Heisenberg double as braided commutative Yetter–Drinfel’d module algebra over Drinfel’d double in multiplier Hopf algebra case

Abstract

Based on a pairing of two regular multiplier Hopf algebras A and B, Heisenberg double H is the smash product A#B with respect to the left regular action of B on A. Let D = A ⊲⊳ B be the Drinfel'd double, then Heisenberg double H is a Yetter-Drinfel'd D-module algebra, and it is also braided commutative by the braiding of Yetter-Drinfel'd module, which generalizes the results in [10] to some infinite dimensional cases.