ICANEWS

Natural Sciences

Affine and Cyclotomic Brauer Categorification via $\imath$-Kac--Moody $2$-Categories

This research establishes a categorical connection between affine Brauer-type representation theory and the half-integral quasi-split type $\operatorname{AIII}$

Overview

This research establishes a categorical connection between affine Brauer-type representation theory and the half-integral quasi-split type $\operatorname{AIII}$ $\imath$-Kac--Moody $2$-category, specifically the one developed by Bao, Shan, Wang, and Webster. The work focuses on developing a framework to understand cyclotomic Brauer or cyclotomic Nazarov--Wenzl algebras within the context of $\imath$-Kac--Moody $2$-categories.

Research Context

Cyclotomic Brauer or cyclotomic Nazarov--Wenzl algebras are significant in higher Schur--Weyl dualities. These dualities involve parabolic categories $\mathcal O$ for Lie algebras of types $B$, $C$, and $D$. The connection of these algebras with Kac--Moody-type categorification is noted as being less developed compared to the type $A$ theory corresponding to cyclotomic Hecke algebras.

Approach

The research constructs a categorical bridge by investigating actions between different mathematical structures. Specifically, an action of the affine Brauer category on a locally Schurian category is considered. This specific action is characterized by a dot spectrum identified as $\frac{1}{2}+\mathbb Z$. This action determines a generalized nilpotent $2$-representation of the even component $\mathfrak U^{\imath}_{+}$.

Conversely, the approach also examines nilpotent $2$-subrepresentations. Any nilpotent $2$-subrepresentation of an ambient locally Schurian $2$-representation of $\mathfrak U^{\imath}_{+}$ is shown to carry a compatible affine Brauer action. The dot spectrum for this compatible action is contained within $\frac{1}{2}+\mathbb Z$.

These constructions are then applied to cyclotomic quotients. Through this application, the research proves an isomorphism between two types of locally unital algebras: one attached to a $\mathbf u$-admissible cyclotomic Brauer category and another attached to the corresponding cyclotomic quotient of the principal $2$-representation of $\mathfrak U^{\imath}_{+}$.

Findings

  • A categorical bridge between affine Brauer-type representation theory and the half-integral quasi-split type $\operatorname{AIII}$ $\imath$-Kac--Moody $2$-category of Bao--Shan--Wang--Webster was constructed.
  • An action of the affine Brauer category on a locally Schurian category, with a dot spectrum of $\frac{1}{2}+\mathbb Z$, determines a generalized nilpotent $2$-representation of the even component $\mathfrak U^{\imath}_{+}$.
  • Every nilpotent $2$-subrepresentation of an ambient locally Schurian $2$-representation of $\mathfrak U^{\imath}_{+}$ carries a compatible affine Brauer action whose dot spectrum is contained in $\frac{1}{2}+\mathbb Z$.
  • Applying these constructions to cyclotomic quotients, it was proven that the locally unital algebra attached to a $\mathbf u$-admissible cyclotomic Brauer category is isomorphic to the locally unital algebra attached to the corresponding cyclotomic quotient of the principal $2$-representation of $\mathfrak U^{\imath}_{+}$.
  • Consequently, the associated cyclotomic Brauer (or cyclotomic Nazarov--Wenzl) algebras acquire natural $\mathbb Z$-gradings.
  • This work represents the first categorical realization of $\mathbf u$-admissible half-integral cyclotomic Brauer algebras using an $\imath$-Kac--Moody $2$-category.

Why This Matters

This work provides a categorical and graded framework. It is anticipated to contribute to a Brauer-type extension of the Brundan--Kleshchev--Ariki theory. In this extended theory, coideal algebras and $\imath$-canonical bases are expected to replace ordinary quantum groups and canonical bases.

Key Findings

  • Construction of a categorical bridge between affine Brauer-type representation theory and the half-integral quasi-split type $\operatorname{AIII}$ $\imath$-Kac--Moody $2$-category.
  • An action of the affine Brauer category on a locally Schurian category (with dot spectrum $\frac{1}{2}+\mathbb Z$) determines a generalized nilpotent $2$-representation of $\mathfrak U^{\imath}_{+}$.
  • Nilpotent $2$-subrepresentations of $\mathfrak U^{\imath}_{+}$ carry compatible affine Brauer actions with dot spectrum contained in $\frac{1}{2}+\mathbb Z$.
  • Proof that the locally unital algebra attached to a $\mathbf u$-admissible cyclotomic Brauer category is isomorphic to that of the corresponding cyclotomic quotient of the principal $2$-representation of $\mathfrak U^{\imath}_{+}$.
  • Associated cyclotomic Brauer (or cyclotomic Nazarov--Wenzl) algebras acquire natural $\mathbb Z$-gradings.