The Drinfel'd 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 Â,A is a pairing of multiplier Hopf algebras. We consider the Drinfel'd double, D=ÂA^cop, associated to this pair. We prove that D is a quasitriangular multiplier Hopf algebra. More precisely, we show that the pair Â,A has a “canonical multiplier” WM(ÂA). The image of W in M(DD) is a generalized R-matrix for D. We use this image of W to deform the product of the dual multiplier Hopf algebra via the right action of D on which defines the pair . As expected from the finite-dimensional case, we find that the deformation of the product in is related to the Heisenberg double A#Â.