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, Rei Inoue Department of Mathematics and Informatics, Faculty of Science, Chiba University , Chiba 263-8522, Japan Search for other works by this author on: Oxford Academic Atsuo Kuniba Institute of Physics, Graduate School of Arts and Sciences, University of Tokyo , Komaba, 153-8902, Tokyo, Japan Search for other works by this author on: Oxford Academic Yuji Terashima Graduate School of Science, Tohoku University , 6-3, Aoba, Aramaki-aza, Aoba-ku, Sendai, 980-8578, Japan Correspondence to be sent to: e-mail: yujiterashima@tohoku.ac.jp Search for other works by this author on: Oxford Academic
International Mathematics Research Notices, rnae128, https://doi.org/10.1093/imrn/rnae128
Published:
17 June 2024
Article history
Received:
22 December 2023
Revision received:
22 December 2023
Accepted:
22 May 2024
Published:
17 June 2024
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Rei Inoue, Atsuo Kuniba, Yuji Terashima, Quantum Cluster Algebras and 3D Integrability: Tetrahedron and 3D Reflection Equations, International Mathematics Research Notices, 2024;, rnae128, https://doi.org/10.1093/imrn/rnae128
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Abstract
We construct a new solution to the tetrahedron equation and the three-dimensional (3D) reflection equation by extending the quantum cluster algebra approach by Sun and Yagi concerning the former. We consider the Fock–Goncharov quivers associated with the longest elements of the Weyl groups of type |$A$| and |$C$|, and investigate the cluster transformations corresponding to changing a reduced expression into a “most distant” one. By devising a new realization of the quantum |$y$|-variables in terms of |$q$|-Weyl algebra, the solutions are extracted as the operators whose adjoint actions yield the cluster transformations of the quantum |$y$|-variables. Explicit formulas of their matrix elements are also derived for some typical representations.
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