BRAIDED GROUPS OF HOPF-ALGEBRAS OBTAINED BY TWISTING

It is known that every quasiitriangular Hopf algebra H can be converte d by a process of transmutation into a braided group B(H, H). The latt er is a certain braided-cocommutative Hopf algebra in the braided mono idal category of H-modules. We use this transmutation construction to relate two approaches to the quantization of enveloping algebras. Spec ifically, we compute B(($) over tilde H, ($) over tilde H) in the case when ($) over tilde H is the quasitriangular Hopf algebra (quantum gr oup) obtained by Drinfeld's twisting construction on a co commutative Hopf algebra H. In the case when ($) over tilde H is triangular we rec over the S-Hopf algebra H-F previously obtained as a deformation-quant ization of H. Here H-F is a Hopf algebra in a symmetric monoidal categ ory. We thereby extend the definition of H-F to the braided case where ($) over tilde H is strictly quasitriangular. We also compute its str ucture to lowest order in a quantization parameter h. In this !way we low that B(U-q(g), U-q(g)) is the quantization of a certain generalize d Poisson bracket associated to the Drinfeld-Jimbo solution elf the cl assical Yang-Baxter equations.