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Dual and double canonical bases of quantum groups

Ming Lu, Xiaolong Pan

Abstract

Qin established the geometric realization of entire quantum groups via perverse sheaves, which further give rise to dual canonical bases with integral and positive structure constants for quantum groups of type ADE. In this paper, we prove that the dual canonical bases of (Drinfeld double) quantum groups coincide with Berenstein--Greenstein's double canonical bases, by reinterpreting their intricate algebraic construction via the geometry of NKS quiver varieties. This result settles several conjectures therein, including those on positivity and invariance under braid group actions.

Dual and double canonical bases of quantum groups

Abstract

Qin established the geometric realization of entire quantum groups via perverse sheaves, which further give rise to dual canonical bases with integral and positive structure constants for quantum groups of type ADE. In this paper, we prove that the dual canonical bases of (Drinfeld double) quantum groups coincide with Berenstein--Greenstein's double canonical bases, by reinterpreting their intricate algebraic construction via the geometry of NKS quiver varieties. This result settles several conjectures therein, including those on positivity and invariance under braid group actions.
Paper Structure (12 sections, 42 theorems, 177 equations)

This paper contains 12 sections, 42 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Quantum groups
  3. Quantum groups via quiver varieties
  4. NKS categories
  5. NKS quiver varieties
  6. Quantum Grothendieck rings
  7. An algebraic construction of dual canonical bases
  8. Double canonical basis
  9. Berenstein-Greenstein's constructions
  10. $\mathbb{C}^*$-actions on NKS quiver varieties
  11. Coincidence with dual canonical basis
  12. Dual canonical basis of $\widetilde{{\mathbf U}}_v(\mathfrak{sl}_2)$

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 (73)

  • Conjecture 1.1
  • Remark 2.1
  • Lemma 2.2: cf. BG17
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5: Lus90a
  • Definition 3.1
  • Proposition 3.2: SS16; see also VV,Qin
  • Lemma 3.3: Qin
  • ...and 63 more