On Lie algebra weight systems for 3-graphs

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 ${\phi }_{\mathfrak{g}}$ obtained from a metric Lie algebra $\mathfrak{g}$ by taking the structure tensor $c$ of $\mathfrak{g}$ 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 $f={\phi }_{\mathfrak{g}}$ for some complex metric Lie algebra $\mathfrak{g}$, then $f={\phi }_{{\mathfrak{g}}^{\mathrm{\prime }}}$ for some unique complex reductive metric Lie algebra ${\mathfrak{g}}^{\mathrm{\prime }}$. Basic tool throughout is geometric invariant theory.