# Pre-Lie algebra

In mathematics, a pre-Lie algebra is an algebraic structure on a vector space that describes some properties of objects such as rooted trees and vector fields on affine space.

The notion of pre-Lie algebra has been introduced by Murray Gerstenhaber in his work on deformations of algebras.

Pre-Lie algebras have been considered under some other names, among which one can cite left-symmetric algebras, right-symmetric algebras or Vinberg algebras.

## Definition

A pre-Lie algebra ${\displaystyle (V,\triangleleft )}$ is a vector space ${\displaystyle V}$ with a bilinear map ${\displaystyle \triangleleft :V\otimes V\to V}$, satisfying the relation ${\displaystyle (x\triangleleft y)\triangleleft z-x\triangleleft (y\triangleleft z)=(x\triangleleft z)\triangleleft y-x\triangleleft (z\triangleleft y).}$

This identity can be seen as the invariance of the associator ${\displaystyle (x,y,z)=(x\triangleleft y)\triangleleft z-x\triangleleft (y\triangleleft z)}$ under the exchange of the two variables ${\displaystyle y}$ and ${\displaystyle z}$.

Every associative algebra is hence also a pre-Lie algebra, as the associator vanishes identically.

## Examples

If we denote by ${\displaystyle f(x)\partial _{x}}$ the vector field ${\displaystyle x\mapsto f(x)}$, and if we define ${\displaystyle \triangleleft }$ as ${\displaystyle f(x)\triangleleft g(x)=f'(x)g(x)}$, we can see that the operator ${\displaystyle \triangleleft }$ is exactly the application of the ${\displaystyle g(x)\partial _{x}}$ field to ${\displaystyle f(x)\partial _{x}}$ field. ${\displaystyle (g(x)\partial _{x})(f(x)\partial _{x})=g(x)\partial _{x}f(x)\partial _{x}=g(x)f'(x)\partial _{x}}$

If we study the difference between ${\displaystyle (x\triangleleft y)\triangleleft z}$ and ${\displaystyle x\triangleleft (y\triangleleft z)}$, we have ${\displaystyle (x\triangleleft y)\triangleleft z-x\triangleleft (y\triangleleft z)=(x'y)'z-x'y'z=x'y'z+x''yz-z'y'z=x''yz}$ which is symmetric on y and z.

• Rooted trees

Let ${\displaystyle \mathbb {T} }$ be the vector space spanned by all rooted trees.

One can introduce a bilinear product ${\displaystyle \curvearrowleft }$ on ${\displaystyle \mathbb {T} }$ as follows. Let ${\displaystyle \tau _{1}}$ and ${\displaystyle \tau _{2}}$ be two rooted trees.

${\displaystyle \tau _{1}\curvearrowleft \tau _{2}=\sum _{s\in \mathrm {Vertices} (\tau _{1})}\tau _{1}\circ _{s}\tau _{2}}$

where ${\displaystyle \tau _{1}\circ _{s}\tau _{2}}$ is the rooted tree obtained by adding to the disjoint union of ${\displaystyle \tau _{1}}$ and ${\displaystyle \tau _{2}}$ an edge going from the vertex ${\displaystyle s}$ of ${\displaystyle \tau _{1}}$ to the root vertex of ${\displaystyle \tau _{2}}$.

Then ${\displaystyle (\mathbb {T} ,\curvearrowleft )}$ is a free pre-Lie algebra on one generator.