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-Teichmüller 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.

arXiv comment18 pages, second version: typos fixed, Remark 8 was added, and Figures 3 and 4 were added to clarify parts of the proof of Theorem 3