On Lie Algebra Weight Systems for 3-graphs

A 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 weight system is a function f on the collection of 3-graphs which is 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 φg obtained from a metric Lie algebra g by taking the structure tensor c of 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 = φg for some complex metric Lie algebra g, then f = φg′ for some unique complex reductive metric Lie algebra g ′. Basic tool throughout is geometric invariant theory.