Combinatorial Formulas for Classical Lie Weight Systems on Arrow Diagrams

TL;DR

This paper provides explicit combinatorial formulas for classical Lie algebra weight systems on arrow diagrams, connecting tensor constructions with combinatorial methods, and compares oriented and unoriented cases.

Contribution

It introduces combinatorial formulas for Lie algebra weight systems using Manin triples, bridging tensor and diagrammatic approaches, and relates them to existing unoriented weight systems.

Findings

Explicit formulas for classical Lie weight systems on arrow diagrams.

Equivalence of oriented and unoriented weight systems via averaging.

Demonstrations through example calculations.

Abstract

In 2002 Haviv gave a way of assigning Lie tensors to directed trivalent graphs. Weight systems on oriented chord idagrams modulo 6T can then be constructed from such tensors. In this paper we give explicit combinatorial formulas of weight systems using Manin triples constrcted from classical Lie algebras. We then compose these oriented weight systems with the averaging map to get weight systems on unoriented chord diagrams and show that they are the same as the ones obtained by Bar-Natan in 1991. In the last section we carry out calculations on certain examples.