Monoidal Categorification and Quantization of Braid Varieties

Abstract

Let G be a simple and simply connected algebraic group of simply-laced type. For each positive braid word β, and for the complete strong duality datum attached to a Q-datum, we construct an explicit based monoidal categorification of the quantum cluster algebra of the braid variety X(β). We identify this algebra with both a localized quantum Grothendieck ring and a localized level-≥1 subalgebra of the bosonic extension algebra. Under these identifications, quantum cluster monomials correspond simultaneously to real simple modules and normalized global basis elements. We construct the specialization homomorphism at q^1/2=1 and prove that it recovers CC[X(β)]; in particular, the resulting integral form is a flat quantum deformation. We also give intrinsic Lusztig parameters for cluster variables attached to double strings, identify the corresponding quantum grid minors, and establish generalized quantum T-systems.

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BibTeX RIS

BibTeX

@misc{bi2026monoidal,
  title = {{Monoidal Categorification and Quantization of Braid Varieties}},
  author = {Yingjin Bi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.02962},
  eprint = {2609.02962},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02962},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Monoidal Categorification and Quantization of Braid Varieties
AU  - Yingjin Bi
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.02962
UR  - https://arxiv.org/abs/2609.02962
AB  - Let G be a simple and simply connected algebraic group of simply-laced type. For each positive braid word β, and for the complete strong duality datum attached to a Q-datum, we construct an explicit based monoidal categorification of the quantum cluster algebra of the braid variety X(β). We identify this algebra with both a localized quantum Grothendieck ring and a localized level-≥1 subalgebra of the bosonic extension algebra. Under these identifications, quantum cluster monomials correspond simultaneously to real simple modules and normalized global basis elements. We construct the specialization homomorphism at q^1/2=1 and prove that it recovers CC[X(β)]; in particular, the resulting integral form is a flat quantum deformation. We also give intrinsic Lusztig parameters for cluster variables attached to double strings, identify the corresponding quantum grid minors, and establish generalized quantum T-systems.
ER  -

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