Quantum Bases in Uq(g)

Abstract

This paper is devoted to analyze, inside the ∞-many possible bases of a Quantum Universal Enveloping Algebra U q (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 U q (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 (g z ′ , ∆ zz ′ ), that describes U q (g).Three cases are particularly relevant: the analytical basis g z , where z ′ = z, the Lie basis g 0 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 U q (su(2)).