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Mathematics > Quantum Algebra
Title: Solutions of tetrahedron equation from quantum cluster algebra associated with symmetric butterfly quiver
(Submitted on 12 Feb 2024 (v1), last revised 24 Mar 2024 (this version, v2))
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, incorporating a total of seven parameters. 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 ('94) utilizing the quantized coordinate ring, Bazhanov-Mangazeev-Sergeev ('10) from a quantum geometry perspective, Kuniba-Matsuike-Yoneyama ('23) linked with the quantized six-vertex model, and Inoue-Kuniba-Terashima ('23) 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.
Submission history
From: Rei Inoue [view email][v1] Mon, 12 Feb 2024 13:50:51 GMT (1159kb,D)
[v2] Sun, 24 Mar 2024 06:27:10 GMT (1159kb,D)
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