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Faithful action of braid group on bosonic extensions

Masaki Kashiwara, Myungho Kim, Se-jin Oh, Euiyong Park

TL;DR

The paper proves that the braid group action on the bosonic extension $\widehat{\mathcal{A}}$ of a finite-type quantum group is faithful, extending Lusztig's braid symmetries to this setting. It develops the automorphisms $\mathbf{T}_i$ (and $\mathbf{T}_i^*$), establishes their braid relations, and analyzes the induced action on weight-graded subalgebras via PBW-type bases and the Garside structure. A nondegenerate symmetric bilinear form $(\cdot,\cdot)_{\widehat{\mathcal{A}}}$ together with adjoint operators $\mathrm{E}_{i,m}$ and $\mathrm{E}^*_{i,m}$ provides a mechanism to detect nontrivial braid elements, culminating in a separation argument that proves faithfulness. This result lays the groundwork for deeper connections between braid group actions and the algebraic structure of bosonic extensions, with potential applications to categorification and quantum coordinate algebras.

Abstract

The braid group action on the bosonic extension of the quantum group has been introduced in recent works, and it can be regarded as a generalization of Lusztig's symmetries on the quantum group. In this notes, we prove the faithfulness of this braid group action.

Faithful action of braid group on bosonic extensions

TL;DR

The paper proves that the braid group action on the bosonic extension of a finite-type quantum group is faithful, extending Lusztig's braid symmetries to this setting. It develops the automorphisms (and ), establishes their braid relations, and analyzes the induced action on weight-graded subalgebras via PBW-type bases and the Garside structure. A nondegenerate symmetric bilinear form together with adjoint operators and provides a mechanism to detect nontrivial braid elements, culminating in a separation argument that proves faithfulness. This result lays the groundwork for deeper connections between braid group actions and the algebraic structure of bosonic extensions, with potential applications to categorification and quantum coordinate algebras.

Abstract

The braid group action on the bosonic extension of the quantum group has been introduced in recent works, and it can be regarded as a generalization of Lusztig's symmetries on the quantum group. In this notes, we prove the faithfulness of this braid group action.

Paper Structure

This paper contains 9 sections, 11 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Bosonic extension
  3. Cartan matrix and associated data
  4. Bosonic extensions
  5. Bilinear form and homomorphisms
  6. Braid group action on $\widehat{\mathcal{A}}$
  7. Braid and Weyl groups
  8. Braid group actions on $\mathcal{U}_q(\mathfrak{g})$ and $\widehat{\mathcal{A}}$
  9. Faithfulness

Key Result

Theorem 2.2

For any $m\in\mathbb{Z}\mspace{1mu}$, the subalgebra $\widehat{\mathcal{A}}[m]$ is isomorphic to the negative half $\mathcal{U}_q^-(\mathfrak{g})$ of the quantum group $\mathcal{U}_q(\mathfrak{g})$ associated with $\mathsf{C}$. Here $\mathcal{U}_q(\mathfrak{g})$ ( resp. $\mathcal{U}_q^-(\mathfrak{g} defined by $x_b \otimes x_{b-1} \otimes \cdots \otimes x_{a+1} \otimes x_a \mapsto x_b x_{b-1} \cdo

Theorems & Definitions (16)

  • Definition 2.1
  • Theorem 2.2: KKOP24
  • Definition 2.3: KKOP24
  • Theorem 2.4: KKOP24
  • Lemma 3.1: see OP24
  • Proposition 3.2: Gar69 and see also KT08
  • Theorem 3.3: Garside left normal form (see WPEM94)
  • Theorem 3.4: KKOP24B
  • Lemma 3.5: JLO2, OP24
  • Lemma 3.6: KKOP24B
  • ...and 6 more