On the modules of a Drinfel'd double multiplier Hopf (*-)algebra

Abstract

When H is a finite-dimensional Hopf algebra, it has been shown by S. Majid that the modules of the Drinfel'd double D(H) are given by H-crossed bimodules. We study the modules over the Drinfel'd double of a pair of multiplier Hopf algebras (A, B). Under appropriate conditions on the pairing, we prove that there is a strong connection between left A-modules and right B-comodules. By using this connection, we characterize the left modules over the Drinfel'd double of the given pairing. We obtain that the result of S. Majid stays valid in this general framework. We also give results in the *-case.