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| Summary: | 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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| ISSN: | 1815-0659 |