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Hypertrees and embedding of the $\mathrm{FMan}$ operad

Paul Laubie

Abstract

The operad $\mathrm{FMan}$ encodes the algebraic structure on vector fields of Frobenius manifolds, in the same way as the operad $\mathrm{Lie}$ encodes the algebraic structure on vector fields of a smooth manifold. It is well known that the operad $\mathrm{Lie}$ admits an embedding in the operad $\mathrm{PreLie}$ encoding pre-Lie algebras. We prove a conjecture of Dotsenko stating that the operad $\mathrm{FMan}$ admits an embedding in the operad $\mathrm{ComPreLie}$. The operad $\mathrm{ComPreLie}$ 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 $\mathrm{PreLie}$. We then give a combinatorial description of the operad $\mathrm{ComPreLie}$ \emph{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.

Hypertrees and embedding of the $\mathrm{FMan}$ operad

Abstract

The operad encodes the algebraic structure on vector fields of Frobenius manifolds, in the same way as the operad encodes the algebraic structure on vector fields of a smooth manifold. It is well known that the operad admits an embedding in the operad encoding pre-Lie algebras. We prove a conjecture of Dotsenko stating that the operad admits an embedding in the operad . The operad 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 . We then give a combinatorial description of the operad \emph{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.
Paper Structure (6 sections, 38 theorems, 82 equations, 5 figures)

This paper contains 6 sections, 38 theorems, 82 equations, 5 figures.

Table of Contents

  1. Rooted trees and Rooted Greg trees
  2. The operad $\mathrm{Greg}$ and the operadic twisting of $\mathrm{PreLie}$
  3. Forests of rooted hypertrees and the operad $\mathrm{ComPreLie}$
  4. Forests of rooted Greg hypertrees
  5. Reduced Rooted Greg hypertrees
  6. The embedding of $\mathrm{FMan}$ in $\mathrm{ComPreLie}$

Key Result

Theorem 1

The morphism of operads $\mathrm{FMan}\to\mathrm{ComPreLie}$ is injective.

Figures (5)

  • Figure 1: The set $\mathcal{G}(\underline{2})$
  • Figure 2: Example of computation of $d_\mathrm{MC}$ on some trees.
  • Figure 3: Composition of the arity $1$ degree $1$ elements.
  • Figure 4: The set $\mathcal{FH}(\underline{2})$
  • Figure 5: The sequential axiom for $\mathcal{FH}$

Theorems & Definitions (95)

  • Theorem : Th. \ref{['thm:main']}
  • Theorem : Th. \ref{['thm:CPLK']}
  • Proposition 1
  • Proposition 2
  • Definition 1
  • Definition 2
  • Proposition 3
  • Theorem 4
  • Definition 3
  • Remark 1
  • ...and 85 more