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Journal of Lie Theory 30 (2020), No. 1, 223--238
Copyright Heldermann Verlag 2020

Poincaré-Birkhoff-Witt Theorem for Pre-Lie and Post-Lie Algebras

Vsevolod Gubarev
Sobolev Institute of Mathematics, 630090 Novosibirsk, Russia
and: Faculty of Mathematics, University of Vienna, 1090 Vienna, Austria

We construct the universal enveloping preassociative and postassociative algebra for a pre-Lie and a post-Lie algebra, respectively. We show that the pairs (pre-Lie, pre-As) and (post-Lie, post-As) are Poincaré-Birkhoff-Witt-pairs; for the first this is a reproof of the result of V. Dotsenko and P. Tamaroff.

Keywords: Rota-Baxter operator, Groebner-Shirshov basis, pre-Lie algebra, post-Lie algebra, preassociative algebra, dendriform algebra, postassociative algebra.

MSC: 16W99,17D25.

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