Pre-Lie Algebras in Positive Characteristic

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 (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.