Endofunctors and Poincar\'e-Birkhoff-Witt theorems

TL;DR

This paper develops a categorical framework using endofunctors to understand Poincaré-Birkhoff-Witt theorems, unifying various results and proving new ones, including for dendriform algebras.

Contribution

It introduces a universal categorical approach to PBW theorems via monad transformations, enabling new proofs and results in algebraic structures.

Findings

A natural transformation of monads has PBW property iff it makes its codomain a free right module.

Unified approach proves several old PBW theorems and new results for dendriform and pre-Lie algebras.

Answers Loday's question by establishing a PBW theorem for universal enveloping dendriform algebras.

Abstract

We determine what appears to be the bare-bones categorical framework for Poincar\'e-Birkhoff-Witt type theorems about universal enveloping algebras of various algebraic structures. Our language is that of endofunctors; we establish that a natural transformation of monads enjoys a Poincar\'e-Birkhoff-Witt property only if that transformation makes its codomain a free right module over its domain. We conclude with a number of applications to show how this unified approach proves various old and new Poincar\'e-Birkhoff-Witt type theorems. In particular, we prove a PBW type result for universal enveloping dendriform algebras of pre-Lie algebras, answering a question of Loday.