Frobenius functors and one-sided Hopf algebras in braided monoidal categories
Abstract
Evidence suggests a tight connection between the existence of antipode-like maps on bialgebra-like structures, and the Frobenius property for the associated free Hopf module functor. In this paper, we prove that in any braided monoidal category satisfying few mild assumptions, a bimonoid is a one-sided Hopf monoid and the antipode is a bimonoid anti-homomorphism, if and only if the free Hopf module functor is Frobenius. Our interest in these structures stems from having found genuine examples of one-sided Hopf monoids in contexts where the braiding is non-trivial. In fact, we provide a construction of the free one-sided Hopf monoid over a comonoid in any symmetric monoidal category with some non-restrictive additional assumptions. We present several examples of this construction, and describe the resulting one-sided (sometimes two-sided) Hopf monoids.
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BibTeX
@misc{bottegoni2026frobenius,
title = {{Frobenius functors and one-sided Hopf algebras in braided monoidal categories}},
author = {Lucrezia Bottegoni and Davide Ferri and Paolo Saracco},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13028},
eprint = {2609.13028},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13028},
}
RIS
TY - UNPB
TI - Frobenius functors and one-sided Hopf algebras in braided monoidal categories
AU - Lucrezia Bottegoni
AU - Davide Ferri
AU - Paolo Saracco
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.13028
UR - https://arxiv.org/abs/2609.13028
AB - Evidence suggests a tight connection between the existence of antipode-like maps on bialgebra-like structures, and the Frobenius property for the associated free Hopf module functor. In this paper, we prove that in any braided monoidal category satisfying few mild assumptions, a bimonoid is a one-sided Hopf monoid and the antipode is a bimonoid anti-homomorphism, if and only if the free Hopf module functor is Frobenius. Our interest in these structures stems from having found genuine examples of one-sided Hopf monoids in contexts where the braiding is non-trivial. In fact, we provide a construction of the free one-sided Hopf monoid over a comonoid in any symmetric monoidal category with some non-restrictive additional assumptions. We present several examples of this construction, and describe the resulting one-sided (sometimes two-sided) Hopf monoids.
ER -