A pre-Lie algebra ${\displaystyle (V,\triangleleft )}$  is a vector space ${\displaystyle V}$  with a linear 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.

Vector fields on an affine space edit

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 edit

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.

• 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.