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".