A pairing between graphs and trees

TL;DR

This paper introduces a canonical pairing between trees and graphs that aids in understanding Lie and Poisson operads, providing duals, bases, and algebraic structures with potential for future developments.

Contribution

It develops a new canonical pairing between graphs and trees, offering a novel approach to analyze Lie and Poisson operads and their duals.

Findings

Established a perfect pairing between trees and graphs.

Reproved standard facts about Lie modules and bases.

Developed duals and algebraic structures for free Lie algebras.

Abstract

We develop a canonical pairing between trees and graphs, which passes to their quotients by Jacobi identities. This pairing is an effective and simple tool for understanding the Lie and Poisson operads, providing canonical duals. In the course of showing that this pairing is perfect we reprove some standard facts about the modules Lie(n), establishing standard bases as well as giving a new means to reduce to those bases. We then move on to define duals to free Lie algebras and to develop product, coproduct and operad structures. We give a brief account here to be built on in a number of different directions in future work.