Affine and cyclotomic Brauer categorification via imath-Kac--Moody $2$-categories: the integral type AIII case
Abstract
Continuing our work on the half-integral case, we construct generalized nilpotent $2$-representations of the integral quasi-split type AIII imath-Kac--Moody $2$-category from affine Brauer actions with dot spectrum exactly ℤ on locally Schurian ℂ-linear categories. Conversely, nilpotent $2$-representations on such categories yield affine Brauer actions with spectrum contained in ℤ. At the fixed vertex $0$, we directly define corrected generators and verify the nil-Brauer and mixed relations. For u-admissible integral parameters at arbitrary positive level, we prove an isomorphism between the locally unital algebras of the cyclotomic Brauer category and a cyclotomic quotient of the principal $2$-representation. This gives compatible ℤ-gradings on the category and its endomorphism algebras, providing a Brauer analogue of the graded realization of type A cyclotomic Hecke algebras. Over algebraically closed fields of odd characteristic p, we also obtain both constructions with spectrum contained in 𝔽p, for the mixed affine datum restricted to its spectral weight subgroup.
Related
- Affine and cyclotomic Brauer categorification via imath-Kac--Moody $2$-categories: the half-integral type AIII case
- Monoidal Gröbner systems and categories of affine Brauer type
- Brauer and iquantum categorification
- Brauer webs of finite and affine type
- Quasi-cellular categories and the structure of the Brauer category
BibTeX
@misc{gao2026affine,
title = {{Affine and cyclotomic Brauer categorification via $\imath$-Kac--Moody $2$-categories: the integral type $\operatornameAIII$ case}},
author = {Mengmeng Gao and Hebing Rui and Linliang Song and Jun Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32589},
eprint = {2610.32589},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32589},
}
RIS
TY - UNPB
TI - Affine and cyclotomic Brauer categorification via imath-Kac--Moody $2$-categories: the integral type AIII case
AU - Mengmeng Gao
AU - Hebing Rui
AU - Linliang Song
AU - Jun Wu
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32589
UR - https://arxiv.org/abs/2610.32589
AB - Continuing our work on the half-integral case, we construct generalized nilpotent $2$-representations of the integral quasi-split type AIII imath-Kac--Moody $2$-category from affine Brauer actions with dot spectrum exactly ℤ on locally Schurian ℂ-linear categories. Conversely, nilpotent $2$-representations on such categories yield affine Brauer actions with spectrum contained in ℤ. At the fixed vertex $0$, we directly define corrected generators and verify the nil-Brauer and mixed relations. For u-admissible integral parameters at arbitrary positive level, we prove an isomorphism between the locally unital algebras of the cyclotomic Brauer category and a cyclotomic quotient of the principal $2$-representation. This gives compatible ℤ-gradings on the category and its endomorphism algebras, providing a Brauer analogue of the graded realization of type A cyclotomic Hecke algebras. Over algebraically closed fields of odd characteristic p, we also obtain both constructions with spectrum contained in 𝔽_p, for the mixed affine datum restricted to its spectral weight subgroup.
ER -