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Register a new userSome remarks for the Akivis algebras and the Pre-Lie algebras
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In this paper, by using the Composition-Diamond lemma for non-associative algebras invented by A. I. Shirshov in 1962, we give Gr{o}bner-Shirshov bases for free Pre-Lie algebras and the universal enveloping non-associative algebra of an Akivis algebra, respectively. As applications, we show I.P. Shestakovs result that any Akivis algebra is linear and D. Segals result that the set of all good words in $X^{**}$ forms a linear basis of the free Pre-Lie algebra $PLie(X)$ generated by the set $X$. For completeness, we give the details of the proof of Shirshovs Composition-Diamond lemma for non-associative algebras.
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We prove that free pre-Lie algebras, when considered as Lie algebras, are free. Working in the category of S-modules, we define a natural filtration on the space of generators. We also relate the symmetric group action on generators with the structure of the anticyclic PreLie operad.
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