Solutions of Tetrahedron Equation from Quantum Cluster Algebra Associated with Symmetric Butterfly Quiver
We construct a new solution to the tetrahedron equation by further pursuing the quantum cluster algebra approach in our previous works. The key ingredients include a symmetric butterfly quiver attached to the wiring diagrams for the longest element of type Weyl groups and the implementation of quan...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2024 |
| Main Authors: | , , , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2024
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/212779 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Solutions of Tetrahedron Equation from Quantum Cluster Algebra Associated with Symmetric Butterfly Quiver. Rei Inoue, Atsuo Kuniba, Xiaoyue Sun, Yuji Terashima and Junya Yagi. SIGMA 20 (2024), 113, 45 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862746380091523072 |
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| author | Inoue, Rei Kuniba, Atsuo Sun, Xiaoyue Terashima, Yuji Yagi, Junya |
| author_facet | Inoue, Rei Kuniba, Atsuo Sun, Xiaoyue Terashima, Yuji Yagi, Junya |
| citation_txt | Solutions of Tetrahedron Equation from Quantum Cluster Algebra Associated with Symmetric Butterfly Quiver. Rei Inoue, Atsuo Kuniba, Xiaoyue Sun, Yuji Terashima and Junya Yagi. SIGMA 20 (2024), 113, 45 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We construct a new solution to the tetrahedron equation by further pursuing the quantum cluster algebra approach in our previous works. The key ingredients include a symmetric butterfly quiver attached to the wiring diagrams for the longest element of type Weyl groups and the implementation of quantum -variables through the -Weyl algebra. The solution consists of four products of quantum dilogarithms. By exploring both the coordinate and momentum representations, along with their modular double counterparts, our solution encompasses various known three-dimensional (3D) -matrices. These include those obtained by Kapranov-Voevodsky (1994) utilizing the quantized coordinate ring, Bazhanov-Mangazeev-Sergeev (2010) from a quantum geometry perspective, Kuniba-Matsuike-Yoneyama (2023) linked with the quantized six-vertex model, and Inoue-Kuniba-Terashima (2023) associated with the Fock-Goncharov quiver. The 3D -matrix presented in this paper offers a unified perspective on these existing solutions, coalescing them within the framework of quantum cluster algebra.
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| first_indexed | 2026-03-21T18:28:30Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-212779 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T18:28:30Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
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| spelling | Inoue, Rei Kuniba, Atsuo Sun, Xiaoyue Terashima, Yuji Yagi, Junya 2026-02-11T10:37:27Z 2024 Solutions of Tetrahedron Equation from Quantum Cluster Algebra Associated with Symmetric Butterfly Quiver. Rei Inoue, Atsuo Kuniba, Xiaoyue Sun, Yuji Terashima and Junya Yagi. SIGMA 20 (2024), 113, 45 pages 1815-0659 2020 Mathematics Subject Classification: 82B23; 81R12; 13F60 arXiv:2403.08814 https://nasplib.isofts.kiev.ua/handle/123456789/212779 https://doi.org/10.3842/SIGMA.2024.113 We construct a new solution to the tetrahedron equation by further pursuing the quantum cluster algebra approach in our previous works. The key ingredients include a symmetric butterfly quiver attached to the wiring diagrams for the longest element of type Weyl groups and the implementation of quantum -variables through the -Weyl algebra. The solution consists of four products of quantum dilogarithms. By exploring both the coordinate and momentum representations, along with their modular double counterparts, our solution encompasses various known three-dimensional (3D) -matrices. These include those obtained by Kapranov-Voevodsky (1994) utilizing the quantized coordinate ring, Bazhanov-Mangazeev-Sergeev (2010) from a quantum geometry perspective, Kuniba-Matsuike-Yoneyama (2023) linked with the quantized six-vertex model, and Inoue-Kuniba-Terashima (2023) associated with the Fock-Goncharov quiver. The 3D -matrix presented in this paper offers a unified perspective on these existing solutions, coalescing them within the framework of quantum cluster algebra. The authors would like to thank Vladimir Bazhanov, Vladimir Mangazeev, Sergey Sergeev, and Akihito Yoneyama for stimulating discussions. The authors also thank the anonymous referees for their careful reading and valuable comments. RI is supported by JSPS KAKENHI Grant Numbers 19K03440 and 23K03048. AK is supported by JSPS KAKENHI Grant Number 24K06882. YT is supported by JSPS KAKENHI Grant Numbers JP21K03240 and 22H01117. JY and XS are supported by NSFC Grant Number 12375064. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Solutions of Tetrahedron Equation from Quantum Cluster Algebra Associated with Symmetric Butterfly Quiver Article published earlier |
| spellingShingle | Solutions of Tetrahedron Equation from Quantum Cluster Algebra Associated with Symmetric Butterfly Quiver Inoue, Rei Kuniba, Atsuo Sun, Xiaoyue Terashima, Yuji Yagi, Junya |
| title | Solutions of Tetrahedron Equation from Quantum Cluster Algebra Associated with Symmetric Butterfly Quiver |
| title_full | Solutions of Tetrahedron Equation from Quantum Cluster Algebra Associated with Symmetric Butterfly Quiver |
| title_fullStr | Solutions of Tetrahedron Equation from Quantum Cluster Algebra Associated with Symmetric Butterfly Quiver |
| title_full_unstemmed | Solutions of Tetrahedron Equation from Quantum Cluster Algebra Associated with Symmetric Butterfly Quiver |
| title_short | Solutions of Tetrahedron Equation from Quantum Cluster Algebra Associated with Symmetric Butterfly Quiver |
| title_sort | solutions of tetrahedron equation from quantum cluster algebra associated with symmetric butterfly quiver |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212779 |
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