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

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, MR 1827084.
Szczesny, M. (2010), Pre-Lie algebras and incidence categories of colored rooted trees, vol. 1007, p. 4784, arXiv:1007.4784, Bibcode:2010arXiv1007.4784S.

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