On $[A,A]/[A,[A,A]]$ and on a $W_n$-action on the consecutive commutators of free associative algebras

Abstract

We consider the lower central series of the free associative algebra $A_n$ with $n$ generators as a Lie algebra. We consider the associated graded Lie algebra. It is shown that this Lie algebra has a huge center which belongs to the cyclic words, and on the quotient Lie algebra by the center there acts the Lie algebra $W_n$ of polynomial vector fields on $\mathbb{C}^n$. We compute the space $[A_n,A_n]/[A_n,[A_n,A_n]]$ and show that it is isomorphic to the space $\Omega^2_{closed}(\mathbb{C}^n)\oplus\Omega^4_{closed}(\mathbb{C}^n)\oplus\Omega^6_{closed}(\mathbb{C}^n) \oplus\dots$.