On PreLie algebras with divided symmetries

Abstract

We study an analogue of the notion of p-restricted Lie-algebra and of the notion of divided power algebra for PreLie-algebras. We deduce our definitions from the general theory of operads. We consider two variants \Lambda(P,-) and \Gamma(P,-) of the monad S(P,-) which governs the category of algebras classically associated to an operad P. For the operad of PreLie-algebras P=PreLie, we prove that the category of algebras over the monad \Lambda(PreLie,-) is identified with an already defined category of p-restricted PreLie-algebras introduced by A. Dzhumadil'daev. We give an explicit description of the structure an algebra over the monad \Gamma (PreLie,-) in terms of brace-type operations and we compute the relations between these generating operations. We prove that classical examples of PreLie-algebras occurring in deformation theory actually form \Gamma (PreLie,-)-algebras.