On universal Lie nilpotent associative algebras

We Study the quotient Q(i)(A) of a free algebra A by the ideal M-i(A) generated by the ith commutator of any elements. In particular, we completely describe such quotient for i = 4 (for i <= 3 this was done previously by Feigin and Shoikhet). We also Study properties of the ideals M-i(A), e.g. when M-i(A)M-j(A) is contained in Mi+j-1 (A) (by a result Of Gupta and Levin, it is always contained ill Mi+j-2(A)). (C) 2008 Elsevier Inc. All rights reserved.