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

Abstract

We study the lower central series filtration Lk for a symplectic quotient A=A2n/<w> of the free algebra A2n on 2n generators, where w=Σ [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 B1(A)= A/(L2 + AL3) and the second component B2=L2/L3, and we conjecture a description for the third component B3=L3/L4.