ON THE CONSECUTIVE COMMUTATORS OF FREE ASSOCIATIVE ALGEBRAS

@inproceedings{Feigin2007ONTC,
title={ON THE CONSECUTIVE COMMUTATORS OF FREE ASSOCIATIVE ALGEBRAS},
author={Boris Feigin and Boris Shoikhet},
year={2007}
}

We consider the lower central series of the free associative algebra An 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 Wn of polynomial vector fields on Cn. We compute the space [An, An]/[An, [An, An]] and show that it is isomorphic to the space Ωclosed(C n)⊕ Ωclosed(C n)⊕ Ωclosed(C n)⊕ . . . .

Appendix E by Mara O. Ronco. Second edition. Chapter 13 by the author in collaboration with Teimuraz Pirashvili, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 301, Springer-Verlag, Berlin • 1998