Endofunctors and Poincaré-Birkhoff-Witt theorems
Preprint
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.Keywords
All Related Versions
This publication has 21 references indexed in Scilit:
- Algebraic structures of F-manifolds via pre-Lie algebrasAnnali di Matematica Pura ed Applicata (1923 -), 2018
- Hopf algebras in non-associative Lie theoryBulletin of Mathematical Sciences, 2014
- Algebras, hyperalgebras, nonassociative bialgebras and loopsAdvances in Mathematics, 2007
- Dialgebras and Related Operads2001
- Free Left-Symmetrical Algebras and an Analogue of the Poincaré-Birkhoff-Witt TheoremJournal of Algebra, 1994
- Universal enveloping algebras of Leibniz algebras and (co)homologyMathematische Annalen, 1993
- The equivalence of certain categories of twisted Lie and Hopf algebras over a commutative ringJournal of Pure and Applied Algebra, 1993
- Toposes, Triples and Theories1985
- Are colimits of algebras simple to construct?Journal of Algebra, 1980
- The diamond lemma for ring theoryAdvances in Mathematics, 1978