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 [math]\displaystyle{ (V,\triangleleft) }[/math] is a vector space [math]\displaystyle{ V }[/math] with a linear map [math]\displaystyle{ \triangleleft : V \otimes V \to V }[/math], satisfying the relation [math]\displaystyle{ (x \triangleleft y) \triangleleft z - x \triangleleft (y \triangleleft z) = (x \triangleleft z) \triangleleft y - x \triangleleft (z \triangleleft y). }[/math]

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

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 [math]\displaystyle{ x \triangleleft y - y \triangleleft x }[/math] is a Lie bracket. In particular, the Jacobi identity for the commutator follows from cycling the [math]\displaystyle{ x,y,z }[/math] terms in the defining relation for pre-Lie algebras, above.

Examples

Vector fields on an affine space

Let [math]\displaystyle{ U \subset \mathbb{R}^n }[/math] be an open neighborhood of [math]\displaystyle{ \mathbb{R}^n }[/math], parameterised by variables [math]\displaystyle{ x_1,\cdots,x_n }[/math]. Given vector fields [math]\displaystyle{ u= u_i \partial_{x_i} }[/math], [math]\displaystyle{ v=v_j \partial_{x_j} }[/math] we define [math]\displaystyle{ u \triangleleft v = v_j \frac{\partial u_i}{\partial x_j} \partial_{x_i} }[/math].

The difference between [math]\displaystyle{ (u \triangleleft v) \triangleleft w }[/math] and [math]\displaystyle{ u \triangleleft (v \triangleleft w) }[/math], is [math]\displaystyle{ (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} }[/math] which is symmetric in [math]\displaystyle{ v }[/math] and [math]\displaystyle{ w }[/math]. Thus [math]\displaystyle{ \triangleleft }[/math] defines a pre-Lie algebra structure.

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

Rooted trees

Let [math]\displaystyle{ \mathbb{T} }[/math] be the free vector space spanned by all rooted trees.

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

[math]\displaystyle{ \tau_1 \curvearrowleft \tau_2 = \sum_{s \in \mathrm{Vertices}(\tau_1)} \tau_1 \circ_s \tau_2 }[/math]

where [math]\displaystyle{ \tau_1 \circ_s \tau_2 }[/math] is the rooted tree obtained by adding to the disjoint union of [math]\displaystyle{ \tau_1 }[/math] and [math]\displaystyle{ \tau_2 }[/math] an edge going from the vertex [math]\displaystyle{ s }[/math] of [math]\displaystyle{ \tau_1 }[/math] to the root vertex of [math]\displaystyle{ \tau_2 }[/math].

Then [math]\displaystyle{ (\mathbb{T}, \curvearrowleft) }[/math] 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 .