The left greedy Lie algebra basis and star graphs

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 and 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.