Symmetric Lie models of a triangle

TL;DR

This paper extends Lie models from intervals to triangles and graphs, incorporating symmetry actions of the symmetric groups, and demonstrates the existence of stable Maurer-Cartan elements under graph automorphisms.

Contribution

It introduces a Lie model for a triangle with symmetric group actions compatible with geometric symmetries, generalizing previous interval models.

Findings

Constructed a Lie model for a triangle with $\, ext{S}_3$ symmetry.

Proved the existence of stable Maurer-Cartan elements in graph models.

Extended symmetry-compatible Lie models to graphs with circuits.

Abstract

R. Lawrence and D. Sullivan have constructed a Lie model for an interval from the geometrical idea of flat connections and flows of gauge transformations. Their model supports an action of the symmetric group $\Sigma_2$ reflecting the geometrical symmetry of the interval. In this work, we present a Lie model of the triangle with an action of the symmetric group $\Sigma_3$ compatible with the geometrical symmetries of the triangle. We also prove that the model of a graph consisting of a circuit with $k$ vertices admits a Maurer-Cartan element stable by the automorphisms of the graph.