Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra over $\mathbb{C}$ and let $U_q(\hat{\mathfrak{g}})$ be the corresponding quantum affine algebra (here $q$ is not a root of unity). We know that, if $V$ is a finite-dimensional $U_q(\hat{\mathfrak{g}})$-module, then $W:=V\otimes V^*$ has the trivial simple module as submodule, that is, we have a monomorphism $\mathbb{C}\hookrightarrow W$.
Are there sufficient conditions to ensure that this is the only nontrivial submodule of $W$? Maybe if $W$ is highest-weight... I don't know. Thanks in advance for answers and references. For my purposes, we can assume that $V$ is simple, but some more general answers are also welcome...