Trivial representation of a lie Algebra?

Question

Can someone explain why if $\rho:L \rightarrow \text{End}(\mathbb{C})$ is a Lie algebra representation of a semisimple Lie algebra $L$ then it must be that $\rho(x)=0\ \forall \ x\in L$.

Answer 1

The main observation to make is that if $\rho: \mathfrak{g} \to \mathfrak{h}$ is a homomorphism of lie algebras then $\rho([\mathfrak{g},\mathfrak{g}]) \subset [\mathfrak{h},\mathfrak{h}]$. In your case $\mathfrak{g}=L$ is semisimple, so $[\mathfrak{g},\mathfrak{g}]=\mathfrak{g}$, and $\mathfrak{h} = \mathbb{C}$, so $[\mathfrak{h},\mathfrak{h}]=0$. Hence $\rho(\mathfrak{g}) \subset 0$.