The Drinfelʹd double versus the Heisenberg double for an algebraic quantum group

Let A be a regular multiplier Hopf algebra with integrals. The dual of A, denoted by Â, is a multiplier Hopf algebra so that left angle bracketÂ,Aright-pointing angle bracket is a pairing of multiplier Hopf algebras. We consider the Drinfelʹd double, D=ÂbowtieAcop, associated to this pair. We prove that D is a quasitriangular multiplier Hopf algebra. More precisely, we show that the pair left angle bracketÂ,Aright-pointing angle bracket has a “canonical multiplier” Wset membership, variantM(Âcircle times operatorA). The image of W in M(Dcircle times operatorD) is a generalized R-matrix for D. We use this image of W to deform the product of the dual multiplier Hopf algebra image via the right action of D on image which defines the pair image. As expected from the finite-dimensional case, we find that the deformation of the product in image is related to the Heisenberg double A#Â.