This probably has an easy answer, but is there any sense in which the following statement is true:
Let $U$ be a quantum enveloping algebra with universal $R$-matrix denoted by $R$, then $R$ is unique.
So I am wondering if (1) there is a suitable definition of this uniqueness and (2) where I can find a proof of this statement.
R matrices are not unique. See Chari's book for a classification of equivalence.