Noncommutative higher commutators (Q2068128)

Let \(A\) be an algebra. For subspaces \(U\) and \(V\) of \(A\), \([U, V]\) denotes the linear span of \(A\) generated by all \([a, b]\), where \(a\in U\) and \(b\in V\), and \([a, b]=ab-ba\) is the usual commutator (or Lie product). A higher commutator \(H\) of \(A\) with length \(n\) is defined to be \begin{itemize} \item \(A\) for \(n=1\); \item \([A, A]\) for \(n=2\); \item \([U, V]\) for \(n>2\), where \(U\) and \(V\) are higher commutators of \(A\) with, respectively, lengths \(p\) and \(q\) such that \(p+q=n\). \end{itemize} The main purpose of this paper is to characterize higher commutators of \(A\), where \(A\) is a unital algebra over a field of characteristic different from 2. It is shown that if \(H\) is a noncommutative higher commutator of \(A\) and contains the unity, then \(H\) is equal to either \(A\) or \([A, A]\). Consequently, if there exist \(x, y\in A\) such that \(1=[x, y]\), then the only higher commutators of \(A\) are \(A\) and \([A, A]\). It is also proved that if \(L=[U, A]\) is noncommutative and contains the unity, where \(U\) is a Lie ideal of \(A\), then \(L=[A, A]\) and \([A, A]\subseteq U\).