Quantum group deformations and quantum $ R $-(co)matrices vs. Quantum Duality Principle

TL;DR

This paper explores how twist and 2-cocycle deformations affect quantum groups and their semiclassical limits, introduces generalized deformation procedures via polar twists, and examines their relation to the Quantum Duality Principle and $R$-(co)matrices.

Contribution

It introduces the concept of polar twists and 2-cocycles, extending deformation procedures and connecting them to the Quantum Duality Principle in quantum group theory.

Findings

Polar twists and 2-cocycles can be used to generalize quantum group deformations.

The Quantum Duality Principle explains the relation between polar and standard twists.

New symmetries for dual Poisson groups are identified through these constructions.

Abstract

In this paper we describe the effect on quantum groups -- namely, both QUEA's and QFSHA's -- of deformations by twist and by 2-cocycles, showing how such deformations affect the semiclassical limit. As a second, more important task, we discuss how these deformation procedures can be "stretched" to a new extent, via a formal variation of the original recipes, using "polar twists" and "polar 2-cocycles". These recipes seemingly should make no sense at all, yet we prove that they actually work, thus providing well-defined, more general deformation procedures. Later on, we explain the underlying reason that motivates such a result in light of the "Quantum Duality Principle", through which every "polar twist/2-cocycle" for a given quantum group can be seen as a standard twist/2-cocycle for another quantum group, associated to the original one via the appropriate Drinfeld functor. As a third task, we consider standard constructions involving $R$-(co)matrices in the general theory of Hopf algebras. First we adapt them to quantum groups, then we show that they extend to the case of "polar $R$-(co)matrices", and finally we discuss how these constructions interact with the Quantum Duality Principle. As a byproduct, this yields new special symmetries (isomorphisms) for the underlying pair of dual Poisson (formal) groups that one gets by specialization.