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

This identity can be seen as the invariance of the associator $(x,y,z)=(x◃y)◃z-x◃(y◃z)$ 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 $x◃y-y◃x$ 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.

Examples

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 $u◃v=v_j\frac{\partial u_i}{\partial x_j}{\partial }_{x_i}$.

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

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

${\tau }_1\curvearrowleft {\tau }_2=\sum _{s\in \mathrm{V}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{e}\mathrm{s}({\tau }_1)}{\tau }_1{\circ }_s{\tau }_2$

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.

References

• Chapoton, F.; Livernet, M. (2001), "Pre-Lie algebras and the rooted trees operad", International Mathematics Research Notices 2001 (8): 395–408, doi:10.1155/S1073792801000198 .
• Szczesny, M. (2010), Pre-Lie algebras and incidence categories of colored rooted trees, 1007, pp. 4784, Bibcode: 2010arXiv1007.4784S .