The Role of the Canonical Element in the Quantized Algebra of Differential Operators $\A\rtimes\U$

TL;DR

This paper explores the canonical element in the quantized algebra of differential operators, revealing its role in coactions, invariance, and the construction of bicovariant vector fields within the algebraic framework.

Contribution

It introduces and analyzes the properties of the canonical element in the cross product algebra of dually paired Hopf algebras, providing new tools for invariance and bicovariance.

Findings

The canonical element enables coactions on the algebra.

A vacuum operator projects elements onto invariant objects.

The framework facilitates the study of bicovariant vector fields.

Abstract

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.