Formal multiparameter quantum groups, deformations and specializations

TL;DR

This paper introduces formal multiparameter quantum universal enveloping algebras (FoMpQUEA), explores their deformation properties, and establishes their relationship with multiparameter Lie bialgebras, showing how quantization and specialization processes commute.

Contribution

It defines FoMpQUEA, proves their stability under twists and 2-cocycle deformations, and links them with multiparameter Lie bialgebras through quantization and specialization.

Findings

FoMpQUEA are closed under toral twists and 2-cocycle deformations.

Any FoMpQUEA is a deformation of Drinfeld's standard QUEA.

Semiclassical limits of FoMpQUEA are MpLbA, and vice versa.

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.