Recall that $U_q(sl_2)$ is the quotient of the free associative $\mathbb{Q}(q)$-algebra on generators $E$, $F$, $K^{\pm 1}$ such that $KE = q^2 EK$, $KF = q^{-2} FK$, and $[E,F] = \frac{K - K^{-1}}{q - q^{-1}}$. It turns out that there is a Hopf algebra structure on $U_q(sl_2)$ such that $\Delta(K) = K \otimes K$, $\Delta(E) = E \otimes K + 1 \otimes E$, and $\Delta(F) = F \otimes 1 + K^{-1} \otimes F$. My question is the following: suppose I only declare that the coproduct must satisfy $\Delta(K) = K \otimes K$. Does this uniquely determine the above formulas for the Hopf algebra structure on $U_q(sl_2)$? In other words, does specifying $\Delta(K)$ uniquely specify $\Delta(E)$ and $\Delta(F)$?
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