Semi-simple Lie algebra representation of dimension $1$

Question

For any semi-simple Lie algebra $\frak{g}$, defined over a field of characteristic zero, one can conclude from the Weyl character formula that $\frak{g}$ has no non-trivial representations of dimension $1$. Is there a more direct way to prove this?

Answer 1

Let $L$ be a semisimple Lie algebra and $\rho\colon L\rightarrow \mathfrak{gl}(V)$ a finite-dimensional representation. Because of $[L,L]=L$ we have $\rho(L)=[\rho(L),\rho(L)]$, so that all $\rho(x)$ have zero trace, i.e., we have $\rho\colon L\rightarrow \mathfrak{sl}(V)$. But for a $1$-dimensional space $V$ over $K$ we have $\mathfrak{sl}_1(K)=0$. Hence every $1$-dimensional representation is trivial.