Are compact Lie algebras necessarily compact as a set of matrices?

Question

I'm reading through a paper and came across something confusing; my limited experience with Lie theory is a bit of a hindrance:

The author starts with a compact set of matrices (in the usual Euclidean topology) $\{A_i\}\subset \mathbb{R}^{n \times n}$ and generates a Lie algebra with Levi decomposition $\mathfrak{g} = \mathfrak{s}\oplus\mathfrak{r}$. Under the assumption that $\mathfrak{s}$ is a compact Lie algebra, I'm curious about the compactness of the set $\{e^{-s}r_i e^{s}\,:\,s\in \mathfrak{s}, A_i = s_i + r_i\}$. Clearly the set of $r_i$ is compact (in the usual Euclidean topology), but it seems to me that as a set, $\mathfrak{s}$ could possibly be non-compact (in Euclidean sense) even though it's declared a compact Lie algebra (since in my reading of compact Lie algebras, I haven't come across any of the familiar "closed and bounded" type assertions).

Any help would be greatly appreciated, thanks!

Answer 1

Compact Lie groups are topological groups whose topology is compact. A compact Lie algebra is the Lie algebra of a compact Lie group, therefore the name "compact". Lie algebras are vector spaces, and all finite-dimensional Lie algebras are linear, i.e., subalgebra of the Lie algebra of matrices. For Lie algebras "compact" just means that they are reductive with negative-definite Killing form.

Answer 2

Lie algebras are vector spaces, in this case subspaces of the vector space of matrices. Over the real and complex numbers, vector spaces are never compact unless they are trivial.