Pre-Lie algebra explained

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 ⊗ 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). (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 R ^{n} be an open neighborhood of R ^{n} , parameterised by variables x _{1, … ,x} _{n} . Given vector fields u = u _{i} \partial _{x _{i}} , v = v _{j} \partial _{x _{j}} we define u \triangleleft v = v _{j} (\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} (\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 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

T be the free vector space spanned by all rooted trees.

One can introduce a bilinear product

\curvearrowleft on T as follows. Let \tau _{1} and \tau _{2} be two rooted trees.\tau _{1} \curvearrowleft \tau _{2} = \sum _{s \in 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

( 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

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