Leibniz algebras can be seen as a non-skew-symmetric generalization of Lie algebras. I have already taken a look at some papers related to Leibniz algebras and extending main results of Lie algebras to Leibniz algebras. Here I want to ask you about the Poincaré-Birkhoff-Witt (PBW) theorem for free Leibniz algebras? Does the PBW theorem allow us to view a Leibniz algebra $L$ as a subalgebra of its universal enveloping algebra $U(L)$?
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2$\begingroup$ Googling for it: Un Théorème de Poincaré–Birkhoff–Witt pour les Algèbres de Leibniz by Aymon and Grivel in Commun. Alg. 31 (2003); Poincaré–Birkhoff–Witt theorem for Leibniz algebras, by Casas, Insua & Ladra in JSC 42 (2007) $\endgroup$– მამუკა ჯიბლაძეMar 23, 2016 at 13:17
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1$\begingroup$ The second reference should be to sciencedirect.com/science/article/pii/S0747717107000995 ; note that its title has "$n$-algebras" instead of "algebras" (researchgate made a mistake here). $\endgroup$– darij grinbergMar 24, 2016 at 2:59
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1$\begingroup$ Also, Theorem 2.9 in: Jean-Louis Loday, Teimuraz Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology, eudml.org/doc/165079 (also on www-irma.u-strasbg.fr/~loday/PAPERS/93LodayPira(Leibniz).pdf ). Disclaimer: I've read none of the references (and, in fact, was only aware of two of them). $\endgroup$– darij grinbergMar 24, 2016 at 3:01
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$\begingroup$ @darijgrinberg Oh I see, thanks for pointing this out! I was wondering why they repeated the same research after four years :D $\endgroup$– მამუკა ჯიბლაძეMar 24, 2016 at 9:54
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