Quantum cluster algebras and 3D integrability: Tetrahedron and 3D reflection equations

2023/10/23 by Rei Inoue, Inoue, Rei, Atsuo Kuniba +3 · 2 citations
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Advanced Algebra and Geometry #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2310.14493

Abstract

We construct a new solution to the tetrahedron equation and the three-dimensional (3D) reflection equation by extending the quantum cluster algebra approach by Sun and Yagi concerning the former. We consider the Fock-Goncharov quivers associated with the longest elements of the Weyl groups of type A and C, and investigate the cluster transformations corresponding to changing a reduced expression into a `most distant' one. By devising a new realization of the quantum y-variables in terms of q-Weyl algebra, the solutions are extracted as the operators whose adjoint actions yield the cluster transformations of the quantum y-variables. Explicit formulas of their matrix elements are also derived for some typical representations.

Cited by

• On integrability of a q-oscillator lattice with a B-boundary
• Leibniz 2-algebras, linear 2-racks and the Zamolodchikov Tetrahedron equation

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