Products of several commutators in a Lie nilpotent associative algebra

摘要

Let F be a field of characteristic not equal 2,3 and let A be a unital associative F-algebra. Define a left-normed commutator [a(1), a(2),..., a(n)] (a(i). is an element of A) recursively by [a(1), a(2)] = a(1)a(2)-a(2)a(1), [a(1),..., a(n-1), a(n)] = [[a(1),..., a(n-1)], a(n)] (n = 3). For n = 2, let T-(n) (A) be the two-sided ideal in A generated by all commutators [a(1), a(2),..., a(n)] (a(i) is an element of A). Define T-(1) (A) = A.