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

TL;DR

This paper demonstrates that the Heisenberg double constructed from a pairing of regular multiplier Hopf algebras forms a braided commutative Yetter-Drinfel'd module algebra over the Drinfel'd double, extending previous finite-dimensional results.

Contribution

It generalizes the structure of the Heisenberg double as a braided commutative Yetter-Drinfel'd module algebra to infinite-dimensional multiplier Hopf algebra cases.

Findings

Heisenberg double is a Yetter-Drinfel'd module algebra over the Drinfel'd double.

The Heisenberg double is braided commutative in this setting.

Extension of finite-dimensional results to infinite-dimensional multiplier Hopf algebras.

Abstract

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 to the left regular action of $B$ on $A$. Let $\mathscr{D}=A\bowtie B$ be the Drinfel'd double, then Heisenberg double $\mathscr{H}$ is a Yetter-Drinfel'd $\mathscr{D}$-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.