Formal multiparameter quantum groups, deformations and specializations

Abstract: We introduce the notion of formal multiparameter quantum universal enveloping algebras - in short FoMpQUEA - as a straightforward generalization of Drinfeld's quantum group. Then we show that the class of FoMpQUEA's is closed under deformations by ("toral") twists and deformations by ("toral") 2-cocycles: as a consequence, all "multiparameter formal QUEA's" 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-cocycles. 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 FoMpQUEA to a MpLbA) and of "deformation (by toral twist or toral 2-cocycle)" - at the level of FoMpQUEA's or of MpLbA's - do commute with each other.