Co-Addition for free Non-Associative Algebras and the Hausdorff Series

TL;DR

This paper introduces the non-associative Hausdorff series, exploring its properties and primitive elements, and provides a method to construct a basis for the free algebra of primitives in non-associative variables.

Contribution

It defines the non-associative Hausdorff series and characterizes its primitive components, offering a new approach to construct bases for free non-associative algebras.

Findings

The non-associative Hausdorff series H satisfies exp(H)=exp(x)exp(y).

Homogeneous components of H are primitive elements under co-addition.

A procedure for constructing a basis of primitive elements in free non-associative algebras is provided.

Abstract

Generalizations of the series exp and log to noncommutative non-associative and other types of algebras were considered by M.Lazard, and recently by V.Drensky and L.Gerritzen. There is a unique power series exp(x) in one non-associative variable x such that exp(x)exp(x)=exp(2x), exp'(0)=1. We call the unique series H=H(x,y) in two non-associative variables satisfying exp(H)=exp(x)exp(y) the non-associative Hausdorff series, and we show that the homogeneous components of H are primitive elements with respect to the co-addition for non-associative variables. We describe the space of primitive elements for the co-addition in non-associative variables using Taylor expansion and a projector onto the algebra A_0 of constants for the partial derivations. By a theorem of Kurosh, A_0 is a free algebra. We describe a procedure to construct a free algebra basis consisting of primitive elements.