QUANTUM BASES IN Uq(g)

This paper is devoted to analyze, inside the ∞-many possible bases of a Quantum Universal Enveloping Algebra Uq(g), those that can be considered as “more equal then others”, like orthonormal bases in the Euclidean spaces. The only possible element of selection has been found to be a privileged connection with the corresponding bialgebra. A new parameter z ∈ C –independent from the z := log(q) ∈ C that defines Uq(g)– has thus been introduced. Each value of such new parameter z defines one of these bases (we call quantum bases) and determines, independently from z, its commutation relations. Both z and z are, on the contrary, necessary to fix the relations between the basic set and its co-products. We have thus –for each pair {z, z}– one n-dimensional quantum basis (gz′ ,∆zz′), that describes Uq(g). Three cases are particularly relevant: the analytical basis gz, where z ′ = z, the Lie basis g0 obtained for z ′ = 0 (where both the basic set and its co-product close Lie-like commutation relations and the non primitive co-product describe an interaction) and the canonical/crystal basis g∞, limit for z ′ → ∞ in the Riemann plane of the generic quantum basis. To simplify the exposition, we discuss in details the easily generalizable case of Uq(su(2)). MSC: 81R50, 17B35, 17B37, 17B62