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

TL;DR

This paper investigates the structure of the lower central series in a symplectic quotient of a free associative algebra, revealing detailed descriptions of initial components and proposing a conjecture for the next.

Contribution

It introduces an action of Hamiltonian vector fields on the graded components and provides explicit descriptions of the first two components, with a conjecture for the third.

Findings

Complete description of ar{B}_1(A) and B_2 components.

Construction of Hamiltonian vector field action on graded components.

Conjecture on the structure of B_3 component.

Abstract

We study the lower central series filtration L_k for a symplectic quotient A=A_{2n}/<w> of the free algebra A_{2n} on 2n generators, where w=\sum [x_i,x_{i+n}]. We construct an action of the Lie algebra H_{2n} 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 \bar{B}_1(A)= A/(L_2 + AL_3) and the second component B_2=L_2/L_3, and we conjecture a description for the third component B_3=L_3/L_4.