Takeuchi-Schneider equivalence and calculi for homogeneous spaces of Hopf algebroids

TL;DR

This paper introduces a covariant differential calculus framework for Hopf algebroids, establishing categorical equivalences that classify calculi on quantum homogeneous spaces and generalize existing results.

Contribution

It develops a new notion of covariant calculus for Hopf algebroids and proves analogues of fundamental theorems, extending classification results to quantum homogeneous spaces.

Findings

Proves a Takeuchi-Schneider equivalence for Hopf algebroids.

Classifies covariant calculi on Hopf algebroids via substructures of the augmentation ideal.

Provides examples including Ehresmann-Schauenburg Hopf algebroid and scalar extension Hopf algebroids.

Abstract

We develop a notion of covariant differential calculus for Hopf algebroids. As a byproduct, we prove analogues of the fundamental theorem of Hopf modules and a Takeuchi-Schneider equivalence in the realm of Hopf algebroids. The resulting categorical equivalences enable us to classify covariant calculi on Hopf algebroids and, more in general, covariant calculi on quantum homogeneous spaces in this context, in terms of substructures of the augmentation ideal. This generalises the well-known classification results of Woronowicz and Hermisson. A particular focus is given on examples, including covariant calculi on the Ehresmann-Schauenburg Hopf algebroid of a faithfully flat Hopf-Galois extension, and covariant calculi on scalar extension Hopf algebroids, as well as homogeneous space variants of the latter.