Exact calculability, semigroup of representations, and the stability property for representations of the algebra of functions on the quantum group $SU_{q}(2)$

An operation of a coproduct of representations of a bialgebra is defined. The coproduct operation for representations of the Hopf algebra of functions on the quantum group $SU_{q}(2)$ is investigated. A notion of a stable representation $\Pi$ is introduced. This means that the representation $\Pi$ is invariant under coproduct by arbitrary representation. Formula for the trace in the representation $\Pi$ is given. The invariant integral of Woronovich on $SU_{q}(2)$ will take the form $\int f d\mu = (1-q^{2}){\rm tr}\,(fcc^{*})$.