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Journal of Lie Theory 23 (2013), No. 4, 937--952
Copyright Heldermann Verlag 2013

Pre-Lie Algebras in Positive Characteristic

Ioannis Dokas
Konstantinoupoleos 22, Patras 26441, Greece

In prime characteristic we introduce the notion of restricted pre-Lie algebras. We prove in the pre-Lie context the analogue to Jacobson's theorem for restricted Lie algebras. In particular, we prove that any dendriform algebra over a field of positive characteristic is a restricted pre-Lie algebra. Thus we obtain that Rota-Baxter algebras and quasitriangular algebras are restricted pre-Lie algebras. Moreover, we prove that the free Γ(preLie)-algebra is a restricted pre-Lie algebra, where "preLie" denotes the pre-Lie operad. Finally, we define the notion of restricted enveloping dendriform algebra and we construct a left adjoint functor for the functor (-)p-preLie: Dend --> p-preLie .

Keywords: Restricted Lie algebra, dendriform algebra, pre-Lie algebra, algebras with divided powers over an operad.

MSC: 17D25, 17B50, 18C15

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