The left-greedy Lie algebra basis and star graphs

TL;DR

This paper introduces a new basis for free Lie algebras called the left-greedy basis, constructed via a bracketing algorithm on Lyndon-Shirshov words, and establishes its duality with star graph coalgebra bases.

Contribution

It presents a novel left-greedy bracketing method for free Lie algebra bases and demonstrates their duality with star graph coalgebra bases using a configuration pairing.

Findings

Left-greedy brackets form a basis for free Lie algebras.

The dual monomial Lie coalgebra basis is given by star graphs.

Application example of the dual basis in Lie algebra computations.

Abstract

We construct a basis for free Lie algebras via a ``left-greedy'' bracketing algorithm on Lyndon-Shirshov words. We use a new tool -- the configuration pairing between Lie brackets and graphs of Sinha-Walter -- to show that the left-greedy brackets form a basis. Our constructions further equip the left-greedy brackets with a dual monomial Lie coalgebra basis of ``star'' graphs. We end with a brief example using the dual basis of star graphs in a Lie algebra computation.