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. Although weaker than associativity, the defining relation of a pre-Lie algebra still implies that the commutator ${\displaystyle x\triangleleft y-y\triangleleft x}$ is a Lie bracket. In particular, the Jacobi identity for the commutator follows from cycling the ${\displaystyle x,y,z}$ terms in the defining relation for pre-Lie algebras, above.

Examples

Vector fields on an affine space

Let ${\displaystyle U\subset \mathbb {R} ^{n}}$ be an open neighborhood of ${\displaystyle \mathbb {R} ^{n}}$, parameterised by variables ${\displaystyle x_{1},\cdots ,x_{n}}$. Given vector fields ${\displaystyle u=u_{i}\partial _{x_{i}}}$, ${\displaystyle v=v_{j}\partial _{x_{j}}}$ we define ${\displaystyle u\triangleleft v=v_{j}{\frac {\partial u_{i}}{\partial x_{j}}}\partial x_{i}}$.

The difference between ${\displaystyle (u\triangleleft v)\triangleleft w}$ and ${\displaystyle u\triangleleft (v\triangleleft w)}$, is ${\displaystyle (u\triangleleft v)\triangleleft w-u\triangleleft (v\triangleleft w)=v_{j}w_{k}{\frac {\partial ^{2}u_{i}}{\partial x_{j}\partial x_{k}}}\partial _{x_{i}}}$ which is symmetric in ${\displaystyle v}$ and ${\displaystyle w}$. Thus ${\displaystyle \triangleleft }$ defines a pre-Lie algebra structure.

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

Rooted trees

Let ${\displaystyle \mathbb {T} }$ be the free 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. 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, 8 (8): 395–408, doi:10.1155/S1073792801000198, MR 1827084.
• Szczesny, M. (2010), Pre-Lie algebras and incidence categories of colored rooted trees, 1007, p. 4784, arXiv:1007.4784, Bibcode:2010arXiv1007.4784S.