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Dual canonical bases of quantum groups and $\imath$quantum groups

Ming Lu, Xiaolong Pan

TL;DR

This work develops a comprehensive framework unifying $ extit{i}$Hall algebras and quantum Grothendieck rings via NKS quiver varieties to construct and study dual canonical bases for ADE-type $ extit{i}$quantum groups, showing these bases are bar-invariant, integral, and braid-group invariant. It introduces Fourier transforms that preserve these bases and demonstrates the positivity of transition matrices from Hall and PBW bases to the dual canonical basis. A key achievement is proving the coincidence of Berenstein–Greenstein’s double canonical basis with the dual canonical basis in the diagonal (quantum group) case, resolving conjectures in BG17. The results extend the dual canonical basis to $ extit{i}$quivers with integral forms, offer a robust geometric- Hall-algebraic picture, and connect to perverse sheaf frameworks, paving the way for broader type extensions and cluster-algebra links.

Abstract

The $\imath$quantum groups have two realizations: one via the $\imath$Hall algebras and the other via the quantum Grothendieck rings of quiver varieties, as developed by the first author and Wang. Perverse sheaves provide the dual canonical bases for $\imath$quantum groups of type ADE with integral and positive structure constants. In this paper, we present a new construction of the dual canonical bases in the setting of $\imath$Hall algebras. We also introduce Fourier transforms for both $\imath$Hall algebras and quantum Grothendieck rings, and prove the invariance of the dual canonical bases under braid group actions and Fourier transforms. Additionally, we establish the positivity of the transition matrix coefficients from the Hall basis (and PBW basis) to the dual canonical basis. As quantum groups can be regarded as $\imath$quantum groups of diagonal type, we demonstrate that the dual canonical bases of Drinfeld double quantum groups coincide with the double canonical bases defined by Berenstein and Greenstein, resolving several conjectures therein.

Dual canonical bases of quantum groups and $\imath$quantum groups

TL;DR

This work develops a comprehensive framework unifying Hall algebras and quantum Grothendieck rings via NKS quiver varieties to construct and study dual canonical bases for ADE-type quantum groups, showing these bases are bar-invariant, integral, and braid-group invariant. It introduces Fourier transforms that preserve these bases and demonstrates the positivity of transition matrices from Hall and PBW bases to the dual canonical basis. A key achievement is proving the coincidence of Berenstein–Greenstein’s double canonical basis with the dual canonical basis in the diagonal (quantum group) case, resolving conjectures in BG17. The results extend the dual canonical basis to quivers with integral forms, offer a robust geometric- Hall-algebraic picture, and connect to perverse sheaf frameworks, paving the way for broader type extensions and cluster-algebra links.

Abstract

The quantum groups have two realizations: one via the Hall algebras and the other via the quantum Grothendieck rings of quiver varieties, as developed by the first author and Wang. Perverse sheaves provide the dual canonical bases for quantum groups of type ADE with integral and positive structure constants. In this paper, we present a new construction of the dual canonical bases in the setting of Hall algebras. We also introduce Fourier transforms for both Hall algebras and quantum Grothendieck rings, and prove the invariance of the dual canonical bases under braid group actions and Fourier transforms. Additionally, we establish the positivity of the transition matrix coefficients from the Hall basis (and PBW basis) to the dual canonical basis. As quantum groups can be regarded as quantum groups of diagonal type, we demonstrate that the dual canonical bases of Drinfeld double quantum groups coincide with the double canonical bases defined by Berenstein and Greenstein, resolving several conjectures therein.
Paper Structure (47 sections, 114 theorems, 482 equations)

This paper contains 47 sections, 114 theorems, 482 equations.

Table of Contents

  1. Introduction
  2. Background on Hall algebras
  3. Background on quiver varieties
  4. Goal
  5. Main results
  6. Dual canonical bases via $\imath$Hall algebras
  7. Comparison of dual canonical bases
  8. Fourier transforms
  9. Double canonical bases of quantum groups
  10. Perspectives
  11. Organization
  12. Quantum groups and $\imath$quantum groups
  13. Quantum groups
  14. The $\imath$quantum groups
  15. $\imath$Quiver algebras and $\imath$Hall algebras
  16. ...and 32 more sections

Key Result

Lemma 2.2

There exists an anti-involution $u\mapsto \overline{u}$ on $\widehat{{\mathbf U}}$ (also $\widetilde{{\mathbf U}}$, ${\mathbf U}$) given by $\overline{v^{1/2}}=v^{-1/2}$, $\overline{E_i}=E_i$, $\overline{F_i}=F_i$, and $\overline{K_i}=K_i$, $\overline{K_i'}=K_i'$, for $i\in\mathbb{I}$.

Theorems & Definitions (203)

  • Remark 2.1
  • Lemma 2.2: cf. BG17
  • proof
  • Remark 2.3
  • Proposition 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6: Lus90a
  • Lemma 2.7
  • Example 2.8
  • ...and 193 more