Labeled binary planar trees and quasi-Lie algebras

Let H be a finitely-generated free abelian group and L(H) the graded free Lie algebra on H . There is a natural homomorphism H ⊗ L(H)→ L(H) defined by bracketing, whose kernel is denoted D(H). If H supports a non-singular symplectic form, eg H = H1(Σ), where Σ is a closed orientable surface, with symplectic basis {xi, yi}, then D(H) is, in fact, a Lie algebra. It can be identified with the Lie subalgebra of D(L(H)) (the graded Lie algebra of derivations of L(H)) consisting of those derivations which vanish on the element ∑ i[xi, yi] ∈ L2(H).