Are fundamental representations of Lie algebras faithful? Let $L$ be a semisimple algebra, $\alpha_1, \cdots, \alpha_l$ a base of the root system $\Phi$,  $\omega_1, \cdots, \omega_l$ the dual basis relative to the inner product (such that
$ \langle \omega_i, \alpha_j \rangle = \delta_{ij}$). Is the irreducible representation $L(\omega_k)$ of highest weight $\omega_k$ faithful?

To summarize the discussion in the comments: this is true iff $L$ is simple. Writing $L$ as the direct sum of its simple factors $L_i$, the simple roots of $L$ are the disjoint union of the simple roots of each $L_i$, and similarly for the fundamental weights. So the fundamental representations of $L$ are the fundamental representations of each $L_i$, and in particular they are faithful representations of exactly one of the simple factors of $L$ at a time. 
