Induction for representations of coideal doubles, with an application to quantum $SL(2,\mathbb{R})$

TL;DR

This paper develops an induction theory for representations of coideal doubles in quantum groups and applies it to analyze the regular representation of quantum SL(2,R), revealing its irreducible components.

Contribution

It introduces a new induction framework for coideal doubles and explicitly decomposes the regular representation of quantum SL(2,R).

Findings

Decomposition of the regular representation into irreducibles.

Application of induction theory to quantum SL(2,R).

Advancement in understanding quantum group representations.

Abstract

We investigate the theory of induction in the setting of doubles of coideal $*$-subalgebras of compact quantum group Hopf $*$-algebras. We then exemplify parts of this theory in the particular case of quantum $SL(2,\mathbb{R})$, and compute the decomposition of the regular representation for quantum $SL(2,\mathbb{R})$ into irreducibles.