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

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

2006-10-12

arxiv.org/abs/math/0610410v2

Mathematics

Quantum Algebra

18 pages, 4 eps Figures, v2: minor corrections are made

Scientific paper

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 $\Omega^2_{closed}(\mathbb{C}^n) \oplus \Omega^4_{closed}(\mathbb{C}^n) \oplus \Omega^6_{closed}(\mathbb{C}^n) \oplus ...$.

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