The hidden group structure of quantum groups: Strong duality,rigidity and preferred deformations

A notion of well-behaved Hopf algebra is introduced; reflexivity (for strong duality) between Hopf algebras of Drinfeld-type and their duals, algebras of coefficients of compact semi-simple groups, is proved. A hidden classical group structure is clearly indicated for all generic models of quantum groups. Moyal-product-like deformations are naturally found for all FRT-models on coefficients andC-functions. Strong rigidity (H_bi²={0}) under deformations in the category of bialgebras is proved and consequences are deduced.