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Endofunctors and Poincaré–Birkhoff–Witt Theorems
International Mathematics Research Notices ( IF 1.291 ) Pub Date : 2020-02-06 , DOI: 10.1093/imrn/rnz369
Dotsenko V, Tamaroff P.

We determine what appears to be the bare-bones categorical framework for Poincaré–Birkhoff–Witt (PBW)-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 PBW 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 PBW-type theorems. In particular, we prove a PBW-type result for universal enveloping dendriform algebras of pre-Lie algebras, answering a question of Loday.
更新日期:2020-02-07

 

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