Pre-Lie algebras in positive characteristic

Abstract

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 $\Gamma(\calligra{preLie})$-algebra is a restricted pre-Lie algebra, where $\calligra{preLie}$ 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 $(-)_{p-preLie}: Dend \rightarrow p-preLie$.