# Functorial PBW theorems for post-Lie algebras

## TL;DR

This paper establishes functorial PBW theorems for various algebraic structures, including post-Lie algebras and Rota-Baxter algebras, using a categorical approach and rewriting theory for shuffle operads.

## Contribution

It introduces a new categorical method for proving functorial PBW theorems for post-Lie and related algebras, differing from previous approaches.

## Key findings

- Proves PBW theorems for universal enveloping Rota-Baxter algebras of tridendriform algebras
- Establishes PBW theorems for Rota--Baxter Lie algebras of post-Lie algebras
- Demonstrates PBW theorems for tridendriform algebras of post-Lie algebras

## Abstract

Using the categorical approach to Poincar\'e-Birkhoff-Witt type theorems from our previous work with Tamaroff, we prove three such theorems: for universal enveloping Rota-Baxter algebras of tridendriform algebras, for universal enveloping Rota--Baxter Lie algebras of post-Lie algebras, and for universal enveloping tridendriform algebras of post-Lie algebras. Similar results, though without functoriality of the PBW isomorphisms, were recently obtained by Gubarev. Our methods are completely different and mainly rely on methods of rewriting theory for shuffle operads.