There is a little question haunted me for few days. I will be grateful to anyone who can give me any clue how to solve it.

Let $V$ be a nontrivial module of $\mathrm{U}_q(\mathfrak{g})$ (the quantum group of adjoint type for a simple finite dimensional Lie algebra $\mathfrak{g}$), $0\neq x\in\mathrm{U}_q(\mathfrak{g})$. Is $x.\mathrm{T}(V)\neq 0$ where $\mathrm{T}(V)$ is a tensor product module?

I only guess it is right. In fact maybe not.