Heisenberg double as braided commutative Yetter-Drinfel'd module algebra over Drinfel'd double in multiplier Hopf algebra case

摘要

Based on a pairing of two regular multiplier Hopf algebras $A$ and $B$,Heisenberg double $\mathscr{H}$ is the smash product $A \# B$ with respect tothe left regular action of $B$ on $A$. Let $\mathscr{D}=A\bowtie B$ be theDrinfel'd double, then Heisenberg double $\mathscr{H}$ is a Yetter-Drinfel'd$\mathscr{D}$-module algebra, and it is also braided commutative by thebraiding of Yetter-Drinfel'd module, which generalizes the results in [10] tosome infinite dimensional cases.