An algebraic framework for the Drinfeld double based on infinite groupoids

Abstract: The Drinfeld double associated to the weak multiplier Hopf ($$-) algebra pairing $\left\langle A, B\right\rangle$ is constructed. We show that the Drinfeld double is again a weak multiplier Hopf ($$-) algebra. If $A$ and $B$ are algebraic quantum groupoids, then so does the double. We also prove the correspondence between modules over the Drinfeld double and Yetter-Drinfeld modules. Finally, we prove that the double is a quasitriangular weak multiplier Hopf algebra.