Quantum anchor : $\uqs$ case

We introduce the tangent space $ T({\rm H}_q) $ on the quantum hyperboloid ($ {\cal A}_{0,q}^c $) and equip it with an action on $ {\cal A}_{0,q}^c $ being a deformation of the action of vectors fields on functions. An embedding $ \displaystyle sl(2)_q \longrightarrow T({\rm H}_q) $ of $q$-deformed Lie algebra $ sl(2) $ being an analogue of the anchor $ \displaystyle sl(2) \longrightarrow \Vect({\rm H}) $ is called "quantum anchor".