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

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2024
• Main Authors: Inoue, Rei, Kuniba, Atsuo, Sun, Xiaoyue, Terashima, Yuji, Yagi, Junya
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2024
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/212779
• 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

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.