Pre-Lie algebra

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

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