  We review the construction of the cross product algebra $\A\rtimes\U$ from
two dually paired Hopf algebras $\U$ and $\A$. The canonical element in $\U
\otimes \A$ is then introduced, and its properties examined. We find that it is
useful for giving coactions on $\A\rtimes\U$, and it allows the construction of
objects with specific invariance properties under these coactions. A ``vacuum
operator'' is found which projects elements of $\A\rtimes\U$ onto said objects.
We then discuss bicovariant vector fields in the context of the canonical
element.
