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Relative braid group symmetries on modified iquantum groups and their modules

Weiqiang Wang, Weinan Zhang

TL;DR

The article advances the theory of braid group symmetries for quasi-split iquantum groups by deriving three explicit rank-one formulas for the relative braid group actions on integrable U^-modules, expressed via idivided powers and iweights. It establishes compatibility with the relative braid group framework through quasi K-matrices, and connects these actions to Lusztig’s symmetries, providing a bridge between iquantum and quantum group settings. By introducing rescalings, it extends these symmetries to integral forms of modified iquantum groups and their modules, and proves the relative braid relations in this integral framework. The work also develops new bases and monomial constructions (notably in l_3]-settings) to facilitate these identities, potentially enabling iRickard-type constructions and broader categorical applications. Overall, the paper delivers a comprehensive, rank-one–driven foundation for explicit, integrality-preserving relative braid group actions across split, diagonal, and quasi-split types in the quasi-split iquantum context, with significant implications for representation theory and quantum symmetric pairs.

Abstract

We present a comprehensive generalization of Lusztig's braid group symmetries for quasi-split iquantum groups. Specifically, we give 3 explicit rank one formulas for symmetries acting on integrable modules over a quasi-split iquantum group of arbitrary Kac-Moody type with general parameters. These symmetries are formulated in terms of idivided powers and iweights of the vectors being acted upon. We show that these symmetries are consistent with the relative braid group symmetries on iquantum groups, and they are also related to Lusztig's symmetries via quasi $K$-matrices. Furthermore, through appropriate rescaling, we obtain compatible symmetries for the integral forms of modified iquantum groups and their integrable modules. We verify that these symmetries satisfy the relative braid group relations.

Relative braid group symmetries on modified iquantum groups and their modules

TL;DR

The article advances the theory of braid group symmetries for quasi-split iquantum groups by deriving three explicit rank-one formulas for the relative braid group actions on integrable U^-modules, expressed via idivided powers and iweights. It establishes compatibility with the relative braid group framework through quasi K-matrices, and connects these actions to Lusztig’s symmetries, providing a bridge between iquantum and quantum group settings. By introducing rescalings, it extends these symmetries to integral forms of modified iquantum groups and their modules, and proves the relative braid relations in this integral framework. The work also develops new bases and monomial constructions (notably in l_3]-settings) to facilitate these identities, potentially enabling iRickard-type constructions and broader categorical applications. Overall, the paper delivers a comprehensive, rank-one–driven foundation for explicit, integrality-preserving relative braid group actions across split, diagonal, and quasi-split types in the quasi-split iquantum context, with significant implications for representation theory and quantum symmetric pairs.

Abstract

We present a comprehensive generalization of Lusztig's braid group symmetries for quasi-split iquantum groups. Specifically, we give 3 explicit rank one formulas for symmetries acting on integrable modules over a quasi-split iquantum group of arbitrary Kac-Moody type with general parameters. These symmetries are formulated in terms of idivided powers and iweights of the vectors being acted upon. We show that these symmetries are consistent with the relative braid group symmetries on iquantum groups, and they are also related to Lusztig's symmetries via quasi -matrices. Furthermore, through appropriate rescaling, we obtain compatible symmetries for the integral forms of modified iquantum groups and their integrable modules. We verify that these symmetries satisfy the relative braid group relations.

Paper Structure

This paper contains 53 sections, 83 theorems, 308 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. What is achieved in this paper?
  4. New rank one formulas
  5. Compatible braid group actions
  6. Formulas for braid group actions on idivided powers
  7. Relative braid group actions and integral forms
  8. Organization
  9. Preliminaries
  10. Quantum groups
  11. The iroot datum and iquantum groups
  12. Relative Weyl groups
  13. Quasi $K$-matrix and relative braid group action
  14. $\mathrm{i}$Divided powers and integrable $\mathbf{U}^\imath$-modules
  15. Formulas for relative braid group symmetries
  16. ...and 38 more sections

Key Result

Theorem A

Let $M$ be an integrable $\mathbf{U}^\imath$-module, and let $i\in \mathbb{I}^{\mathrm{fin}}$. The following identities hold, for any $x\in \mathbf{U}^\imath,v\in M$:

Theorems & Definitions (157)

  • Theorem A: \ref{['thm:split', 'thm:diag', 'thm:qs-split', 'prop:split2', 'prop:diag2', 'prop:quasisplit2']}
  • Theorem B: \ref{['thm:res-split', 'thm:res-diag', 'thm:res-qssplit']}
  • Theorem C: \ref{['thm:split-T1DP', 'thm:diag-T1DP', 'thm:qsplit-T1DP']}
  • Theorem D: \ref{['thm:integralTi']}
  • Theorem E: \ref{['thm:TTd_braid', 'cor:braid']}
  • Lemma 2.1
  • Proposition 2.2: Lus90a
  • Lemma 2.3
  • Lemma 2.4: cf. WZ23
  • Proposition 2.5
  • ...and 147 more