Pre Lie algebra

Definition

A pre-Lie algebra ( V , ◃ ) is a vector space V with a bilinear map ◃ : V ⊗ V → V , satisfying the relation ( x ◃ y ) ◃ z − x ◃ ( y ◃ z ) = ( x ◃ z ) ◃ y − x ◃ ( z ◃ y ) .

This identity can be seen as the invariance of the associator ( x , y , z ) = ( x ◃ y ) ◃ z − x ◃ ( y ◃ 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

If we denote by f ( x ) ∂ x the vector field x ↦ f ( x ) , and if we define ◃ as f ( x ) ◃ g ( x ) = f ′ ( x ) g ( x ) , we can see that the operator ◃ is exactly the application of the g ( x ) ∂ x field to f ( x ) ∂ x field. ( g ( x ) ∂ x ) ( f ( x ) ∂ x ) = g ( x ) ∂ x f ( x ) ∂ x = g ( x ) f ′ ( x ) ∂ x

If we study the difference between ( x ◃ y ) ◃ z and x ◃ ( y ◃ z ) , we have ( x ◃ y ) ◃ z − x ◃ ( y ◃ z ) = ( x ′ y ) ′ z − x ′ y ′ z = x ′ y ′ z + x ″ y z − z ′ y ′ z = x ″ y z which is symmetric on y and z.

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

One can introduce a bilinear product ↶ on T as follows. Let τ 1 and τ 2 be two rooted trees.

τ 1 ↶ τ 2 = ∑ s ∈ V e r t i c e s ( τ 1 ) τ 1 ∘ s τ 2

where τ 1 ∘ s τ 2 is the rooted tree obtained by adding to the disjoint union of τ 1 and τ 2 an edge going from the vertex s of τ 1 to the root vertex of τ 2 .

Then ( T , ↶ ) is a free pre-Lie algebra on one generator.