仿射与分圆 Brauer 范畴化通过 $\imath$-Kac--Moody $2$-范畴：半整数型 $\operatorname{AIII}$ 情形
Affine and cyclotomic Brauer categorification via $\imath$-Kac--Moody $2$-categories: the half-integral type $\operatorname{AIII}$ case
浏览论文内容
中文总结 AI 辅助
本文通过 $\imath$-Kac--Moody $2$-范畴实现半整数型 AIII 仿射与分圆 Brauer 代数的范畴化，建立仿射 Brauer 作用与幂零 2-表示的双向对应，并证明分圆商代数同构，首次给出此类代数的自然 Z-分次。
中文摘要 AI 辅助
分圆 Brauer 代数或分圆 Nazarov--Wenzl 代数出现在涉及类型 $B,C$ 和 $D$ 的李代数的抛物范畴 $\mathcal O$ 的高阶 Schur--Weyl 对偶中。它们与 Kac--Moody 型范畴化的联系远不如分圆 Hecke 代数对应的类型 $A$ 理论那样发达。我们构建了仿射 Brauer 型表示论与 Bao--Shan--Wang--Webster 的半整数拟分裂型 $\operatorname{AIII}$ $\imath$-Kac--Moody $2$-范畴之间的范畴桥梁。更精确地说，仿射 Brauer 范畴在局部 Schurian 范畴上的作用，其点谱恰为 $\frac{1}{2}+\mathbb Z$，决定了偶分量 $\mathfrak U^{\imath}_{+}$ 的广义幂零 $2$-表示。反之，$\mathfrak U^{\imath}_{+}$ 的环境局部 Schurian $2$-表示的每个幂零 $2$-子表示都承载一个相容的仿射 Brauer 作用，其点谱包含于 $\frac{1}{2}+\mathbb Z$ 中。将这些构造应用于分圆商，我们证明了与 $\mathbf u$-容许分圆 Brauer 范畴相关的局部幺半代数同构于与 $\mathfrak U^{\imath}_{+}$ 的主 $2$-表示的相应分圆商相关的局部幺半代数。因此，相关的分圆 Brauer（或分圆 Nazarov--Wenzl）代数获得了自然的 $\mathbb Z$-分次。据我们所知，这是首次通过 $\imath$-Kac--Moody $2$-范畴实现对 $\mathbf u$-容许半整数分圆 Brauer 代数的此类范畴实现。它为 Brundan--Kleshchev--Ariki 理论的 Brauer 型推广提供了范畴和分次框架，在该推广中，余理想代数和 $\imath$-典范基有望取代普通量子群和典范基。
英文摘要
Cyclotomic Brauer or cyclotomic Nazarov--Wenzl algebras arise in higher Schur--Weyl dualities involving parabolic categories $\mathcal O$ 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 $\operatorname{AIII}$ $\imath$-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 $\frac{1}{2}+\mathbb Z$, determines a generalized nilpotent $2$-representation of the even component $\mathfrak U^{\imath}_{+}$. Conversely, 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, we prove 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. To our knowledge, this is the first such categorical realization of $\mathbf u$-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.
发表机构
- Tongji University（同济大学）
机构由 AI 辅助整理，请以论文原文为准。
