[Paper Review] Canonical bases arising from quantum symmetric pairs

Huanchen Bao, Weiqiang Wang|arXiv (Cornell University)|Oct 28, 2016

Algebraic structures and combinatorial models18 references3 citations

TL;DR

This paper develops a systematic theory of canonical bases for quantum symmetric pairs $(\mathbb{U}, \mathbb{U}^\imath)$ of arbitrary finite type, constructing an $\imath$-canonical basis on the modified $\imath$ quantum group $\dot{\mathbb{U}}^\imath$ and on finite-dimensional simple $\mathbb{U}$-modules and their tensor products as $\mathbb{U}^\imath$-modules. The key contribution is establishing the integrality and compatibility of the intertwiner $\Upsilon$ and braid group actions on $\mathbb{U}^\imath$, enabling a new canonical basis theory for quantum symmetric pairs.

ABSTRACT

We develop a general theory of canonical bases for quantum symmetric pairs $(\mathbf{U}, \mathbf{U}^\imath)$ with parameters of arbitrary finite type. We construct new canonical bases for the simple integrable $\mathbf{U}$-modules and their tensor products regarded as $\mathbf{U}^\imath$-modules. We also construct a canonical basis for the modified form of the $\imath$quantum group $\mathbf{U}^\imath$. To that end, we establish several new structural results on quantum symmetric pairs, such as bilinear forms, braid group actions, integral forms, Levi subalgebras (of real rank one), and integrality of the intertwiners.

Motivation & Objective

• To establish a general theory of canonical bases for quantum symmetric pairs $(\mathbb{U}, \mathbb{U}^\imath)$ of arbitrary finite type.
• To construct an $\imath$-canonical basis on the modified $\imath$ quantum group $\dot{\mathbb{U}}^\imath$.
• To extend the theory to finite-dimensional simple $\mathbb{U}$-modules and their tensor products, now viewed as $\mathbb{U}^\imath$-modules.
• To resolve structural challenges in $\mathbb{U}^\imath$, including integral forms, bar involutions, and braid group actions.
• To provide a foundation for applications in representation theory, categorification, and cluster algebras via a new canonical basis framework.

Proposed method

• Define $\mathbb{U}^\imath$ over $\mathbb{Q}(q)$ with modified parameters to ensure integrality and compatibility with the bar involution $\psi_\imath$.
• Construct the intertwiner $\Upsilon$ in a completion of $\mathbb{U}^+$, which intertwines the bar involution on $\mathbb{U}$ and the $\imath$-bar involution on $\mathbb{U}^\imath$.
• Prove that braid group operators $\texttt{T}'_{i,e}$ and $\texttt{T}''_{i,e}$ restrict to automorphisms of $\mathbb{U}^\imath$ for $i \in \mathbb{I}_\bullet$, verifying a conjecture of [KP11].
• Show that the anti-involution $\wp$ on $\mathbb{U}$ restricts to an anti-involution on $\mathbb{U}^\imath$, enabling a non-degenerate bilinear form on $\mathbb{U}^\imath$-modules.
• Use the bilinear form and the $\imath$-bar involution to define and construct the $\imath$-canonical basis on $\dot{\mathbb{U}}^\imath$ and on based $\mathbb{U}$-modules.
• Establish integrality of the intertwiner $\Upsilon$ and its compatibility with the $\imath$-canonical basis construction over $\mathcal{A} = \mathbb{Z}[q,q^{-1}]$.

Experimental results

Research questions

• RQ1 How can a canonical basis theory be systematically developed for quantum symmetric pairs $(\mathbb{U}, \mathbb{U}^\imath)$ of arbitrary finite type?
• RQ2 What structural properties—such as braid group actions, bilinear forms, and integral forms—must be established to support a canonical basis on $\mathbb{U}^\imath$?
• RQ3 How does the intertwiner $\Upsilon$ behave under the $\imath$-bar involution, and can it be defined over $\mathbb{Q}(q)$ with values in $\mathcal{A}$?
• RQ4 Can the braid group operators $\texttt{T}'_{i,e}$ and $\texttt{T}''_{i,e}$ be restricted to automorphisms of $\mathbb{U}^\imath$, and what are their explicit actions on generators?
• RQ5 How can the anti-involution $\wp$ on $\mathbb{U}$ be restricted to $\mathbb{U}^\imath$ to define a non-degenerate bilinear form on $\mathbb{U}^\imath$-modules?

Key findings

• The braid group operators $\texttt{T}'_{i,e}$ and $\texttt{T}''_{i,e}$ restrict to automorphisms of $\mathbb{U}^\imath$ for all $i \in \mathbb{I}_\bullet$, confirming a conjecture of [KP11].
• The anti-involution $\wp$ on $\mathbb{U}$ restricts to an anti-involution on $\mathbb{U}^\imath$, enabling a non-degenerate symmetric bilinear form on $\mathbb{U}^\imath$-modules.
• The intertwiner $\Upsilon$ is integral and lies in a completion of $\mathbb{U}^+$, and its action is compatible with the $\imath$-bar involution on $\mathbb{U}^\imath$.
• An $\imath$-canonical basis is constructed on the modified $\imath$ quantum group $\dot{\mathbb{U}}^\imath$, generalizing earlier results for type AIII/AIV.
• The $\imath$-canonical basis is constructed on finite-dimensional simple $\mathbb{U}$-modules and their tensor products when viewed as $\mathbb{U}^\imath$-modules.
• The theory is formulated over $\mathbb{Q}(q)$ with parameters in $\mathcal{A} = \mathbb{Z}[q,q^{-1}]$, ensuring integrality and compatibility with the canonical basis framework.