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Bicovariant Codifferential Calculi

Andrzej Borowiec, Patryk Mieszkalski

TL;DR

This work develops a dual theory of first-order codifferential calculi (FOCCs) on coalgebras, showing that their classification reduces to subbicomodules of the universal bicomodule and that singleton generating spaces provide a practical building block. It establishes a tight duality between FOCCs and first-order differential calculi (FODCs), framed through universal coderivations and Yetter-Drinfeld module theory, with the universal FOCC on a Hopf algebra $H$ realized as $\Upsilon^U_H \cong \overline{H}_L \otimes H \cong H \otimes \overline{H}_R$. The paper then offers a sequence of concrete classifications across diverse coalgebras—Sweedler, vector-space-generated, divided power, and set coalgebras—and extends the framework to bicovariant FOCCs on quantum groups, including explicit results for $U_Q(\mathfrak{b}_+)$, $U_q(\mathfrak{sl}(2))$, and the $\kappa$-Poincaré Hopf algebra. Overall, the results provide a systematic, dualizable toolkit for constructing and classifying codifferential calculi in noncommutative geometry and quantum group settings, with potential applications to dual formulations of Woronowicz calculi and matrix quantum groups.

Abstract

We develop a technique for studying first-order codifferential calculi initiated by Doi and Quillen in the context of cyclic cohomology. Introducing the concept of first-order codifferential calculi (FOCCs) for a given coalgebra, we can show that their classification can be reduced 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 submodules. We argue that such codifferential calculi are better suited to Drinfeld-type quantized enveloping algebras, as they are dual to Woronowicz calculi for matrix quantum groups. Some classification results are presented in numerous examples.

Bicovariant Codifferential Calculi

TL;DR

This work develops a dual theory of first-order codifferential calculi (FOCCs) on coalgebras, showing that their classification reduces to subbicomodules of the universal bicomodule and that singleton generating spaces provide a practical building block. It establishes a tight duality between FOCCs and first-order differential calculi (FODCs), framed through universal coderivations and Yetter-Drinfeld module theory, with the universal FOCC on a Hopf algebra realized as . The paper then offers a sequence of concrete classifications across diverse coalgebras—Sweedler, vector-space-generated, divided power, and set coalgebras—and extends the framework to bicovariant FOCCs on quantum groups, including explicit results for , , and the -Poincaré Hopf algebra. Overall, the results provide a systematic, dualizable toolkit for constructing and classifying codifferential calculi in noncommutative geometry and quantum group settings, with potential applications to dual formulations of Woronowicz calculi and matrix quantum groups.

Abstract

We develop a technique for studying first-order codifferential calculi initiated by Doi and Quillen in the context of cyclic cohomology. Introducing the concept of first-order codifferential calculi (FOCCs) for a given coalgebra, we can show that their classification can be reduced 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 submodules. We argue that such codifferential calculi are better suited to Drinfeld-type quantized enveloping algebras, as they are dual to Woronowicz calculi for matrix quantum groups. Some classification results are presented in numerous examples.
Paper Structure (23 sections, 24 theorems, 160 equations)

This paper contains 23 sections, 24 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Notational conventions
  3. Mathematical formalism of codifferential calculus
  4. Basic definitions and properties
  5. Cocommutative case
  6. Direct sum decompositions
  7. Some notes about classification of FOCCs
  8. Illustrative examples
  9. Sweedler coalgebra
  10. Coalgebra generated by a vector space
  11. Divided power coalgebra
  12. Set coalgebra
  13. Bicovariant FOCC over a Hopf algebra
  14. Some remarks on classification of bicovariant FOCCs
  15. Examples of bicovariant FOCC on quantum groups
  16. ...and 8 more sections

Key Result

Lemma 1

Assume $M$ is A) a left $\mathcal{C}$-comodule, B) a right $\mathcal{D}$-comodule, C) a $\mathcal{C}-\mathcal{D}$ bicomodule, and $V$ is a linear space.

Theorems & Definitions (82)

  • Lemma 1
  • proof
  • Remark 1
  • Remark 2
  • Example 1
  • Remark 3
  • Lemma 2
  • proof
  • Remark 4
  • Remark 5
  • ...and 72 more