Quantization of orbit bundles in gl∗(n,C)gl^*(n,C)

Let $G$ be the complex general linear group and $g$ its Lie algebra equipped with a factorizable Lie bialgebra structure; let $U_h$ be the corresponding quantum group. We construct explicit $U_h$-equivariant quantization of Poisson orbit bundles $O_\lambda \to O_\mu$ in $gl(n)*$.Comment: 25 page

Twisting cocycles in fundamental representation and triangular bicrossproduct Hopf algebras

We find the general solution to the twisting equation in the tensor bialgebra $T({\bf R})$ of an associative unital ring ${\bf R}$ viewed as that of fundamental representation for a universal enveloping Lie algebra and its quantum deformations. We suggest a procedure of constructing twisting cocycles belonging to a given quasitriangular subbialgebra ${\cal H}\subset T({\bf R})$. This algorithm generalizes Reshetikhin's approach, which involves cocycles fulfilling the Yang-Baxter equation. Within this framework we study a class of quantized inhomogeneous Lie algebras related to associative rings in a certain way, for which we build twisting cocycles and universal $R$-matrices. Our approach is a generalization of the methods developed for the case of commutative rings in our recent work including such well-known examples as Jordanian quantization of the Borel subalgebra of $sl(2)$ and the null-plane quantized Poincar\'e algebra by Ballesteros at al. We reveal the role of special group cohomologies in this process and establish the bicrossproduct structure of the examples studied.Comment: 20 pages, LaTe