Is a non-zero homomorphism f: V ---> W of L-representations V,W surjective if W is simple?

Question

This could be wrong but: If W is simple then V must be simple and if V is simple then its only subrepresentations are 0 and V which means the Ker(F)=0 and thus F must be injective. But how can I show it it is surjective?

Answer 1

$im(f) \subseteq W$ is a subrepresentation. So if $W$ is simple ...

Answer 2

Let $W=sl(2,\mathbb{R})$, take a non zero element $x$ of $W$, $Vect(x)$ is a commutative algebra, and the canonical imbedding is not surjective.

Answer 3

If $V, W$ are both simple L-modules Schur's lemma states that $f$ is either invertible or zero. However, since $f$ is by assumption non-zero, it must be invertible. Therefore, it is also surjective. However, this does not follow from your argumentation. See Torsten's reply for a (much) simpler argument.