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

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 quasi-twists and quasi-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: this comes from the "Quantum Duality Principle", through which every "quasi-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 quasi- 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.