Quantum hyperboloid and braided modules

When a quantum hyperboloid is realized, as a three - parameter algebra $\ahqc$, in the usual manner, the following problem arises: what is a ``representation theory'' of this algebra? We construct the series of all spin representations of $\ahqc$, and we discuss a braided version of the orbit method, i.e. a correspondence between orbits in $\gggg^*$ and $\gggg$-modules. A braided trace and a braided involution are discussed as well.