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 bilinear 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.

Examples

Vector fields on the affine space

If we denote by $f(x)\partial _{x}$ the vector field $x\mapsto f(x)$ , and if we define $\triangleleft$ as $f(x)\triangleleft g(x)=f'(x)g(x)$ , we can see that the operator $\triangleleft$ is exactly the application of the $g(x)\partial _{x}$ field to $f(x)\partial _{x}$ field. $(g(x)\partial _{x})(f(x)\partial _{x})=g(x)\partial _{x}f(x)\partial _{x}=g(x)f'(x)\partial _{x}$

If we study the difference between $(x\triangleleft y)\triangleleft z$ and $x\triangleleft (y\triangleleft z)$ , we have $(x \triangleleft y) \triangleleft z - x \triangleleft (y \triangleleft z) = (x' y)'z - x'y'z = x'y'z + x''yz - z'y'z = x''yz$ which is symmetric on y and z.

Rooted trees

Let $\mathbb {T}$ be the 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.

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 .

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