On Lie algebra weight systems for 3-graphs

A 3-graph is a connected cubic graph such that each vertex is equipped with a cyclic order of the edges incident with it. A weight system is a function f on the collection of 3-graphs which is antisymmetric  : f(H)=−f(G)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 φgφg obtained from a metric Lie algebra gg by taking the structure tensor c   of gg 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=φgf=φg for some complex metric Lie algebra gg, then f=φg′f=φg′ for some unique complex reductive metric Lie algebra g′g′. Basic tool throughout is geometric invariant theory.