Braided cohomology of quasi-triangular bialgebras and braided Morita invariance
Abstract
In this paper, we introduce the braided cochain complex and the braided cohomology of braided coalgebras in linear monoidal categories, and compare those of braided coalgebras living in different linear monoidal categories using relative morphisms. The symmetric cohomology was introduced for groups by Staic, and was generalized to cocommutative Hopf algebras by Shiba, Sanada, and the second author. This notion involves degreewise actions of the symmetric groups on a cochain complex, which come from the symmetry on the monoidal category of group representations. We generalize this framework by dealing with arbitrary linear monoidal categories, and by replacing symmetries with braidings defined merely on objects. We first describe relative morphisms in this general setting, and apply this result to prove that the braided cochain complex of quasi-triangular bialgebras is a braided Morita invariant under some hypotheses.
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BibTeX
@misc{inoue2026braided,
title = {{Braided cohomology of quasi-triangular bialgebras and braided Morita invariance}},
author = {Shota Inoue and Ayako Itaba},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.09107},
eprint = {2606.09107},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.09107},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Braided cohomology of quasi-triangular bialgebras and braided Morita invariance
AU - Shota Inoue
AU - Ayako Itaba
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2606.09107
UR - https://arxiv.org/abs/2606.09107
AB - In this paper, we introduce the braided cochain complex and the braided cohomology of braided coalgebras in linear monoidal categories, and compare those of braided coalgebras living in different linear monoidal categories using relative morphisms. The symmetric cohomology was introduced for groups by Staic, and was generalized to cocommutative Hopf algebras by Shiba, Sanada, and the second author. This notion involves degreewise actions of the symmetric groups on a cochain complex, which come from the symmetry on the monoidal category of group representations. We generalize this framework by dealing with arbitrary linear monoidal categories, and by replacing symmetries with braidings defined merely on objects. We first describe relative morphisms in this general setting, and apply this result to prove that the braided cochain complex of quasi-triangular bialgebras is a braided Morita invariant under some hypotheses.
ER -