On Lie algebra weight systems for 3-graphs

TL;DR

This paper characterizes complex Lie algebra weight systems on 3-graphs using algebraic equations and shows their uniqueness, employing geometric invariant theory as a key tool.

Contribution

It provides equations that uniquely identify complex Lie algebra weight systems and establishes their correspondence with complex reductive metric Lie algebras.

Findings

Characterization equations for complex Lie algebra weight systems

Uniqueness of the associated reductive Lie algebra

Application of geometric invariant theory

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.