On Lie algebra weight systems for 3-graphs

Abstract

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) 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 ϕ g 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 = ϕ g for some complex metric Lie algebra g, then f = ϕ g for some unique complex reductive metric Lie algebra g . Basic tool throughout is geometric invariant theory.

On partition functions for 3-graphs

“…Furthermore, a 'k-join lemma' is given below that simplifies the proof. The complex case, as studied in [4], [21], demands different conditions and machinery, and requires (so far) the dimension of the vertex model to be specified in the theorem.…”