The book Enveloping Algebras, by Jacques Dixmier has a chapter on inducing representations of universal enveloping algebras (chapter 5) and it goes very simply and smoothly: If $\mathfrak{h}$ is a Lie subalgebras of $\mathfrak{g}$, and $W$ is a $U(\mathfrak{h})$-module, then we can imbue $U(\mathfrak{g}) \otimes_{U(\mathfrak{h})} W$ with a $U(\mathfrak{g})$-module structure ($g` \cdot (g \otimes w) = g`g \otimes w$) called the induced moduled structure.
I need to know how this process works in the quantized case, so my question is: Do you know if there are references in the literature treating the quatized case? Of course, you can just answer this question with the induction procedure in the quantized case, but I would really appreciate references.