Explicit Rieffel induction module for quantum groups

For $\mathbb{G}$ an algebraic (or more generally, a bornological) quantum group and $\mathbb{B}$ a closed quantum subgroup of $\mathbb{G}$, we build in this paper an induction module by explicitly defining an inner product which takes its value in the convolution algebra of $\mathbb{B}$, as in the original approach of Rieffel \cite{Rieffel}. In this context, we study the link with the induction functor defined by Vaes. In the last part we illustrate our result with parabolic induction of complex semi-simple quantum groups with the approach suggested by Clare \cite{Clare}\cite{CCH}.