Lie ideals of $gl_n(K)$

Question

I am looking for some reference where I can find a detailed study of the Lie ideals of the general linear Lie algebra $gl_n(K)$ with the bracket $[A,B]=AB-BA$, where $K$ is a field (if there are restrictions on $K$, I do not mind). Thanks a lot.

Answer 1

You can find this in the classical books on Lie algebras, e.g. N. Jacobson's book. Since $\mathfrak{gl}_n(K)\simeq \mathfrak{sl}_n(K)\oplus K$ is a reductive Lie algebra, and $\mathfrak{sl}_n(K)$ is simple (if the characteristic of $K$ is zero, or not dividing $n$), we know all the ideals. Recall that a simple Lie algebra has only the trivial ideals. The centre is isomorphic to $K$, and this is an ideal.