[Paper Review] Categorification of quantum symmetric pairs I

Huanchen Bao, Peng Shan+2 more|arXiv (Cornell University)|May 12, 2016

Algebraic structures and combinatorial models27 references3 citations

TL;DR

This paper introduces a 2-category categorification of the coideal subalgebra $\dot{U}^\jmath$ of the quantum group $U(\mathfrak{sl}_{2r+1})$, generalizing Khovanov-Lauda-Rouquier categorification to quantum symmetric pairs. It constructs a 2-category $\dot{U}^\jmath$ with new relations at the special index $i = \diamond$, including modified $\jmath$-Serre relations and a novel triple point relation, and proves that self-dual indecomposable 1-morphisms categorify the canonical basis of $\dot{U}^\jmath$. The categorification lifts to categorical actions on Schur categories and category $\mathcal{O}$ of type B/C, extending Schur duality to quantum symmetric pairs.

ABSTRACT

We categorify a coideal subalgebra of the quantum group of $\mathfrak{sl}_{2r+1}$ by introducing a $2$-category à la Khovanov-Lauda-Rouquier, and show that self-dual indecomposable $1$-morphisms categorify the canonical basis of this algebra. This allows us to define a categorical action of this coideal algebra on the categories of modules over cohomology rings of partial flag varieties and on the BGG category $\mathcal{O}$ of type B/C.

Motivation & Objective

• To construct a 2-category $\dot{U}^\jmath$ categorifying the coideal subalgebra $\dot{U}^\jmath$ of $U(\mathfrak{sl}_{2r+1})$.
• To generalize the Khovanov-Lauda-Rouquier categorification of quantum groups to the setting of quantum symmetric pairs.
• To establish that self-dual indecomposable 1-morphisms in $\dot{U}^\jmath$ categorify the canonical basis of $\dot{U}^\jmath$.
• To define a strong categorical action of $\dot{U}^\jmath$ on the category of modules over cohomology rings of partial flag varieties of type B/C and on category $\mathcal{O}$ of type B/C.

Proposed method

• Introduce a 2-category $\dot{U}^\jmath$ analogous to the KLR 2-category $\dot{U}$, with generators $E_i$, $F_i$ for $i = \diamond, \dots, r - \diamond$.
• Define new 2-morphisms and relations for the special index $i = \diamond$, including modified $\jmath$-Serre relations and a triple point relation (3.17).
• Use bubble slide formulas and relations (3.9), (3.17) to prove the categorified $\jmath$-Serre relations in the Grothendieck group.
• Construct a 2-representation on the Schur 2-category via Soergel bimodules and Demazure operators in type B/C.
• Lift the action to a strong categorical action on the category of modules over cohomology rings of partial flag varieties of type B/C.
• Establish a categorical action on category $\mathcal{O}$ of type B/C via Harish-Chandra bimodules and the coideal algebra $U^\jmath$.

Experimental results

Research questions

• RQ1 Can the coideal subalgebra $\dot{U}^\jmath$ of $U(\mathfrak{sl}_{2r+1})$ be categorified via a 2-category structure?
• RQ2 What new relations arise in the categorification of $\dot{U}^\jmath$ at the special index $i = \diamond$, and how do they differ from standard KLR relations?
• RQ3 Do self-dual indecomposable 1-morphisms in the 2-category $\dot{U}^\jmath$ correspond to the canonical basis of $\dot{U}^\jmath$?
• RQ4 Can the categorified $\jmath$-Serre relations be proven using bubble slide identities and new 2-morphism relations?
• RQ5 Does $\dot{U}^\jmath$ admit a strong categorical action on the Schur 2-category and on category $\mathcal{O}$ of type B/C?

Key findings

• The 2-category $\dot{U}^\jmath$ is constructed with generators $E_i$, $F_i$ and new 2-morphisms satisfying modified $\jmath$-Serre relations at $i = \diamond$.
• The categorified $\jmath$-Serre relations are proven via a combination of bubble slide identities and the triple point relation (3.17).
• Self-dual indecomposable 1-morphisms in $\dot{U}^\jmath$ categorify the canonical basis of $\dot{U}^\jmath$, as shown in Theorem 6.1.
• The 2-category $\dot{U}^\jmath$ acts categorically on the Schur 2-category of type B/C, with the action decategorifying to the known action on the Grothendieck group.
• A strong categorical action of $\dot{U}^\jmath$ is constructed on the category of modules over cohomology rings of partial flag varieties of type B/C.
• A categorical action of $\dot{U}^\jmath$ is defined on category $\mathcal{O}$ of type B/C via Harish-Chandra bimodules, extending the weak action to a strong one.