Quasi-cellular categories and the structure of the Brauer category
Abstract
We introduce the notion of a quasi-cellular category, which is a variant of cellular algebras and cellular categories. Many diagram categories, such as the Brauer category and the partition category, are quasi-cellular. We study the Brauer category B over a commutative ring k with parameter δ through its quasi-cellular structure. We construct a linear functor L:B→ mB to the \emph{matrix Brauer category} mB, which has the same objects and hom-spaces as B but a matrix-like composition. For a non-singular parameter δ, we show that L is an isomorphism of linear categories, and that B is Morita equivalent to its subcategory 𝕊 spanned by permutations, whose endomorphism algebras are the group algebras of symmetric groups. These results generalize those of Brown and König and Xi for the Brauer algebras.
Related
- Affine and cyclotomic Brauer categorification via imath-Kac--Moody $2$-categories: the integral type AIII case
- Monoidal Gröbner systems and categories of affine Brauer type
- A Categorical Realization of the (2-)Category of Monoids via Sch{ü}tzenberger Categories and Strict Factorization Systems
- An example of the non-existence of coequalizers in the category of the A∞-categories
- Affine and cyclotomic Brauer categorification via imath-Kac--Moody $2$-categories: the half-integral type AIII case
BibTeX
@misc{habiro2026quasicellular,
title = {{Quasi-cellular categories and the structure of the Brauer category}},
author = {Kazuo Habiro and Mai Katada},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34203},
eprint = {2609.34203},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34203},
}
RIS
TY - UNPB
TI - Quasi-cellular categories and the structure of the Brauer category
AU - Kazuo Habiro
AU - Mai Katada
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34203
UR - https://arxiv.org/abs/2609.34203
AB - We introduce the notion of a quasi-cellular category, which is a variant of cellular algebras and cellular categories. Many diagram categories, such as the Brauer category and the partition category, are quasi-cellular. We study the Brauer category B over a commutative ring k with parameter δ through its quasi-cellular structure. We construct a linear functor L:B→ mB to the \emph{matrix Brauer category} mB, which has the same objects and hom-spaces as B but a matrix-like composition. For a non-singular parameter δ, we show that L is an isomorphism of linear categories, and that B is Morita equivalent to its subcategory 𝕊 spanned by permutations, whose endomorphism algebras are the group algebras of symmetric groups. These results generalize those of Brown and König and Xi for the Brauer algebras.
ER -