Affine and cyclotomic Brauer categorification via imath-Kac--Moody $2$-categories: the half-integral type AIII case

@misc{gao2026affine,
  title = {{Affine and cyclotomic Brauer categorification via $\imath$-Kac--Moody $2$-categories: the half-integral type $\operatornameAIII$ case}},
  author = {Mengmeng Gao and Hebing Rui and Linliang Song},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07018},
  eprint = {2609.07018},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07018},
}

TY  - UNPB
TI  - Affine and cyclotomic Brauer categorification via imath-Kac--Moody $2$-categories: the half-integral type AIII case
AU  - Mengmeng Gao
AU  - Hebing Rui
AU  - Linliang Song
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.07018
UR  - https://arxiv.org/abs/2609.07018
AB  - Cyclotomic Brauer or cyclotomic Nazarov--Wenzl algebras arise in higher Schur--Weyl dualities involving parabolic categories 𝒪 for Lie algebras of types B,C and D. Their connection with Kac--Moody-type categorification is substantially less developed than the corresponding type A theory for cyclotomic Hecke algebras. We construct a categorical bridge between affine Brauer-type representation theory and the half-integral quasi-split type AIII imath-Kac--Moody $2$-category of Bao--Shan--Wang--Webster. More precisely, an action of the affine Brauer category on a locally Schurian category, with dot spectrum exactly 1/2+ℤ, determines a generalized nilpotent $2$-representation of the even component U^imath_+. Conversely, every nilpotent $2$-subrepresentation of an ambient locally Schurian $2$-representation of U^imath_+ carries a compatible affine Brauer action whose dot spectrum is contained in 1/2+ℤ. Applying these constructions to cyclotomic quotients, we prove that the locally unital algebra attached to a u-admissible cyclotomic Brauer category is isomorphic to the locally unital algebra attached to the corresponding cyclotomic quotient of the principal $2$-representation of U^imath_+. Consequently, the associated cyclotomic Brauer (or cyclotomic Nazarov--Wenzl) algebras acquire natural ℤ-gradings. To our knowledge, this is the first such categorical realization of u-admissible half-integral cyclotomic Brauer algebras by means of an imath-Kac--Moody $2$-category. It provides the categorical and graded framework toward a Brauer-type extension of the Brundan--Kleshchev--Ariki theory in which coideal algebras and imath-canonical bases are expected to replace ordinary quantum groups and canonical bases.
ER  -