Boundary from bulk integrability in three dimensions: 3D reflection maps from tetrahedron maps

Abstract

We established a method for obtaining set-theoretical solutions to the 3D reflection equation by using known ones to the Zamolodchikov tetrahedron equation, where the former equation was proposed by Isaev and Kulish as a boundary analog of the latter. By applying our method to Sergeev's electrical solution and a two-component solution associated with the discrete modified KP equation, we obtain new solutions to the 3D reflection equation. Our approach is closely related to a relation between the transition maps of Lusztig's parametrizations of the totally positive part of $S{L}_3$ and $S{O}_5$, which is obtained via folding the Dynkin diagram of $A_3$ into one of $B_2$.

arXiv comment14 pages, 8 figures, v2: updated abstract and minor corrections. For the interactive Figure 6, see the Mathematica notebook obtained from https://www.wolframcloud.com/obj/yoneyama/Published/R20_identity.nb

References in corpus (16)

• Quantum group symmetry in sine-Gordon and affine Toda field theories on the half-line
• Zamolodchikov's Tetrahedron Equation and Hidden Structure of Quantum Groups
• Functional Tetrahedron Equation
• Yang-Baxter maps: dynamical point of view
• The vertex formulation of the Bazhanov-Baxter Model
• Labeling Schemes for Tetrahedron Equations and Dualities between Them
• Tetrahedron and 3D reflection equations from quantized algebra of functions
• Set-theoretical reflection equation: Classification of reflection maps
• Tetrahedron maps and symmetries of three dimensional integrable discrete equations
• Box-ball system with reflecting end
• Tetrahedron Reflection Equation
• Electrical networks and Lie theory
• A Common Structure in PBW Bases of the Nilpotent Subalgebra of $U_q(\mathfrak{g})$ and Quantized Algebra of Functions
• Combinatorial solutions to the reflection equation
• Reflection matrices, coideal subalgebras and generalized Satake diagrams of affine type
• Set-theoretical solutions to the reflection equation associated to the quantum affine algebra of type $A_{n-1}^{(1)}$

Cited by in corpus (4)

• Local Yang--Baxter correspondences and set-theoretical solutions to the Zamolodchikov tetrahedron equation
• Algebraic and differential-geometric constructions of set-theoretical solutions to the Zamolodchikov tetrahedron equation
• Tetrahedron maps, Yang-Baxter maps, and partial linearisations
• Set-theoretical solutions to the Zamolodchikov tetrahedron equation on associative rings and Liouville integrability