The Drinfel'd double versus the Heisenberg double for an algebraic quantum group

Abstract

Let A be a regular multiplier Hopf algebra with integrals. The dual of A, denoted by (A) over cap, is a multiplier Hopf algebra so that <(A) over cap ,A> is a pairing of multiplier Hopf algebras. We consider the Drinfel'd double, D = (A) over cap A(cop), associated to this pair. We prove that D is a quasitriangular multiplier Hopf algebra. More precisely, we show that the pair <(A) over cap ,A> has a "canonical multiplier" W epsilon M((A) over cap circle times A). The image of W in M(D circle times D) is a generalized R-matrix for D. We use this image of W to deform the product of the dual multiplier Hopf algebra D via the right action of D on (D) over cap which defines the pair <(D) over cap ,D>. As expected from the finite-dimensional case, we find that the deformation of the product in (D) over cap is related to the Heisenberg double A#(A) over cap. (C) 2003 Elsevier B.V. All rights reserved.