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
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
Ω
2
closed
(
C
n
)⊕
Ω
4
closed
(
C
n
)⊕
Ω
6
closed
(
C
n
)⊕...
.