Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 113, 45 pages      arXiv:2403.08814      https://doi.org/10.3842/SIGMA.2024.113

Solutions of Tetrahedron Equation from Quantum Cluster Algebra Associated with Symmetric Butterfly Quiver

Rei Inoue a, Atsuo Kuniba b, Xiaoyue Sun c, Yuji Terashima d and Junya Yagi e
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) Department of Mathematical Sciences and Yau Mathematical Sciences Center, Tsinghua University, Haidian District, Beijing, 100084, P.R. China
d) Graduate School of Science, Tohoku University, 6-3, Aoba, Aramaki-aza, Aoba-ku, Sendai, 980-8578, Japan
e) Yau Mathematical Sciences Center, Tsinghua University, Haidian District, Beijing, 100084, P.R. China

Received March 26, 2024, in final form December 05, 2024; Published online December 21, 2024

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.

Key words: tetrahedron equation; quantum cluster algebra; $q$-Weyl algebra.

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