The Drinfel'd double of multiplier Hopf algebras

Abstract

Let <A, B> be a pairing of two regular multiplier Hopf algebras A and B. The Drinfel'd double associated to this pairing is constructed by using appropriate representations of A and B on the same vector space B circle times A. We realize the Drinfel'd double, denoted by D, as an algebra of operators on the vector space B circle times A. In the case that <A, B> is a multiplier Hopf*-algebra pairing, we prove that D is again a multiplier Hopf*-algebra. If A and B carry positive integrals, we prove that D also has a positive integral. This proof is not given before.