Generalized Lie Elements

Abstract

Let λ(ij), i,j = 1, 2, … , m, be m2 elements in a field K of characteristic zero such that λ(ij)λ(ji) = 1 for all i and j, and X1, x2, … , xm non-commutative associative indeterminates over K. Define the elements [xi1Xi2 … xin] inductively by [xi] = xi andAny linear combination of the elementswith coefficients in K will be called a generalized Lie elememt. Generalized Lie elements reduce to ordinary Lie elements if λ(ij) = 1 for all i and j.The purpose of this paper is to generalize to the generalized Lie elements the following: a theorem of Friedrichs, a theorem of Dynkin-Specht-Wever (2), and the Witt formula on the dimension of the space spanned by homogeneous Lie elements of a fixed degree. The set of all generalized Lie elements will be made into an algebra which generalizes the ordinary free Lie algebra. This algebra turns out to be free in a certain sense. We shall also generalize the algebra associated with shuffles in (2).