It is known that every
quasitriangular Hopf algebra H can be converted by a process of transmutation into
a braided group B(H,H). The latter is a certain braided-cocommutative
Hopf algebra in the braided monoidal category of H-modules. We use this
transmutation construction to relate two approaches to the quantization of
enveloping algebras.
Specifically, we compute B(H,H) in the case when H is the quasitriangular Hopf
algebra (quantum group) obtained by Drinfeld’s twisting construction on a
cocommutative Hopf algebra H. In the case when H is triangular we recover the
S-Hopf algebra HF previously obtained as a deformation-quantization of H. Here
HF is a Hopf algebra in a symmetric monoidal category. We thereby extend the
definition of HF to the braided case where H is strictly quasitriangular. We also
compute its structure to lowest order in a quantization parameter ℏ. In this way we
show that B(Uq(g),Uq(g)) is the quantization of a certain generalized Poisson
bracket associated to the Drinfeld-Jimbo solution of the classical Yang-Baxter
equations.
