Isomorphism of $\mathfrak{gl}_n(K)/\mathfrak{sl}_n(K)$ to $K$.

Question

Let $\mathfrak{gl}_n(K)$ be the general linear Lie algebra over a field $K$ and $\mathfrak{sl}_n(K)$ the special linear Lie algebra over $K$.

I have to show that the quotient $\mathfrak{gl}_n(K)/\mathfrak{sl}_n(K)$ is isomorphic to the onedimensional abelian Lie algebra.

I know that this abelian Lie algebra is isomorphic to $K$ itself, so I have to find an isomorphism of the quotient to $K$.

I think it has something to do with the trace $$tr:\mathfrak{gl}_n(K)/\mathfrak{sl}_n(K)\rightarrow K,$$

but I can't see if this is an isomorphism.

Any help or advice is very much appreciated!

Answer 1

The kernel of $\operatorname{tr}\colon\mathfrak{gl}_n(K)\longrightarrow K$ is $\mathfrak{sl}_n(K)$ and its range is $K$. Therefore, it induces a Lie algebra isomorphism from $\mathfrak{gl}_n(K)$ onto $K$.