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