Monoidal structure on Rota-Baxter algebras via augmentation, and free cocycle bialgebras
Abstract
A tensor product is known for commutative Rota--Baxter algebras through a universal construction, but its underlying algebra need not be the ordinary tensor product A⊗ B and can be substantially larger. We show that, after passing to augmented Rota--Baxter algebras, the ordinary tensor product itself carries a natural Rota--Baxter operator which makes the category monoidal with the usual associator and unit constraints. The resulting structure is intrinsically nonsymmetric: over a field of characteristic zero, this operator and its flip-conjugated counterpart are the only possibilities among linear combinations of five natural terms, while over any field the full category admits no braiding. Its comonoid objects are precisely Rota--Baxter cocycle bialgebras. Moreover, the free Rota--Baxter adjunction yields a bimonad and free cocycle bialgebras on ordinary bialgebras, with compatible Hopf structures for connected filtered inputs, in both the associative and commutative settings.
Related
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BibTeX
@misc{guo2026monoidal,
title = {{Monoidal structure on Rota-Baxter algebras via augmentation, and free cocycle bialgebras}},
author = {Li Guo and Yunzhou Xie and Yi Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17525},
eprint = {2610.17525},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17525},
}
RIS
TY - UNPB
TI - Monoidal structure on Rota-Baxter algebras via augmentation, and free cocycle bialgebras
AU - Li Guo
AU - Yunzhou Xie
AU - Yi Zhang
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17525
UR - https://arxiv.org/abs/2610.17525
AB - A tensor product is known for commutative Rota--Baxter algebras through a universal construction, but its underlying algebra need not be the ordinary tensor product A⊗ B and can be substantially larger. We show that, after passing to augmented Rota--Baxter algebras, the ordinary tensor product itself carries a natural Rota--Baxter operator which makes the category monoidal with the usual associator and unit constraints. The resulting structure is intrinsically nonsymmetric: over a field of characteristic zero, this operator and its flip-conjugated counterpart are the only possibilities among linear combinations of five natural terms, while over any field the full category admits no braiding. Its comonoid objects are precisely Rota--Baxter cocycle bialgebras. Moreover, the free Rota--Baxter adjunction yields a bimonad and free cocycle bialgebras on ordinary bialgebras, with compatible Hopf structures for connected filtered inputs, in both the associative and commutative settings.
ER -