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The Role of the Canonical Element in the Quantized Algebra of Differential Operators A×U 1

We review the construction of the cross product algebra A×U from two dually paired Hopf algebras U and A. The canonical element in U ⊗ A is then introduced, and its properties examined. We find that it is useful for giving coactions on A×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×U onto said objects. We then discuss bicovariant vector fields in the context of the canonical element.