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Braid group invariance of quantum tori and applications

  • Francesca Paganelli
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Abstract

We establish that the skew-symmetric bicharacter on the group of ℓ-weights of a quantum affine algebra is invariant under Chari's braid group action. Furthermore, we define an extension of this bicharacter to the group generated by prefundamental ℓ-weights and show that this invariance continues to hold under the extended braid group action of Frenkel-Hernandez. As immediate applications, we obtain braid group actions by automorphisms on the quantum tori associated respectively with the category of finite-dimensional representations of the quantum affine algebra and with the category 𝒪b of its Borel subalgebra. In addition, we derive explicit formulas for quantum QQ-systems in all simply-laced types. Finally, we show that our extended bicharacter yields a quantization matrix compatible with the Hernandez-Leclerc cluster algebra categorified by 𝒪b for all finite Dynkin types, extending Bittmann's construction in the simply-laced setting.

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Cite

BibTeXRIS
BibTeX
@misc{paganelli2026braid,
  title = {{Braid group invariance of quantum tori and applications}},
  author = {Francesca Paganelli},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06384},
  eprint = {2610.06384},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06384},
}
RIS
TY  - UNPB
TI  - Braid group invariance of quantum tori and applications
AU  - Francesca Paganelli
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.06384
UR  - https://arxiv.org/abs/2610.06384
AB  - We establish that the skew-symmetric bicharacter on the group of ℓ-weights of a quantum affine algebra is invariant under Chari's braid group action. Furthermore, we define an extension of this bicharacter to the group generated by prefundamental ℓ-weights and show that this invariance continues to hold under the extended braid group action of Frenkel-Hernandez. As immediate applications, we obtain braid group actions by automorphisms on the quantum tori associated respectively with the category of finite-dimensional representations of the quantum affine algebra and with the category 𝒪^b of its Borel subalgebra. In addition, we derive explicit formulas for quantum QQ-systems in all simply-laced types. Finally, we show that our extended bicharacter yields a quantization matrix compatible with the Hernandez-Leclerc cluster algebra categorified by 𝒪^b for all finite Dynkin types, extending Bittmann's construction in the simply-laced setting.
ER  -