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arXiv · 1509.05599

On PreLie algebras with divided symmetries

Andrea Cesaro
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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 Λ(P,-) and Γ(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 Λ(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 Γ(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 Γ(PreLie,-)-algebras.

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Andrea Cesaro. 2016-07-03. On PreLie algebras with divided symmetries. https://arxiv.org/abs/1509.05599

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