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 $$(V,\triangleleft)$$ is a vector space $$V$$ with a linear map $$\triangleleft : V \otimes V \to V$$, satisfying the relation $$ (x \triangleleft y) \triangleleft z - x \triangleleft (y \triangleleft z) = (x \triangleleft z) \triangleleft y - x \triangleleft (z \triangleleft y). $$

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

Every associative algebra is hence also a pre-Lie algebra, as the associator vanishes identically. Although weaker than associativity, the defining relation of a pre-Lie algebra still implies that the commutator $$ x \triangleleft y - y \triangleleft x $$ is a Lie bracket. In particular, the Jacobi identity for the commutator follows from cycling the $$ x,y,z $$ terms in the defining relation for pre-Lie algebras, above.

Vector fields on an affine space
Let $$U \subset \mathbb{R}^n$$ be an open neighborhood of $$ \mathbb{R}^n$$, parameterised by variables $$ x_1,\cdots,x_n$$. Given vector fields $$ u= u_i \partial_{x_i}$$, $$ v=v_j \partial_{x_j}$$ we define $$ u \triangleleft v = v_j \frac{\partial u_i}{\partial x_j} \partial_{x_i}$$.

The difference between $$(u \triangleleft v) \triangleleft w$$ and $$u \triangleleft (v \triangleleft w)$$, is $$ (u \triangleleft v) \triangleleft w - u \triangleleft (v \triangleleft w) = v_j w_k \frac{\partial^2 u_i}{\partial x_j \partial x_k}\partial_{x_i} $$ which is symmetric in $$ v $$ and $$ w $$. Thus $$ \triangleleft $$ defines a pre-Lie algebra structure.

Given a manifold $$ M $$ and homeomorphisms $$ \phi, \phi'$$ from $$ U,U'\subset \mathbb{R}^n$$ to overlapping open neighborhoods of $$ M $$, they each define a pre-Lie algebra structure $$ \triangleleft, \triangleleft'$$ on vector fields defined on the overlap. Whilst $$ \triangleleft $$ need not agree with $$\triangleleft'$$, their commutators do agree: $$ u \triangleleft v - v \triangleleft u =u \triangleleft' v - v\triangleleft' u =[v,u]$$, the Lie bracket of $$ v $$ and $$ u $$.

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

One can introduce a bilinear product $$\curvearrowleft$$ on $$\mathbb{T}$$ as follows. Let $$\tau_1$$ and $$\tau_2$$ be two rooted trees.

$$\tau_1 \curvearrowleft \tau_2 = \sum_{s \in \mathrm{Vertices}(\tau_1)} \tau_1 \circ_s \tau_2$$

where $$\tau_1 \circ_s \tau_2$$ is the rooted tree obtained by adding to the disjoint union of $$\tau_1$$ and $$\tau_2$$ an edge going from the vertex $$s$$ of $$\tau_1$$ to the root vertex of $$\tau_2$$.

Then $$(\mathbb{T}, \curvearrowleft)$$ is a free pre-Lie algebra on one generator. More generally, the free pre-Lie algebra on any set of generators is constructed the same way from trees with each vertex labelled by one of the generators.