Internally connected graphs and the Kashiwara-Vergne Lie algebra

TL;DR

This paper explores the structure of the Kashiwara-Vergne Lie algebra by using graph complexes to connect it with the Grothendieck-Teichmüller Lie algebra, providing a new perspective on their relationship.

Contribution

It introduces a nested sequence of Lie subalgebras of the Kashiwara-Vergne Lie algebra using graph complexes, interpolating between it and the Grothendieck-Teichmüller Lie algebra.

Findings

Defined a sequence of Lie subalgebras of al{krv}_2

Showed the intersection of these subalgebras is al{grt}_1

Provided a new approach to relate al{krv}_2 and al{grt}_1

Abstract

It is conjectured that the Kashiwara-Vergne Lie algebra $\widehat{\mathfrak{krv}}_2$ is isomorphic to the direct sum of the Grothendieck-Teichm\"uller 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.