Braided groups of Hopf algebras obtained by twisting
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 \(B(\widetilde {H},\widetilde {H})\) in the case when \(\widetilde {H}\) is the quasitriangular Hopf algebra (quantum group) obtained by Drinfeld's twisting construction on a cocommutative Hopf algebra \(H\). In the case when \(\widetilde{H}\) is triangular we recover the \(S\)-Hopf algebra \(H_ F\) previously obtained as a deformation- quantization of \(H\). Here \(H_ F\) is a Hopf algebra in a symmetric monoidal category. We thereby extend the definition of \(H_ F\) to the braided case where \(\widetilde {H}\) is strictly quasitriangular. We also compute its structure to lowest order in a quantization parameter \(\hbar\). In this way we show that \(B(U_ q(g),U_ q(g))\) is the quantization of a certain generalized Poisson bracket associated to the Drinfeld-Jimbo solution of the classical Yang-Baxter equations.
quantization of generalized Poisson bracket
scientific article; zbMATH DE number 540657