Algebraic study of quantum configuration spaces of decorated flags

Abstract

Let G be a connected, simply connected complex simple algebraic group and 𝒜_G=G/U⁺ its base affine space, whose elements are called decorated flags. We introduce the quantum configuration space of decorated flags 𝒪_q(Conf_K 𝒜_G) and initiate its algebraic study, based on the representation theory of quantized enveloping algebras. Our algebra 𝒪_q(Conf_K 𝒜_G) gives a quantum analogue of the configuration space Conf_K 𝒜_G of K decorated flags, which provides local building blocks for the Fock--Goncharov moduli space 𝒜_G,Σ of decorated twisted G-local systems on a marked surface Σ. We establish basic algebraic properties of 𝒪_q(Conf_K 𝒜_G) such as the domain property, quantum normalization of representatives, the quantum cyclic shifts, the quantum Wilson lines, whose classical counterparts have been fundamental in the study of 𝒜_G,Σ. Moreover, we construct quantum seeds for 𝒪_q(Conf₄ 𝒜_G) by transporting the Berenstein--Zelevinsky quantum cluster structure on 𝒪_q(G) via quantum Wilson lines, and prove that 𝒪_q(Conf₄ 𝒜_G) coincides with the corresponding quantum cluster algebra and its upper counterpart after the localization at frozen variables. The exchange matrices for our quantum seeds agree with the Goncharov--Shen exchange matrices. We also show that quantum seeds for 𝒪_q(Conf₄ 𝒜_G) restrict to those for 𝒪_q(Conf₃ 𝒜_G). Finally, we construct quantum seeds for 𝒪_q(Conf_K 𝒜_G), K≥ 5, and prove that the corresponding quantum cluster algebras contain the quantum configuration space 𝒪_q(Conf_K 𝒜_G).

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BibTeX RIS

BibTeX

@misc{ishibashi2026algebraic,
  title = {{Algebraic study of quantum configuration spaces of decorated flags}},
  author = {Tsukasa Ishibashi and Hironori Oya},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.02509},
  eprint = {2609.02509},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02509},
}

RIS

TY  - UNPB
TI  - Algebraic study of quantum configuration spaces of decorated flags
AU  - Tsukasa Ishibashi
AU  - Hironori Oya
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.02509
UR  - https://arxiv.org/abs/2609.02509
AB  - Let G be a connected, simply connected complex simple algebraic group and 𝒜_G=G/U^+ its base affine space, whose elements are called decorated flags. We introduce the quantum configuration space of decorated flags 𝒪_q(Conf_K 𝒜_G) and initiate its algebraic study, based on the representation theory of quantized enveloping algebras. Our algebra 𝒪_q(Conf_K 𝒜_G) gives a quantum analogue of the configuration space Conf_K 𝒜_G of K decorated flags, which provides local building blocks for the Fock--Goncharov moduli space 𝒜_G,Σ of decorated twisted G-local systems on a marked surface Σ. We establish basic algebraic properties of 𝒪_q(Conf_K 𝒜_G) such as the domain property, quantum normalization of representatives, the quantum cyclic shifts, the quantum Wilson lines, whose classical counterparts have been fundamental in the study of 𝒜_G,Σ. Moreover, we construct quantum seeds for 𝒪_q(Conf_4 𝒜_G) by transporting the Berenstein--Zelevinsky quantum cluster structure on 𝒪_q(G) via quantum Wilson lines, and prove that 𝒪_q(Conf_4 𝒜_G) coincides with the corresponding quantum cluster algebra and its upper counterpart after the localization at frozen variables. The exchange matrices for our quantum seeds agree with the Goncharov--Shen exchange matrices. We also show that quantum seeds for 𝒪_q(Conf_4 𝒜_G) restrict to those for 𝒪_q(Conf_3 𝒜_G). Finally, we construct quantum seeds for 𝒪_q(Conf_K 𝒜_G), K≥ 5, and prove that the corresponding quantum cluster algebras contain the quantum configuration space 𝒪_q(Conf_K 𝒜_G).
ER  -

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