Quantum Representation of Affine Weyl Groups and Associated Quantum Curves
We study a quantum (non-commutative) representation of the affine Weyl group, primarily of type ⁽¹⁾₈, where the representation is given by birational actions on two variables, and , with q-commutation relations. Using the tau variables, we also construct quantum ''fundamental''...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2021 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2021
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211347 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Quantum Representation of Affine Weyl Groups and Associated Quantum Curves. Sanefumi Moriyama and Yasuhiko Yamada. SIGMA 17 (2021), 076, 24 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862734001163206656 |
|---|---|
| author | Moriyama, Sanefumi Yamada, Yasuhiko |
| author_facet | Moriyama, Sanefumi Yamada, Yasuhiko |
| citation_txt | Quantum Representation of Affine Weyl Groups and Associated Quantum Curves. Sanefumi Moriyama and Yasuhiko Yamada. SIGMA 17 (2021), 076, 24 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We study a quantum (non-commutative) representation of the affine Weyl group, primarily of type ⁽¹⁾₈, where the representation is given by birational actions on two variables, and , with q-commutation relations. Using the tau variables, we also construct quantum ''fundamental'' polynomials (, ) which completely control the Weyl group actions. The geometric properties of the polynomials (, ) for the commutative case are lifted distinctively in the quantum case to certain singularity structures as the q-difference operators. This property is further utilized as the characterization of the quantum polynomials (, ). As an application, the quantum curve associated with topological strings proposed recently by the first-named author is rederived by the Weyl group symmetry. The cases of type ⁽¹⁾₅, ⁽¹⁾₆, ⁽¹⁾₇ are also discussed.
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| first_indexed | 2026-04-17T15:59:35Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211347 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-04-17T15:59:35Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Moriyama, Sanefumi Yamada, Yasuhiko 2025-12-30T15:53:00Z 2021 Quantum Representation of Affine Weyl Groups and Associated Quantum Curves. Sanefumi Moriyama and Yasuhiko Yamada. SIGMA 17 (2021), 076, 24 pages 1815-0659 2020 Mathematics Subject Classification: 39A06; 39A13 arXiv:2104.06661 https://nasplib.isofts.kiev.ua/handle/123456789/211347 https://doi.org/10.3842/SIGMA.2021.076 We study a quantum (non-commutative) representation of the affine Weyl group, primarily of type ⁽¹⁾₈, where the representation is given by birational actions on two variables, and , with q-commutation relations. Using the tau variables, we also construct quantum ''fundamental'' polynomials (, ) which completely control the Weyl group actions. The geometric properties of the polynomials (, ) for the commutative case are lifted distinctively in the quantum case to certain singularity structures as the q-difference operators. This property is further utilized as the characterization of the quantum polynomials (, ). As an application, the quantum curve associated with topological strings proposed recently by the first-named author is rederived by the Weyl group symmetry. The cases of type ⁽¹⁾₅, ⁽¹⁾₆, ⁽¹⁾₇ are also discussed. We would like to thank our colleagues for their valuable discussions. The work of S.M. is supported by Grant-in-Aid for Scientific Research (C) No. 19K03829. The work of Y.Y. is supported by Grant-in-Aid for Scientific Research (S) No. 17H06127. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Quantum Representation of Affine Weyl Groups and Associated Quantum Curves Article published earlier |
| spellingShingle | Quantum Representation of Affine Weyl Groups and Associated Quantum Curves Moriyama, Sanefumi Yamada, Yasuhiko |
| title | Quantum Representation of Affine Weyl Groups and Associated Quantum Curves |
| title_full | Quantum Representation of Affine Weyl Groups and Associated Quantum Curves |
| title_fullStr | Quantum Representation of Affine Weyl Groups and Associated Quantum Curves |
| title_full_unstemmed | Quantum Representation of Affine Weyl Groups and Associated Quantum Curves |
| title_short | Quantum Representation of Affine Weyl Groups and Associated Quantum Curves |
| title_sort | quantum representation of affine weyl groups and associated quantum curves |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211347 |
| work_keys_str_mv | AT moriyamasanefumi quantumrepresentationofaffineweylgroupsandassociatedquantumcurves AT yamadayasuhiko quantumrepresentationofaffineweylgroupsandassociatedquantumcurves |