On Lie algebra weight systems for 3-graphs

Abstract

A {\em $3$-graph} is a connected cubic graph such that each vertex is is equipped with a cyclic order of the edges incident with it. A {\em weight system} is a function $f$ on the collection of $3$-graphs which is {\em antisymmetric}: $f(H)=-f(G)$ if $H$ arises from $G$ by reversing the orientation at one of its vertices, and satisfies the IHX-equation. Key instances of weight systems are the functions $\varphi_{\frak{g}}$ obtained from a metric Lie algebra $\frak{g}$ by taking the structure tensor $c$ of $\frak{g}$ with respect to some orthonormal basis, decorating each vertex of the $3$-graph by $c$, and contracting along the edges. We give equations on values of any complex-valued weight system that characterize it as complex Lie algebra weight system. It also follows that if $f=\varphi_{\frak{g}}$ for some complex metric Lie algebra $\frak{g}$, then $f=\varphi_{\frak{g}'}$ for some unique complex reductive metric Lie algebra $\frak{g}'$. Basic tool throughout is geometric invariant theory.