Quantum symmetric spaces at roots of 1

Abstract

We study the integral form of quantized coordinate algebras of symmetric spaces and its specializations at roots of unity. Our main tool is an integral localization, whose classical specialization recovers the open Borel orbit in the symmetric space. At a suitable odd root of unity, we prove that the quantum symmetric space is a finite projective module over its Frobenius center, which is canonically isomorphic to the classical coordinate algebra of the symmetric space. We show that the fibre algebras are isomorphic along each double Bruhat cell of the symmetric space. We determine the PI degree and propose a conjectural dimension formula for irreducible modules. In the simply connected case, we introduce a framed Kac-Moody data, which leads to a description of fibre algebras of quantum symmetric spaces via those of quantum Schubert cells.

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BibTeX RIS

BibTeX

@misc{song2026quantum,
  title = {{Quantum symmetric spaces at roots of 1}},
  author = {Jinfeng Song and Weinan Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25624},
  eprint = {2610.25624},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25624},
}

RIS

TY  - UNPB
TI  - Quantum symmetric spaces at roots of 1
AU  - Jinfeng Song
AU  - Weinan Zhang
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.25624
UR  - https://arxiv.org/abs/2610.25624
AB  - We study the integral form of quantized coordinate algebras of symmetric spaces and its specializations at roots of unity. Our main tool is an integral localization, whose classical specialization recovers the open Borel orbit in the symmetric space. At a suitable odd root of unity, we prove that the quantum symmetric space is a finite projective module over its Frobenius center, which is canonically isomorphic to the classical coordinate algebra of the symmetric space. We show that the fibre algebras are isomorphic along each double Bruhat cell of the symmetric space. We determine the PI degree and propose a conjectural dimension formula for irreducible modules. In the simply connected case, we introduce a framed Kac-Moody data, which leads to a description of fibre algebras of quantum symmetric spaces via those of quantum Schubert cells.
ER  -

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