Invariant integrals on coideals and their Drinfeld doubles

TL;DR

This paper studies invariant integrals on coideals within CQG Hopf *-algebras and their Drinfeld doubles, establishing conditions for their existence and exploring the representation theory of the resulting structures.

Contribution

It provides new conditions for the existence of quasi-invariant integrals on coideals and extends these to Drinfeld doubles, with applications to quantum groups like $U_q(sl(2,R))$.

Findings

Existence conditions for quasi-invariant integrals on stabilizer coideals.

Extension of integrals to Drinfeld double coideals.

Monoidal structure in the representation theory of the Drinfeld double coideal.

Abstract

Let $A$ be a CQG Hopf $*$-algebra, i.e. a Hopf $*$-algebra with a positive invariant state. Given a unital right coideal $*$-subalgebra $B$ of $A$, we provide conditions for the existence of a quasi-invariant integral on the stabilizer coideal $B^{\perp}$ inside the dual discrete multiplier Hopf $*$-algebra of $A$. Given such a quasi-invariant integral, we show how it can be extended to a quasi-invariant integral on the Drinfeld double coideal. We moreover show that the representation theory of the Drinfeld double coideal has a monoidal structure. As an application, we determine the quasi-invariant integral for the coideal $*$-algebra $U_q(\mathfrak{sl}(2,\mathbb{R}))$ constructed from the Podle\'{s} spheres.