Quantum Hopf rigidity in symmetric monoidal categories
Abstract
We prove that a coaction of a Hopf algebra on another preserving the latter's associative-algebra structure and comultiplication automatically also preserves the counit, antipode, and tensor-interchange map and in fact factors through the coaction dual to an action of an affine group scheme by Hopf-algebra automorphisms. This generalizes and unifies a number of results to the effect that quantum groups have only classical symmetries, due to Kasprzak et al., Budziński-Kasprzak and Brannan et al. The stated automatic classicality follows from the fact that the antipode, counit and tensorand-interchange of a Hopf algebra internal to any symmetric monoidal category are contained in the (non-symmetric) monoidal subcategory generated by the associative-algebra structure and the comultiplication, assuming a weak form of arrow-retraction invariance.
Related
- Frobenius functors and one-sided Hopf algebras in braided monoidal categories
- On closed symmetric monoidal enriched ∞-categories
- The symmetric skew monoidal structure of Gray-categories
- On pushouts of strict symmetric monoidal categories
- Monoidal Categories associated with Kac-Moody Open Richardson Varieties in Symmetric Type
BibTeX
@misc{chirvasitu2026quantum,
title = {{Quantum Hopf rigidity in symmetric monoidal categories}},
author = {Alexandru Chirvasitu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01541},
eprint = {2609.01541},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01541},
}
RIS
TY - UNPB
TI - Quantum Hopf rigidity in symmetric monoidal categories
AU - Alexandru Chirvasitu
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.01541
UR - https://arxiv.org/abs/2609.01541
AB - We prove that a coaction of a Hopf algebra on another preserving the latter's associative-algebra structure and comultiplication automatically also preserves the counit, antipode, and tensor-interchange map and in fact factors through the coaction dual to an action of an affine group scheme by Hopf-algebra automorphisms. This generalizes and unifies a number of results to the effect that quantum groups have only classical symmetries, due to Kasprzak et al., Budziński-Kasprzak and Brannan et al. The stated automatic classicality follows from the fact that the antipode, counit and tensorand-interchange of a Hopf algebra internal to any symmetric monoidal category are contained in the (non-symmetric) monoidal subcategory generated by the associative-algebra structure and the comultiplication, assuming a weak form of arrow-retraction invariance.
ER -