Functorial PBW theorems for post-Lie algebras

Abstract

Using the categorical approach to Poincaré–Birkhoff–Witt type theorems from our previous work with Tamaroff, we prove three such theorems: for universal enveloping Rota–Baxter algebras of tridendriform algebras, for universal enveloping Rota–Baxter Lie algebras of post-Lie algebras, and for universal enveloping tridendriform algebras of post-Lie algebras. Similar results, though without functoriality of the PBW isomorphisms, were recently obtained by Gubarev. Our methods are completely different and mainly rely on methods of rewriting theory for shuffle operads.

Communicated by Pavel Kolesnikov

Keywords:

• Operad
• PBW-pair
• post-Lie algebra
• Rota–Baxter algebra

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 17B35 (Primary)
• 16B50
• 18D50
• 68Q42 (Secondary)

Acknowledgements

I am grateful to Vsevolod Gubarev for a discussion of results of [14], and to Murray Bremner for his comments on this article [9], and particularly for a query as to how results of that paper may be applied to post-Lie algebras.