Twisted tensor coproduct of multiplier Hopf algebras

We put a non-trivial comultiplication on the natural tensor product algebra of two multiplier Hopf algebras by means of a “cotwisting map.” As a special case we characterize the dual of the Drinfelʹd double of an algebraic quantum group. Because any finite-dimensional Hopf algebra is an algebraic quantum group, our characterization applies to the dual of the Drinfelʹd double of a finite-dimensional Hopf algebra. Then it coincides with a result of Lu.