The notion of algebra of differential operators on a group \(G\) is generalized to the deformed case. The deformed algebra of functions \({\mathfrak A}\equiv \text{Fun} (G_ q)\) is combined with the universal enveloping algebra \({\mathfrak U}\equiv U_ q {\mathfrak g}\) into a new algebra, the cross product \({\mathfrak A} \rtimes {\mathfrak U}\), according to the rule \[ ax\cdot by=ab_{(1)} \langle x_{(1)}, b_{(2)}\rangle x_{(2)}y, \qquad \forall a,b\in {\mathfrak A},\quad \forall x,y\in {\mathfrak U}. \] Here \(\Delta(c)= c_{(1)}\otimes c_{(2)}\) (Sweedler’s notation). Left and right \({\mathfrak A}\)-coactions \({}_{\mathfrak A}\Delta\) and \(\Delta_{\mathfrak A}\) on the cross product are introduced and studied. They are given by \({}_{\mathfrak A}\Delta (by)= b_{(1)}\otimes b_{(2)}y\) and \(\Delta_{\mathfrak A} (by)= \sum_ j (b_{(1)}\cdot e_ j {\overset {\text{ad}} \vartriangleright}y) \otimes b_{(2)} f^ j\), with \(\{e_ j\}\) and \(\{f^ j\}\) being mutually dual bases. The adjoint action “\({\overset {\text{ad}} \vartriangleright}\)” is defined as usual.
Furthermore, invariant maps \(\Phi\) from \(\text{Fun} (G_ q)\) to \(U_ q {\mathfrak g}\) are introduced with the help of elements of the pure braid group \(B_ n\). By definition, \({\mathcal Z}\in \text{span} \{B_ n\}\) iff \({\mathcal Z}\in {\mathfrak U}^{ \widehat{\otimes} n}\) and \({\mathcal Z} \Delta^{(n-1)} (x)= \Delta^{(n-1)} (x){\mathcal Z}\), \(\forall x\in{\mathfrak U}\). Particular examples are constructed using the universal \({\mathcal R}\)- matrix: \({\mathcal Z}={\mathcal R}^{21} {\mathcal R}^{12},\;{\mathcal R}^{21} {\mathcal R}^{31} {\mathcal R}^{13} {\mathcal R}^{12},\ldots\). Let \(A\in \text{Mat}_ n ({\mathfrak A})\) be the defining (vector) corepresentation of \({\mathfrak A}\) and set \({\mathcal Z}={\mathcal R}^{21}{\mathcal R}^{12}\). Then the map \(\Phi\) produces the so called “reflection matrix” \(Y:= \Phi({\mathcal A})\in \text{Mat}_ n ({\mathfrak U})\), with remarkable transformation properties. The vector space spanned by entries of \(Y\) is closed under the adjoint action. This property is considered as a basis for the notion of quantum Lie algebra.