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JOURNALS // Symmetry, Integrability and Geometry: Methods and Applications // Archive

SIGMA, 2024 Volume 20, 113, 45 pp. (Mi sigma2115)

Solutions of Tetrahedron Equation from Quantum Cluster Algebra Associatedwith Symmetric Butterfly Quiver

Rei Inouea, Atsuo Kunibab, Xiaoyue Suncd, Yuji Terashimae, Junya Yagic

a Department of Mathematics and Informatics, Faculty of Science, Chiba University, Chiba, 263-8522, Japan
b Institute of Physics, Graduate School of Arts and Sciences, University of Tokyo, Komaba, Tokyo 153-8902, Japan
c Yau Mathematical Sciences Center, Tsinghua University, Haidian District, Beijing, 100084, P.R. China
d Department of Mathematical Sciences, Tsinghua University, Haidian District, Beijing, 100084, P.R. China
e Graduate School of Science, Tohoku University, 6-3, Aoba, Aramaki-aza, Aoba-ku, Sendai, 980-8578, Japan

Abstract: 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 $A$ Weyl groups and the implementation of quantum $Y$-variables through the $q$-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) $R$-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 $R$-matrix presented in this paper offers a unified perspective on these existing solutions, coalescing them within the framework of quantum cluster algebra.

Keywords: tetrahedron equation, quantum cluster algebra, $q$-Weyl algebra.

MSC: 82B23, 81R12, 13F60

Received: March 26, 2024; in final form December 5, 2024; Published online December 21, 2024

Language: English

DOI: 10.3842/SIGMA.2024.113


ArXiv: 2403.08814


© Steklov Math. Inst. of RAS, 2025