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)).
Categories: math.QA