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 $V\_2\otimes V\_3$.



Simple quotients of a triple tensor product
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 $P \otimes V\_3 \to Q$.


