Lie commutators in a free diassociative algebra

We give a criterion for Leibniz elements in a free diassociative algebra. In the diassociative case one can consider two versions of Lie commutators. We give criterions for elements of diassociative algebras to be Lie under these commutators. One of them corresponds to Leibniz elements. It generalizes the Dynkin-Specht-Wever criterion for Lie elements in a free associative algebra.

Lie Elements and Knuth Relations

A coplactic class in the symmetric group consists of all permutations in with a given Schensted Q-symbol, and may be described in terms of local relations introduced by Knuth. Any Lie element in the group algebra of which is constant on coplactic classes is already constant on descent classes. As a consequence, the intersection of the Lie convolution algebra introduced by Patras and Reutenauer and the coplactic algebra introduced by Poirier and Reutenauer is the direct sum of all Solomon descent algebras.

Engel Congruences in Groups of Prime-Power Exponent

In this paper a simplified proof of a theorem of Sanov (4) is given. No mention is required of Lie elements or of the Baker-Hausdorff formula, both of which played central roles in Sanov's proof.Let G be a group. Define, for all x, y ∈ G,

Generalized Lie Elements

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).