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

Rei Inoue ; Atsuo Kuniba ; Xiaoyue Sun ; Yuji Terashima ; Junya Yagi

Symmetry, integrability and geometry: methods and applications, Tome 20 (2024)

Résumé

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.

Détail

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

Bibliographie

[1] Baxter R.J., Exactly solved models in statistical mechanics, Academic Press, London, 1982 | MR | Zbl

[2] Bazhanov V.V., Mangazeev V.V., Sergeev S.M., “Quantum geometry of 3-dimensional lattices and tetrahedron equation”, XVIth International Congress on Mathematical Physics, World Scientific Publishing, Hackensack, NJ, 2010, 23–44, arXiv: 0911.3693 | DOI | MR | Zbl

[3] Bazhanov V.V., Sergeev S.M., “Zamolodchikov's tetrahedron equation and hidden structure of quantum groups”, J. Phys. A, 39 (2006), 3295–3310, arXiv: hep-th/0509181 | DOI | MR | Zbl

[4] Faddeev L.D., Kashaev R.M., Volkov A.Yu., “Strongly coupled quantum discrete Liouville theory. I Algebraic approach and duality”, Comm. Math. Phys., 219 (2001), 199–219, arXiv: hep-th/0006156 | DOI | MR | Zbl

[5] Fock V.V., Goncharov A.B., “Cluster $\mathcal{X}$-varieties, amalgamation, and Poisson–Lie groups”, Algebraic Geometry and Number Theory, Progr. Math., 253, Birkhäuser Boston, Boston, MA, 2006, 27–68, arXiv: math.RT/0508408 | DOI | MR | Zbl

[6] Fock V.V., Goncharov A.B., “Cluster ensembles, quantization and the dilogarithm”, Ann. Sci. Éc. Norm. Supér., 42 (2009), 865–930, arXiv: math.AG/0311245 | DOI | MR | Zbl

[7] Fock V.V., Goncharov A.B., “The quantum dilogarithm and representations of quantum cluster varieties”, Invent. Math., 175 (2009), 223–286, arXiv: math.QA/0702397 | DOI | MR | Zbl

[8] Fomin S., Zelevinsky A., “Cluster algebras. IV Coefficients”, Compos. Math., 143 (2007), 112–164, arXiv: math.RA/0602259 | DOI | MR | Zbl

[9] Gavrylenko P., Semenyakin M., Zenkevich Y., “Solution of tetrahedron equation and cluster algebras”, J. High Energy Phys., 2021:5 (2021), 103, 33 pp., arXiv: 2010.15871 | DOI | MR | Zbl

[10] Gross M., Hacking P., Keel S., Kontsevich M., “Canonical bases for cluster algebras”, J. Amer. Math. Soc., 31 (2018), 497–608, arXiv: 1411.1394 | DOI | MR | Zbl

[11] Inoue R., Kuniba A., Terashima Y., “Quantum cluster algebras and 3D integrability: tetrahedron and 3D reflection equations”, Int. Math. Res. Not., 2024 (2024), 11549–11581, arXiv: 2310.14493 | DOI | MR | Zbl

[12] Inoue R., Kuniba A., Terashima Y., “Tetrahedron equation and quantum cluster algebras”, J. Phys. A, 57 (2024), 085202, 33 pp., arXiv: 2310.14529 | DOI | MR | Zbl

[13] Isaev A.P., Kulish P.P., “Tetrahedron reflection equations”, Modern Phys. Lett. A, 12 (1997), 427–437, arXiv: hep-th/9702013 | DOI | MR | Zbl

[14] Kapranov M.M., Voevodsky V.A., “$2$-categories and Zamolodchikov tetrahedra equations”, Algebraic Groups and Their Generalizations: Quantum and Infinite-Dimensional Methods, Proc. Sympos. Pure Math., 56, American Mathematical Society, Providence, RI, 1994, 177–259 | DOI | MR | Zbl

[15] Kashaev R.M., Nakanishi T., “Classical and quantum dilogarithm identities”, SIGMA, 7 (2011), 102, 29 pp., arXiv: 1104.4630 | DOI | MR | Zbl

[16] Keller B., “On cluster theory and quantum dilogarithm identities”, Representations of Algebras and Related Topics, EMS Ser. Congr. Rep., European Mathematical Society, Zürich, 2011, 85–116, arXiv: 1102.4148 | DOI | MR | Zbl

[17] Kuniba A., Quantum groups in three-dimensional integrability, Theoret. and Math. Phys., Springer, Singapore, 2022 | DOI | MR | Zbl

[18] Kuniba A., Matsuike S., Yoneyama A., “New solutions to the tetrahedron equation associated with quantized six-vertex models”, Comm. Math. Phys., 401 (2023), 3247–3276, arXiv: 2208.10258 | DOI | MR | Zbl

[19] Nakanishi T., “Periodicities in cluster algebras and dilogarithm identities”, Representations of Algebras and Related Topics, EMS Ser. Congr. Rep., European Mathematical Society, Zürich, 2011, 407–443, arXiv: 1006.0632 | DOI | MR | Zbl

[20] Nakanishi T., “Synchronicity phenomenon in cluster patterns”, J. Lond. Math. Soc., 103 (2021), 1120–1152, arXiv: 1906.12036 | DOI | MR | Zbl

[21] Sergeev S.M., Arithmetic of quantum integrable systems in multidimensional discrete space-time, Unpublished note, 2010

[22] Sergeev S.M., “Quantum $2+1$ evolution model”, J. Phys. A, 32 (1999), 5693–5714, arXiv: solv-int/9811003 | DOI | MR | Zbl

[23] Sun X., Yagi J., “Cluster transformations, the tetrahedron equation and three-dimensional gauge theories”, Adv. Theor. Math. Phys., 27 (2023), 1065–1106, arXiv: 2211.10702 | DOI | MR | Zbl

[24] Zamolodchikov A.B., “Tetrahedra equations and integrable systems in three-dimensional space”, Soviet Phys. JETP, 79 (1980), 325–336 | MR