Bicovariant Codifferential Calculi

TL;DR

This paper develops a classification technique for bicovariant codifferential calculi on Hopf algebras, linking them to Yetter-Drinfeld modules and quantum algebra structures, with various examples illustrating the results.

Contribution

It introduces a new approach to classifying bicovariant codifferential calculi using Yetter-Drinfeld modules, enhancing understanding of quantum algebraic structures.

Findings

Classification reduces to classifying Y-D submodules.

Dual Y-D structures relate to Woronowicz's calculus.

Examples demonstrate the classification results.

Abstract

We develop a technique for studying first-order codifferential calculi (FOCCs) initiated by Doi and Quillen in the context of cyclic cohomology. Their classification, for a given coalgebra, reduces to the classification of subbicomodules in the universal bicomodule. For completing this task, the role of one-dimensional generating spaces (a.k.a. singletons) is found to be useful. We are particularly interested in classifying bicovariant codifferential calculi, which we define over Hopf algebras. This, in turn, can be reduced to classifying Yetter-Drinfeld (Y-D) submodules. In fact, there are two, mutually dual, Y-D structures on arbitrary Hopf algebra: one used by Woronowicz for constructing bicovariant differential calculi, and the another used here for FOCCs and shown to be related with Woronowicz construction of quantum tangent space. This argues that such codifferential calculi are better suited to Drinfeld-Jimbo type quantized enveloping algebras, as they are dual to Woronowicz' bicovariant calculi over matrix quantum groups. Relations with quantum Lie algebras and quantum vector fields are also shown. Some classification results are presented in numerous examples.