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

Abstract

We consider the lower central filtration 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 ${\Omega }_{closed}^2({\mathbb{C}}^n)\oplus {\Omega }_{closed}^4({\mathbb{C}}^n)\oplus {\Omega }_{closed}^6({\mathbb{C}}^n)\oplus ...$.

arXiv comment18 pages, 4 eps Figures, v2: minor corrections are made

Cited by in corpus (5)

• The additive group of a Lie nilpotent associative ring
• Lower central series of a free associative algebra over the integers and finite fields
• The torsion subgroup of the additive group of a Lie nilpotent associative ring of class 3
• Relations in universal Lie nilpotent associative algebras of class 4
• On Lie nilpotent associative algebras