The Lie algebra of rooted planar trees

TL;DR

This paper investigates the Lie algebra structure on rooted planar trees, exploring its properties, finite generation, and relationships with polynomial vector fields, revealing structural limitations and algebraic characteristics.

Contribution

It introduces and analyzes a Lie algebra structure on rooted planar trees, establishing its properties and connections to polynomial vector fields, and proves non-splitting of a key homomorphism.

Findings

The Lie algebra of rooted planar trees is studied for finite generation.

The augmentation homomorphism to polynomial vector fields has no splitting preserving units.

Structural properties of related Lie algebras are characterized.

Abstract

We study a natural Lie algebra structure on the free vector space generated by all rooted planar trees as the associated Lie algebra of the nonsymmetric operad (non-$\Sigma$ operad, preoperad) of rooted planar trees. We determine whether the Lie algebra and some related Lie algebras are finitely generated or not, and prove that a natural surjection called the augmentation homomorphism onto the Lie algebra of polynomial vector fields on the line has no splitting preserving the units.