The configuration basis of a Lie algebra and its dual

TL;DR

This paper introduces new monomial bases for free Lie algebras and coalgebras using the configuration pairing framework, enhancing computational methods and deriving a new basis for the shuffle algebra.

Contribution

It develops a novel left-normed monomial basis for free Lie algebras and a dual basis for Lie coalgebras using the configuration framework, with explicit examples and applications.

Findings

New monomial basis for free Lie algebras

Dual basis for Lie coalgebras derived

A new multiplicative basis for the shuffle algebra

Abstract

We use the Lie coalgebra and configuration pairing framework presented previously by Sinha and Walter to derive a new, left-normed monomial basis for free Lie algebras (built from associative Lyndon-Shirshov words), as well as a dual monomial basis for Lie coalgebras. Our focus is on computational dexterity gained by using the configuration framework and basis. We include several explicit examples using the dual coalgebra basis and configuration pairing to perform Lie algebra computations. As a corollary of our work, we get a new multiplicative basis for the shuffle algebra.