# Poincare-Birkhoff-Witt theorem for pre-Lie and postLie algebras

## TL;DR

This paper extends the Poincare-Birkhoff-Witt theorem to pre-Lie and postLie algebras by constructing their universal enveloping algebras and establishing PBW pairs, confirming known results with new proofs.

## Contribution

It constructs universal enveloping algebras for pre-Lie and postLie algebras and proves they form PBW pairs, providing new proofs for existing results.

## Key findings

- Established PBW pairs for pre-Lie and postLie algebras
- Constructed universal enveloping preassociative and postassociative algebras
- Reproved known PBW result for pre-Lie algebras

## Abstract

We construct the universal enveloping preassociative and postassociative algebra for a pre-Lie and a postLie algebra respectively. We show that the pairs $(\mathrm{preLie},\mathrm{preAs})$ and $(\mathrm{postLie},\mathrm{postAs})$ are Poincare-Birkhoff-Witt-pairs, for the first one it's a reproof of the result of V. Dotsenko and P. Tamaroff.