Communications in algebra vol:34 issue:8 pages:2811-2842
Let < A, B > be a pairing of two regular multiplier Hopf algebras A and B. One mthod of constructing the Drinfel'd double D = A vertical bar x vertical bar B-cop is by the use of an invertible twist map R : B circle times -> A circle times B defining an associative product on A circle times B. In Delvaux (2003) and Drabant and Van Deale (2001), the authors construct R by R = sigma o R-2 o R-1(-1), where R-2 and R-1(-1) is only related to the module actions between A and B. Another way is given in Delvaux and Van Daele (2204a) in which the authors also just consider the module actions and then construct the Drinfel'd double D as an algebra of operators on the vector space B circle times A. In this article we will give two different points of view of constructing the Drinfel'd double D for multiplier Hopf algebras. The first is that the Drinfel'd double D associated to the pairing < A, B > is constructed by using not only the module actions but also the comodule coactions, i.e., the Drinfeld double D is given in the framework of a special twisted tensor product algebra structure A boxed dot B-Aop circle times A. The second is as follows: If P is a multiplier Hopf algebra and a reduced (A,B)-bicomodule algebra (an A-long module algebra), then we present a twisting construction of the product of P via the coactions of A and B on P, and we show that the Drinfel'd double D is isomorphic as a multiplier Hopf algebra to the opposite twisting of (A(op,cop) circle times B-op,B-cop). As an application of our theory, we consider the case of group-cograded multiplier Hopf algebras and the case of Hopf group-coalgebras.