The lower central series of the symplectic quotient of a free associative algebra

We study the lower central series filtration L for a symplectic quotient A=A /〈ω〉 of the free algebra A on 2n generators, where ω=∑[x , x ]. We construct an action of the Lie algebra H of Hamiltonian vector fields on the associated graded components of the filtration, and use this action to give a complete description of the reduced first component B̄1(A)=A/(L2+AL3) and the second component B =L /L , and we conjecture a description for the third component B =L /L .