On the $\mathfrak{grt}$ hexagon symmetry

TL;DR

This paper demonstrates how to project onto solutions of the $rak{grt}$ hexagon equation, introduces generalized symmetry equations, and constructs canonical solutions using symmetrization, linking these to associated differentials.

Contribution

It introduces a method to project onto solutions of the $rak{grt}$ hexagon equation and constructs canonical solutions for generalized symmetry equations via symmetrization.

Findings

Projection onto $rak{grt}$ hexagon solutions is possible.

Canonical solutions for generalized equations are constructed.

Associated differentials are introduced for symmetry analysis.

Abstract

In this paper we show that it is possible to project onto the solutions of the $\mathfrak{grt}$ hexagon equation. We also consider in some sense generalized hexagon equations and other symmetry equations for multiple argument maps between groups or torsors as source and show that for these equations we can construct at least some canonical solutions by symmetrization procedures. With help of this solutions we finally introduce a bunch of differentials associated to the considered symmetries.