Twisting cocycles in fundamental representaion and triangular bicrossproduct Hopf algebras

We find the general solution to the twisting equation in the tensor bialgebra T (R) of an associative unital ring 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 H ⊂ T (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 Poincaré 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.