1 0 Ju l 2 00 2 CO - ADDITION FOR FREE NON - ASSOCIATIVE ALGEBRAS AND THE HAUSDORFF SERIES

Generalizations of the series exp and log to noncommutative non-associative and other types of algebras were regarded 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 H n 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 w.r.t. the partial derivations, which contains all non-linear primitive elements. By a theorem of Kurosh, A 0 is a free algebra. We describe a procedure to construct a free algebra basis which consists of primitive elements.