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 ⊗ A. We realize the Drinfel'd double, denoted by D, as an algebra of operators on the vector space B ⊗ 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. © 2004 Elsevier Inc. All rights reserved.
Delvaux, L., & Van Daele, A. (2004). The Drinfel’d double of multiplier Hopf algebras. Journal of Algebra, 272(1), 273–291. https://doi.org/10.1016/j.jalgebra.2003.03.003