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 $Γ(\calligra{preLie})$-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 $(-{)}_{p-preLie}:Dend\to p-preLie$.

14 pages

References in corpus (2)

• Dyer-Lashof-Cohen operations in Hochschild cohomology
• Lie elements in pre-Lie algebras, trees and cohomology operations

Cited by in corpus (3)

• Embedding of post-Lie algebras into postassociative algebras
• On PreLie algebras with divided symmetries
• Embedding of pre-Lie algebras into preassociative algebras