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Relative braid group symmetries on $\imath$quantum groups of Kac-Moody type

Weinan Zhang

TL;DR

This work generalizes relative braid group actions from finite-type to Kac-Moody-type $\imath$quantum groups by constructing renormalized braids on the universal pair $\big(\widetilde{\mathbf U},\widetilde{\mathbf U}^\imath\big)$ and introducing root vectors in recursive and $\imath$-divided power forms. The core contribution is a uniform higher-rank framework that expresses the action of $\widetilde{\mathbf T}'_{i,-1}$ and $\widetilde{\mathbf T}''_{i,+1}$ via root vectors, including explicit formulas for rank-one and rank-two interactions and their intertwiners with quasi $K$-matrices. The authors prove that these symmetries map root vectors to root vectors and satisfy relative braid relations in the quasi-split and Kac-Moody settings, enabling a braided Drinfeld-type presentation and central reductions to traditional $\imath$quantum groups. This advances the structural understanding of quantum symmetric pairs beyond finite type and paves the way for future Drinfeld-type realizations and representation-theoretic applications in the affine and indefinite regimes.

Abstract

Recently, relative braid group actions on $\imath$quantum groups of arbitrary finite types have been constructed by Wang and the author. In this paper, we extend that construction to $\imath$quantum groups of Kac-Moody type. We formulate root vectors for $\imath$quantum groups in both recursive forms and closed $\imath$divided power forms. We show that the relative braid group symmetries send root vectors to root vectors.

Relative braid group symmetries on $\imath$quantum groups of Kac-Moody type

TL;DR

This work generalizes relative braid group actions from finite-type to Kac-Moody-type quantum groups by constructing renormalized braids on the universal pair and introducing root vectors in recursive and -divided power forms. The core contribution is a uniform higher-rank framework that expresses the action of and via root vectors, including explicit formulas for rank-one and rank-two interactions and their intertwiners with quasi -matrices. The authors prove that these symmetries map root vectors to root vectors and satisfy relative braid relations in the quasi-split and Kac-Moody settings, enabling a braided Drinfeld-type presentation and central reductions to traditional quantum groups. This advances the structural understanding of quantum symmetric pairs beyond finite type and paves the way for future Drinfeld-type realizations and representation-theoretic applications in the affine and indefinite regimes.

Abstract

Recently, relative braid group actions on quantum groups of arbitrary finite types have been constructed by Wang and the author. In this paper, we extend that construction to quantum groups of Kac-Moody type. We formulate root vectors for quantum groups in both recursive forms and closed divided power forms. We show that the relative braid group symmetries send root vectors to root vectors.
Paper Structure (40 sections, 50 theorems, 173 equations)

This paper contains 40 sections, 50 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Goal
  4. Motivation
  5. Main results
  6. New symmetries on $\imath$quantum groups $\widetilde{{\mathbf U}}^\imath$
  7. Higher rank formulas for relative braid group symmetries
  8. $\widetilde{\mathbf{T}}'_{i,-1}$ and root vectors
  9. Relative braid group actions on $\imath$quantum groups $\widetilde{{\mathbf U}}^\imath$
  10. Application
  11. Organization
  12. Preliminaries
  13. Drinfeld doubles and quantum symmetric pairs
  14. Braid group action on Drinfeld double $\widetilde{{\mathbf U}}$
  15. Quantum symmetric pairs
  16. ...and 25 more sections

Key Result

Proposition 2.3

Let $\mathbf{a}=(a_i)_{i\in \mathbb{I}} \in ( \mathbb{F}^\times )^\mathbb{I}$. There exist an automorphism $\widetilde{\Psi}_{\mathbf{a}}$ on the $\mathbb{F}$-algebra $\widetilde{{\mathbf U}}$ such that

Theorems & Definitions (109)

  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3: cf. WZ22
  • Proposition 2.4: cf. WZ22
  • Remark 2.5
  • Lemma 2.6: cf. Lus94
  • Proposition 2.7: cf. WZ22
  • Proposition 2.8: WZ22
  • Proposition 2.9: WZ22
  • Remark 2.10
  • ...and 99 more