Invariant Integrals on Coideals and Their Drinfeld Doubles

Abstract

Abstract Let $A$ be a CQG Hopf *-algebra, that is, 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 relatively invariant integral on the stabilizer coideal $B^{\perp }$ inside the dual discrete multiplier Hopf *-algebra of $A$. Given such a relatively invariant integral, we show how it can be extended to a relatively invariant integral on the Drinfeld double coideal. We moreover show that the representation category of the Drinfeld double coideal has a monoidal structure. As an application, we determine the relatively invariant integral for the Drinfeld double coideal *-algebra ${\mathcal{U}}_{q}(\mathfrak{s}\mathfrak{l}(2,{\mathbb{R}}))$ constructed from the Podleś spheres.