n,\mathbb{R})$. And, no, there is no classification.