Explicit Rieffel induction modules for quantum groups

TL;DR

This paper constructs explicit Rieffel induction modules for quantum groups, linking them with Vaes' induction functor and illustrating with parabolic induction in complex semi-simple quantum groups.

Contribution

It explicitly defines Rieffel induction modules for quantum groups using an inner product valued in the convolution algebra, connecting with existing induction functors.

Findings

Explicit construction of Rieffel induction modules for quantum groups.

Establishment of the link between Rieffel induction and Vaes' induction functor.

Application to parabolic induction in complex semi-simple quantum groups.

Abstract

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}.