# Algebraic structures of $F$-manifolds via pre-Lie algebras

## TL;DR

This paper explores the algebraic structures of $F$-manifolds by connecting the operad controlling their tangent sheaf structures to the operad of pre-Lie algebras, revealing a graded relationship.

## Contribution

It establishes a link between the operad for $F$-manifold tangent sheaves and the operad of pre-Lie algebras, showing the associated graded structure is FMan.

## Key findings

- The operad FMan is related to the operad PreLie via a filtration.
- The associated graded object of PreLie by this filtration is FMan.
- Provides a new algebraic perspective on $F$-manifolds.

## Abstract

We relate the operad FMan controlling the algebraic structure on the tangent sheaf of an $F$-manifold (weak Frobenius manifold) defined by Hertling and Manin to the operad PreLie of pre-Lie algebras: for the filtration of PreLie by powers of the ideal generated by the Lie bracket, the associated graded object is FMan.