The Campbell–Hausdorff near-ring over [formula omitted]—A topological view

Abstract

The famous Baker–Campbell–Hausdorff series defines a group composition on the set of the so called grouplike elements of a completed free Lie algebra as we can find in P. Cartier’s paper on the Campbell–Hausdorff formula in 1956. According to A. Baider and R.C. Churchill, we call this group the Campbell–Hausdorff group over the alphabetX of the free Lie algebra. By a certain coproduct and certain bialgebra endomorphisms of the free associative algebra over a set X which should be at most countably infinite, we get an alternative realisation of this Campbell–Hausdorff group. This realisation is much more comfortable to deal with than the classical one. The usual composition of endomorphisms gives a near-ring structure to this group. In this paper we consider a metric on the Campbell–Hausdorff near-ring over the alphabet N and the compatibility of this metric with the near-ring compositions and we make some topological remarks.