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Dual canonical bases of quantum groups and $\imath$quantum groups II: geometry

Ming Lu, Xiaolong Pan

Abstract

The $\imath$quantum groups admit 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. Based on these two realizations, we establish the dual canonical bases for $\imath$quantum groups of type ADE in two distinct ways, using perverse sheaves and Hall algebras respectively. In this paper, we prove that these two dual canonical bases coincide, thereby proving their invariance under braid group actions, and that their structural constants are integral and positive. Furthermore, we establish the positivity of the coefficients of the transition matrix from the Hall basis (and PBW basis) to the dual canonical basis.

Dual canonical bases of quantum groups and $\imath$quantum groups II: geometry

Abstract

The quantum groups admit 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. Based on these two realizations, we establish the dual canonical bases for quantum groups of type ADE in two distinct ways, using perverse sheaves and Hall algebras respectively. In this paper, we prove that these two dual canonical bases coincide, thereby proving their invariance under braid group actions, and that their structural constants are integral and positive. Furthermore, we establish the positivity of the coefficients of the transition matrix from the Hall basis (and PBW basis) to the dual canonical basis.
Paper Structure (32 sections, 55 theorems, 238 equations)

This paper contains 32 sections, 55 theorems, 238 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Goal
  4. Main results
  5. Perspectives
  6. Organization
  7. Quantum groups and $\imath$quantum groups
  8. Quantum groups
  9. The $\imath$quantum groups
  10. $\imath$Quiver algebras and $\imath$Hall algebras
  11. Notations for quivers and representations
  12. The $\imath$quivers and doubles
  13. $\imath$Hall algebras
  14. Generic $\imath$Hall algebras
  15. Dual canonical bases of $\imath$Hall aglebras
  16. ...and 17 more sections

Key Result

Lemma 2.1

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 (89)

  • Lemma 2.1: cf. BG17
  • Example 2.2
  • Lemma 2.3
  • Example 3.1: $\imath$quivers of diagonal type
  • Lemma 3.2: LW19
  • Lemma 3.3: cf. LW19
  • Lemma 3.4: cf. LW19
  • Lemma 3.5: Bridgeland's Theorem reformulated
  • Lemma 3.6: LP25
  • Theorem 3.7: LP25
  • ...and 79 more