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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.

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

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