One can find in some sources (e.g. Pavel Etingof, Lectures on representation theory and KZ equation, page 91) formula for $R$ matrix (for $\mathfrak{sl}_2$ case)
$$R = q^{\frac{1}{2} h \otimes h} \sum_{n \geq 0} q^{ \frac{n(n-1)}{2} } \frac{(q-q^{-1})^n e^n \otimes f^n}{[n]!} $$
Question. What is $q^{\frac{1}{2}h \otimes h}$ ?
Remark. There is no such thing as $h$ for $U_q ( \mathfrak{sl}_2 )$. There is just $q^h$ (some authors denote it as $K$). Of course you may say that that $K$ is diagonalizable, so there is no problem to find logarithm. I would reply, that logarithm is not defined uniquely. At least, if $q = e^{\hbar}$, then you may replace $h \rightarrow h + \frac{2 \pi i}{\hbar}$. It is important $$e^{\frac{1}{2} \hbar h \otimes h} \neq e^{\frac{1}{2} \hbar (h + \frac{2 \pi i}{\hbar}) \otimes h}$$
You see, it is well defined for representations of even highest weight. It is defined up to sign for representations with odd highest weight. And it is not well defined for Verma module of generic highest weight.