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

@article{Feigin2006OnA,
  title={On \$[A,A]/[A,[A,A]]\$ and on a \$W_n\$-action on the consecutive commutators of free associative algebras},
  author={B. Feigin and B. Shoikhet},
  journal={Mathematical Research Letters},
  year={2006},
  volume={14},
  pages={781-795}
}
  • B. Feigin, B. Shoikhet
  • Published 2006
  • Mathematics
  • Mathematical Research Letters
  • 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… CONTINUE READING
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