Hypertrees and Embedding of the FMan Operad

The operad FMan encodes the algebraic structure on vector fields of Frobenius manifolds—the same way as the operad Lie encodes the algebraic structure on vector fields of a smooth manifold. It is well known that the operad Lie admits an embedding in the operad PreLie encoding pre-Lie algebras. We prove a conjecture of Dotsenko stating that the operad FMan admits an embedding in the operad ComPreLie. The operad ComPreLie is the operad encoding pre-Lie algebras with an additional commutative product such that right pre-Lie multiplications act as derivations. To prove this result, we first remark a link between the Greg trees and the so-called operadic twisting of PreLie. We then give a combinatorial description of the operad ComPreLie à la Chapoton–Livernet with forests of rooted hypertrees. We generalize this construction to forests of rooted Greg hypertrees, and then use operadic twisting techniques to prove the conjecture.