On the lower central series of an associative algebra: (with an appendix by Pavel Etingof)

For an associative algebra A, define its lower central series L0(A)=A, Li(A)=[A,Li−1(A)], and the corresponding quotients Bi(A)=Li(A)/Li+1(A). In this paper, we study the structure of Bi(An) for a free algebra An. We construct a basis for B2(An) and determine the structure of B3(A2) and B4(A2). In Appendix A, we study the structure of B2(A) for any associative algebra A over C.