On Braided Differential Calculi and Quantum G-structures
Abstract
We develop a theory of first order differential calculi in braided monoidal categories and classify braided covariant and bicovariant calculi on braided Hopf algebras. We show that, under certain conditions, bicovariant calculi can be transmuted to braided bicovariant calculi. For Radford--Majid biproducts, we combine bicovariant calculi on a Hopf algebra and braided bicovariant calculi on the corresponding braided Hopf algebra to covariant smash product calculi. The associated Maurer--Cartan form is shown to decompose into a direct sum of the Maurer--Cartan forms of the structure Hopf algebra of the quantum principal bundle and the braided Hopf algebra on the base. Geometrically, this construction realises the quantum affine extension of a given Hopf algebra, and we prove that the resulting quantum principal bundle is equipped with a frame resolution induced by the quantum Maurer--Cartan form. Building on this correspondence, we introduce and develop the notion of quantum G-structure, proving that quantum G-structures are quantum frame resolutions on the reduction. The theory is illustrated by examples based on transmutations of higher analogues of Sweedler's Hopf algebra and on the braided quantum plane, seen as a Yetter--Drinfeld module of Oq(GL2).
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BibTeX
@misc{donno2026braided,
title = {{On Braided Differential Calculi and Quantum G-structures}},
author = {Antonio Del Donno and Giovanni Gava and Emanuele Latini and Thomas Weber},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01401},
eprint = {2609.01401},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01401},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - On Braided Differential Calculi and Quantum G-structures
AU - Antonio Del Donno
AU - Giovanni Gava
AU - Emanuele Latini
AU - Thomas Weber
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01401
UR - https://arxiv.org/abs/2609.01401
AB - We develop a theory of first order differential calculi in braided monoidal categories and classify braided covariant and bicovariant calculi on braided Hopf algebras. We show that, under certain conditions, bicovariant calculi can be transmuted to braided bicovariant calculi. For Radford--Majid biproducts, we combine bicovariant calculi on a Hopf algebra and braided bicovariant calculi on the corresponding braided Hopf algebra to covariant smash product calculi. The associated Maurer--Cartan form is shown to decompose into a direct sum of the Maurer--Cartan forms of the structure Hopf algebra of the quantum principal bundle and the braided Hopf algebra on the base. Geometrically, this construction realises the quantum affine extension of a given Hopf algebra, and we prove that the resulting quantum principal bundle is equipped with a frame resolution induced by the quantum Maurer--Cartan form. Building on this correspondence, we introduce and develop the notion of quantum G-structure, proving that quantum G-structures are quantum frame resolutions on the reduction. The theory is illustrated by examples based on transmutations of higher analogues of Sweedler's Hopf algebra and on the braided quantum plane, seen as a Yetter--Drinfeld module of O_q(GL_2).
ER -