Quantum Bases in Uq(g)

This paper is devoted to analize inside the infinitely many possible bases of Uq(g), same that can be considered "more equal then others". The element of selection has been a privileged relation with the bialgebra. A new parameter z' has been found that determines the commutation relations, independent from the z=log(q) that defines Uq(g). Both z and z' are necessary to fix the relations between the basic set and its coproducts. Three cases are particularly relevant: the analytical set with z'=z, the Lie set with Lie-like commutation relations (for z'=0) and the canonical/crystal basis with z' infinity. To simplify the exposition, we discuss in details the easy generalizable case of Uq(su(2)).