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arXiv · 1804.06485

Endofunctors and Poincaré-Birkhoff-Witt theorems

Vladimir Dotsenko·Pedro Tamaroff
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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.

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Vladimir Dotsenko, Pedro Tamaroff. 2019-12-17. Endofunctors and Poincaré-Birkhoff-Witt theorems. https://doi.org/10.1093/imrn%2Frnz369

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