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Let $\mathcal{H}$ be a Hopf algebra over $\mathbb{C}$. Let also $V_1, V_2, V_3$ finite-dimensional simple modules over $\mathcal{H}$ and $Q$ be a simple quotient of $V_1\otimes V_2\otimes V_3$. Is it possible to show that one of the following statements is true? Is there any counterexample?

i) $Q$ is a quotient of $N\otimes V_3$, for some simple quotient $N$ of $V_1\otimes V_2$;

ii) $Q$ is a quotient of $V_1\otimes P$, for some simple quotient $P$ of $V_2\otimes V_3$.

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Both of these statements are true (at least if $H$ is semisimple). It suffices to prove the first one. By hypothesis there is a nonzero map $V_1 \otimes V_2 \otimes V_3 \to Q$. It dualizes to a nonzero map $V_1 \otimes V_2 \to Q \otimes V_3^{\ast}$ (I don't know if I need to distinguish between left and right duals here if $H$ isn't cocommutative but I don't think it matters, just whichever dual makes this true), which factors through its image $P \to Q \otimes V_3^{\ast}$. Dualizing again we get a nonzero map $P \otimes V_3 \to Q$.

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  • $\begingroup$ Thank you very much! However, in this case, if $f: V_1\otimes V_2\rightarrow Q\otimes V_3^*$, then $Im(f)=P$ is not necessarily simple. Am I right? $\endgroup$
    – cl4y70n____
    Commented Oct 9, 2020 at 18:00
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    $\begingroup$ @cl4y70n____: ah, you're right, I thought I didn't need semisimplicity as a hypothesis but I forgot that you asked for $Q$ to be simple. I don't know what happens if $H$ isn't semisimple. $\endgroup$
    – Qiaochu Yuan
    Commented Oct 9, 2020 at 18:03

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