Endofunctors and Poincaré-Birkhoff-Witt theorems

Abstract

We determine what appears to be the bare-bones categorical framework for Poincaré-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 Poincaré-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 Poincaré-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.

18 pages, final version before submission for peer review, to appear in IMRN