Wn-action on the consecutive commutators of free associative algebra

We consider the lower central filtration 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 algebraWn of polynomial vector fields on C. We compute the space [An, An]/[An, [An, An]] and show that it is isomorphic to the space Ω closed (C)⊕ Ω closed (C)⊕ Ω closed (C)⊕ . . . . Introduction Let A be an associative algebra. A free resolution R of A is a free graded differential algebra R = ⊕ R, i ∈ Z≤0, the differential Q has degree +1 and the cohomology of Q is only in degree zero and is canonically isomorphic to A as algebra. Such a resolution can be used for calculation of ”higher derived functors” for A. For example, higher cyclic homology of A is the higher derived functor for the functor