Extending Doi--Takeuchi correspondence to bimodules and noncommutative covers
Abstract
Doi and Takeuchi, and independently Blattner, Cohen and Montgomery, have exhibited a correspondence between cleft extensions of an algebra U by a Hopf algebra H and H-measurings on U equipped with a compatible normalized 2-cocycle. Each cleft extension is isomorphic as an algebra, a left U-module and a right H-comodule to the crossed product algebra constructed from the measuring and the cocycle. Cleft H-extensions are a special case of H-Hopf-Galois extensions, describing noncommutative principal H-bundles with affine total and affine base space. Corings are used to describe noncommutative space covers, allowing a description of possibly nonaffine noncommutative spaces. We propose to describe the total space of a noncommutative H-principal bundle by an internal E-coring in H-comodules where E is an H-comodule algebra which is itself H-Hopf-Galois. This article focuses on the cleft case. As corings involve bimodules and morphisms of bimodules we extend the Doi--Takeuchi correspondence to bimodule H-comodules between cleft extensions on one side and bimodules of H-coinvariants equipped with bimodule H-measurings compatible with 2-cocycles on the other side. In fact, the correspondences for algebras and for bimodules are parts of a biequivalence of suitable bicategories defined in this article. As a corollary one obtains a correspondence between cleft H-extensions of corings and corings equipped with a nonabelian Čech 1-cocycle covalued in H.
Related
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- Global Mizohata-Takeuchi for Indicator Weights of Bounded Complexity in ℝ2
- Hopf bimodules for bialgebroids
- Classification of Finite-Dimensional Simple Novikov Algebras and Their Irreducible Bimodules in Positive Characteristic
- Simple Harish-Chandra bimodules in positive characteristics
BibTeX
@misc{skoda2026extending,
title = {{Extending Doi--Takeuchi correspondence to bimodules and noncommutative covers}},
author = {Zoran Škoda},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29619},
eprint = {2610.29619},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29619},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Extending Doi--Takeuchi correspondence to bimodules and noncommutative covers
AU - Zoran Škoda
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29619
UR - https://arxiv.org/abs/2610.29619
AB - Doi and Takeuchi, and independently Blattner, Cohen and Montgomery, have exhibited a correspondence between cleft extensions of an algebra U by a Hopf algebra H and H-measurings on U equipped with a compatible normalized 2-cocycle. Each cleft extension is isomorphic as an algebra, a left U-module and a right H-comodule to the crossed product algebra constructed from the measuring and the cocycle. Cleft H-extensions are a special case of H-Hopf-Galois extensions, describing noncommutative principal H-bundles with affine total and affine base space. Corings are used to describe noncommutative space covers, allowing a description of possibly nonaffine noncommutative spaces. We propose to describe the total space of a noncommutative H-principal bundle by an internal E-coring in H-comodules where E is an H-comodule algebra which is itself H-Hopf-Galois. This article focuses on the cleft case. As corings involve bimodules and morphisms of bimodules we extend the Doi--Takeuchi correspondence to bimodule H-comodules between cleft extensions on one side and bimodules of H-coinvariants equipped with bimodule H-measurings compatible with 2-cocycles on the other side. In fact, the correspondences for algebras and for bimodules are parts of a biequivalence of suitable bicategories defined in this article. As a corollary one obtains a correspondence between cleft H-extensions of corings and corings equipped with a nonabelian Čech 1-cocycle covalued in H.
ER -