# Pre- and Post-Lie Algebras: The Algebro-Geometric View

## TL;DR

This paper explores the algebraic geometry underlying pre- and post-Lie algebras, connecting them to Hopf algebras and infinite-dimensional varieties, and describing their automorphism groups.

## Contribution

It establishes a geometric framework for understanding pre- and post-Lie algebras through algebraic varieties and Hopf algebra structures.

## Key findings

- Connes-Kreimer and MKW Hopf algebras as coordinate rings of tree series varieties
- Description of automorphism groups governing substitution laws
- Linking algebraic structures to infinite-dimensional geometric objects

## Abstract

We relate composition and substitution in pre- and post-Lie algebras to algebraic geometry. The Connes-Kreimer Hopf algebras, and MKW Hopf algebras are then coordinate rings of the infinite-dimensional affine varieties consisting of series of trees, resp.\ Lie series of ordered trees. Furthermore we describe the Hopf algebras which are coordinate rings of the automorphism groups of these varieties, which govern the substitution law in pre- and post-Lie algebras.