In non-semisimple category $M \otimes N \not\cong N \otimes M$?

Question

Let $\mathcal{C}$ be the category of all finite dimensional $U_q(\hat{g})$-modules, where $U_q(\hat{g})$ is an quantum affine algebra. In the category $\mathcal{C}$, usually $M \otimes N \not\cong N \otimes M$ ($M,N$ are two finite dimensional $U_q(\hat{g})$-modules)? Thank you very much.

Answer 1

Yes, in general $M\otimes N$ is not isomorphic to $N\otimes M$ in $\mathcal{C}$, because the category $\mathcal{C}$ is not braided - see the answers here.