Twisting cocycles in fundamental representation and triangular bicrossproduct Hopf algebras

Abstract

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.