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.

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 $u \triangleleft v = v_j \frac{\partial u_i}{\partial x_j} \partial_{x_i}$ .

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.

References

{{citation

| last1 = Chapoton | first1 = F.

| last2 = Livernet | first2 = M.

| year = 2001

| mr = 1827084

| journal = International Mathematics Research Notices

| title = Pre-Lie algebras and the rooted trees operad

| doi = 10.1155/S1073792801000198

| pages = 395–408

| volume = 2001

| issue = 8| doi-access = free

}}.

{{citation

| last = Szczesny | first = M.

| year = 2010

| mr =

| journal =

| title = Pre-Lie algebras and incidence categories of colored rooted trees

| volume =1007

| bibcode = 2010arXiv1007.4784S

| pages = 4784

| arxiv = 1007.4784 }}.

Category:Lie groups

Category:Non-associative algebra