Symmetric Lie models of a triangle

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.