Enriched pre-Lie operads and freeness theorems

Abstract

In this paper, we study the operad of C-enriched pre-Lie algebras, defined for any Hopf cooperad C; it slightly generalises a similar notion defined by Calaque and Willwacher to produce conceptual constructions of the operads acting on various deformation complexes. Maps between Hopf cooperads lead to maps between the corresponding operads of enriched pre-Lie algebras; we prove a criterion for the module action of the domain on the codomain to be free, on the left and on the right. In particular, this implies a new functorial Poincaré-Birkhoff-Witt type theorem for universal enveloping brace algebras of pre-Lie algebras.