TITLE: Simple quotients of a triple tensor product
QUESTION [5 upvotes]: 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 $V_2\otimes V_3$.

REPLY [7 votes]: 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 $P \otimes V_3 \to Q$.