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

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.

• What is the monoidal structure here? – Qiaochu Yuan Mar 7 '17 at 19:21
• @Qiaochu Yuan: i guess OP implies the usual tensor product of modules. – KonKan Mar 9 '17 at 23:36
• But what do you mean "usually" ? are you interested in either examples or counterexamples ? – KonKan Mar 9 '17 at 23:37

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.