Invariant Integrals on Coideals and Their Drinfeld Doubles

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⊥ 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 Uq(sl(2,R)) constructed from the Podleś spheres.