On PreLie algebras with divided symmetries

TL;DR

This paper explores p-restricted and divided power structures in PreLie algebras, connecting operad theory with algebraic structures relevant in deformation theory, and introduces new variants of monads governing these algebras.

Contribution

It defines and analyzes two variants of monads for PreLie-algebras, linking them to p-restricted structures and providing explicit descriptions of their algebraic operations.

Findings

Identification of mbda(PreLie,-) with p-restricted PreLie-algebras

Explicit description of mma(PreLie,-) in terms of brace operations

Classical deformation theory PreLie-algebras are mma(PreLie,-)-algebras

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.