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

TL;DR

This paper introduces a method to derive solutions to the 3D reflection equation from known tetrahedron maps, expanding the set of solutions and linking algebraic structures through folding of Dynkin diagrams.

Contribution

It presents a novel approach to generate 3D reflection maps from tetrahedron maps, connecting algebraic parametrizations with boundary integrability.

Findings

New solutions to the 3D reflection equation were obtained.

The method relates transition maps of Lusztig's parametrizations via diagram folding.

The approach links boundary solutions to algebraic structures in Lie theory.

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 $SL_3$ and $SO_5$, which is obtained via folding the Dynkin diagram of $A_3$ into one of $B_2$.