Internally connected graphs and the Kashiwara-Vergne Lie algebra

Abstract

It is conjectured that the Kashiwara-Vergne Lie algebra $$\widehat{\mathfrak {krv}}_2$$ is isomorphic to the direct sum of the Grothendieck-Teichmuller Lie algebra $$\mathfrak {grt}_1$$ and a one-dimensional Lie algebra. In this paper, we use the graph complex of internally connected graphs to define a nested sequence of Lie subalgebras of $$\widehat{\mathfrak {krv}}_2$$ whose intersection is $$\mathfrak {grt}_1$$ , thus giving a way to interpolate between these two Lie algebras.