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

We study the lower central series filtration Lk for a symplectic quotient A=A2n/〈ω〉 of the free algebra A2n on 2n generators, where ω=∑[xi,xi+n]. We construct an action of the Lie algebra H2n 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 and the second component B2=L2/L3, and we conjecture a description for the third component B3=L3/L4.