Formal multiparameter quantum groups, deformations and specializations

We introduce the notion of formal multiparameter QUEA — in short FoMpQUEA — as a straightforward generalization of Drinfeld’s quantum group U_ℏ(g) . Then we show that the class of FoMpQUEAs is closed under deformations by (“toral”) twists and deformations by (“toral”) 2–cocycles: as a consequence, all “multiparameter formal QUEAs” considered so far are recovered, as falling within this class. In particular, we prove that any FoMpQUEA is isomorphic to a suitable deformation, by twist or by 2–cocycle, of Drinfeld’s standard QUEA. We introduce also multiparameter Lie bialgebras (in short, MpLbA’s), and we consider their deformations, by twist and by 2–cocycle. The semiclassical limit of every FoMpQUEA is a suitable MpLbA, and conversely each MpLbA can be quantized to a suitable FoMpQUEA. In the end, we prove that, roughly speaking, the two processes of “specialization” — of a FoMpQUEA to a MpLbA — and of “deformation (by toral twist or toral 2–cocycle)” — at the level of FoMpQUEAs or of MpLbA’s — do commute with each other.