The Role of the Canonical Element in the Quantized Algebra of Differential Operators ArtimesU

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.